The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics
Item Type text; Electronic Dissertation
Authors Ramalheira-Tsu, Jonathan
Citation Ramalheira-Tsu, Jonathan. (2020). The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics (Doctoral dissertation, University of Arizona, Tucson, USA).
Publisher The University of Arizona.
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Download date 25/09/2021 19:02:03
Link to Item http://hdl.handle.net/10150/648656 The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics
by Jonathan Ramalheira-Tsu
Copyright © Jonathan Ramalheira-Tsu 2020
A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona
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THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
The Kostant-Toda Lattice, Combinatorial Algorithms and Ultradiscrete Dynamics
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Dedication
Para minha m˜ae Dedicated in loving memory to my mum. 4
Acknowledgements
First and foremost, I would like to express my deepest gratitude to my advisor, Nick Ercolani, for the rich and rewarding mathematical journey that culminated in this dissertation. Your exuberance in sharing your love of mathematics was a constant source of motivation, keeping charged the excitement and momentum in our work. You have truly made me feel supported and challenged, and I really cannot empha- sise enough how grateful I am for the privilege and honour of having had you as my advisor.
I wish to thank my committee members: Joceline Lega, Sergey Cherkis and David Glickenstein, for so generously sharing your time and guidance. I have thoroughly enjoyed our interactions; your questions and advice have been crucial in helping to form a great deal of what has been accomplished in this dissertation.
I wish to convey my appreciation for my family. Firstly, to my parents for your love and support. To my mother: your unfaltering faith in my life and mathematical journey have kept me going. Although you are no longer with me, I want you to know how much I miss you and still feel you cheering me on. To my father, thank you for your continuing to be there for me, taking on the mantle of encouraging and supporting me on this journey. I especially thank you for offering such a patient ear when I have needed someone to talk to.
Whilst I may not be able to list everyone else in my family who has supported me during this journey, I will certainly give it a good go! My first thanks go to the aunts and uncles who have supported me throughout the years, either directly in helping me move to the Arizona or supporting me during my stay, or simply for the occasional contact and friendship. I wish to thank Tio Amadeu, Tia Antonia, Tio Artur, Auntie Cathy, Auntie Dominique, Uncle George, Uncle Jim, Uncle Jimmy, Tia Luisa, Auntie 5
Mina, Tio Paulo, Auntie Paulette, Auntie Rita and Auntie Suzanne for your help. To Auntie Rita, Auntie Paulette and Auntie Cathy, I thank you for our many en- tertaining conversations throughout my time at the University of Arizona: you have helped keep me sane, providing lightheartedness when needed. I also wish to thank Cousin Marcelle and Justin for their support in my move to Tucson. Thanks also to Cousin Janina and her family for allowing me to join in with the fun and games at their house, as well as Cousin Nadine and her family for hosting me for my Christmas in the US.
To the rest of my family, I thank you for being part of such a fantastic network of love and care, especially during the difficult times.
Whilst at the University of Arizona, I have been fortunate to have forged great friend- ships, making my time in Tucson so much better. Brandon Tippings, I thank you for being such a wonderful friend, colleague and office mate. Arias Storm Hathway Turner, Breanna Gushiken, Jeff Davis and Michelle Trinh, I thank you all for the great times and close friendship we shared. To Ashley, Bonnie, Elliot and Max (the Klahrs), I thank you for welcoming me into your home for such fun-filled holidays and for providing me with a second family away from home. Lastly, but by no means least, I wish to thank my Tucson dance friends, Erika Raymond, Karen Yee and Terry Daily, to name a few, for helping to provide the fun outside of academia.
Charlotte Dorme and Ruth Lewis, my dearest friends back in the United Kingdom, I thank you for your most cherished friendship and for not letting our busy lives take us out of contact.
Finally, I would like to acknowledge the support for this research from the NSF grant DMS 1615921, for which I am very grateful. 6
Table of Contents
List of Figures ...... 9
Abstract ...... 11
Chapter 1. Introduction ...... 12 1.1. The Tropical Semiring and Maslov Dequantisation...... 13 1.2. The Robinson-Schensted-Knuth Correspondence...... 14 1.3. Geometric RSK and the Discrete-Time Toda Lattice...... 16 1.4. The Continuous-Time Toda Lattice and B¨acklund Transformations. 18 1.5. Box-Ball Systems...... 20 1.6. The Structure of this Dissertation...... 22 1.7. Main Results...... 25 1.7.1. Realisation of the Quantisation of the RSK Algorithm as a Dis- crete Dynamical System...... 25 1.7.2. Interpretation of RSK in Terms of Solitonic Particle Dynamics 26 1.7.3. The Ghost-Box-Ball System...... 27 1.7.4. Direct Integrable Systems Construction of Continuous and Dis- crete Time Geometric RSK...... 28 1.7.5. The Full Toda Lattice...... 29
Chapter 2. General Background ...... 31 2.1. The Toda Lattice...... 31 2.1.1. Definition and Description of the Toda Lattice Dynamical System 31 2.1.2. Integrability of the Toda Lattice...... 33 2.1.3. The Phase-Shift Formulæ for the Toda Lattice...... 36 2.1.4. Explicit Solution of the Toda Lattice: Method of Factorisation 38 2.1.5. Geometry of the Solutions: Embeddings into the Flag Manifold Phase Space...... 39 2.1.6. Symes’s Discrete-Time Matrix Dynamics and the Discrete-Time Toda Lattice...... 41 2.2. Box-Ball Systems...... 45 2.2.1. The Box-Ball Evolution...... 45 2.2.2. Soliton Behaviour and the Sorting Property...... 46 2.2.3. Invariants of the Box-Ball System...... 47 2.2.4. The Box-Ball Phase Shift...... 48 2.2.5. Coordinates on the Box-Ball System...... 51 2.2.6. Ultradiscretisation of Discrete-Time Toda...... 52 2.3. The Robinson-Schensted-Knuth Correspondence...... 54 7
Table of Contents—Continued
2.3.1. Young Tableaux...... 54 2.3.2. The Length of the Longest Increasing Subsequence of a Permu- tation and Patience Sorting...... 56 2.3.3. Schensted Insertion...... 58 2.3.4. The Robinson-Schensted Correspondence...... 60 2.3.5. Semistandard Young Tableaux and the Robinson-Schensted- Knuth Correspondence...... 63 2.3.6. The RSK Equations for Schensted Insertion...... 69 2.3.7. Kirillov’s Geometric Lifting: gRSK...... 76 2.3.8. A Matrix Representation of the Geometric RSK...... 78 2.3.9. Noumi and Yamada’s Observation...... 81
Chapter 3. Geometric RSK and Toda: The Discrete Picture .. 83 3.1. The Factorisation Problem...... 83 3.2. Parametrised Factorisations...... 84 3.3. Factorisations by Generalised Eigenfunctions...... 90 3.4. Geometric RSK as a Degeneration of the Discrete-Time Toda Lattice 93
Chapter 4. RSK and BBS: The Ultradiscrete Picture ...... 103 4.1. Ultradiscretisation of Geometric RSK...... 104 4.2. RSK Insertion and the Box-Ball Coordinates...... 107 4.3. The Ghost-Box-Ball System...... 110 4.4. Exorcism, Soliton Behaviour and the Invariant Shape...... 113 4.5. The Ghost-Box-Ball System and Schensted Insertion...... 116 4.5.1. The RSK Walls in the Ghost-Box-Ball System...... 117 4.6. The Ghost-Box-Ball Evolution and the Extended Box-Ball Coordinate Dynamics...... 124 4.7. Fukuda: Remarks and Distinctions...... 126
Chapter 5. Geometric RSK and the Toda Lattice: The Continuous- Time Picture ...... 128 5.1. Geometric Lifting and Path Operators...... 128 5.2. Lusztig Parameters and Total Positivity...... 131 5.2.1. Lusztig Parameters...... 131 λ 5.2.2. The Flow Rt on P ...... 132
5.3. Tw0 and the Linear Path η(t) = λt ...... 133 5.3.1. Painlev´eBalances...... 135 5.3.2. The Birkhoff Decomposition...... 135 5.3.3. Passage to General Toda: The Crystal Embedding...... 136 5.3.4. Tridiagonality of the Flow Associated to b(t) = eελt ...... 138 8
Table of Contents—Continued
5.4. The gRSK Stroboscope and a Nesting of Toda Lattices...... 139
Chapter 6. The Full Kostant-Toda Lattice ...... 142 6.1. Triangular Arrays and the Gelfand-Tsetlin Parametrisation...... 142 6.2. Continuous-Time gRSK and Dynamics on T and P ...... 144 6.3. The Set Tλ ...... 145 6.3.1. The Connection to the Toda Lattice...... 147 6.4. Flag Manifolds: The Crystal Embedding and the Companion Embedding 149 6.5. Extension to the Full Kostant-Toda Lattice...... 154 6.5.1. The Mapg ˆλ ...... 155 6.6. The Poisson Structure and Symplectic Geometry of Full Kostant-Toda 166 6.6.1. The Arhangelskij Normal Form and Parabolic Casimirs.... 166
Chapter 7. Future Directions ...... 178 7.1. Some Natural Extensions of this Work...... 178 7.2. Generalised Dressing Transformations...... 179 7.3. Geometric RSK and Box-Ball Systems for General Semisimple Lie Al- gebras...... 180 7.4. Geometric Quantisation...... 180
Appendices ...... 182
Appendix A. Fukuda: The Advanced Box-Ball System and the Carrier Algorithm ...... 183 A.1. The Box-Ball System with Labels...... 183 A.2. The Advanced Box-Ball System...... 184 A.3. The Carrier Algorithm...... 185 A.4. Schensted Insertion in the Advanced Box-Ball System...... 188
Index ...... 190
References ...... 192 9
List of Figures
Figure 1.1. Patience sorting a permutation in S8...... 14 Figure 1.2. The box-ball time evolution (iterated five times)...... 21 Figure 1.3. Roadmap of the dissertation, with numbers corresponding to connections between adjacent cells...... 22 Figure 2.1. A box-ball system time evolution (one time step)...... 46 Figure 2.2. The sorting property of the box-ball system...... 47 Figure 2.3. The invariant shape of the box-ball system(s) in Figure 2.2... 48 Figure 2.4. A phase shift interaction between two colliding chains...... 48 Figure 2.5. The box-ball coordinates on a box-ball system and its time evo- lution...... 51 Figure 2.6. Iterative Schensted word insertions building the RSK correspon- dence...... 80 Figure 4.1. A single time-step of the ghost-box-ball evolution, split into its the subroutine that defines it...... 112 Figure 4.2. A single time-step of the ghost-box-ball evolution (without the intermediate steps)...... 113 Figure 4.3. Three iterations of the ghost-box-ball algorithm...... 115 Figure 4.4. Three iterations of the ghost-box-ball evolution (after global ex- orcism)...... 115 Figure 4.5. The invariant shape of the ghost-box-ball system(s) in Figure 4.3 116 Figure 4.6. The initial ghost-box-ball system with its finite regions labelled. 118 Figure 4.7. The evolution of a ghost-box-ball system with walls...... 120 Figure 4.8. The Schensted evolution encoded by Figure 4.7...... 120 Figure 5.1. Iterative Schensted word insertions. In the geometric setting, the y’s here are precisely the y’s in Theorem 5.11...... 141 Figure 6.1. The quiver structure on triangular arrays...... 146 Figure 7.1. Roadmap of the dissertation, with numbers corresponding to connections between adjacent cells...... 178 Figure A.1. A single time-step of the advanced box-ball system, split into the successive movements of each colour/label...... 183 Figure A.2. An advanced box-ball system with carrying capacities...... 184 Figure A.3. The steps of a single time evolution of an advanced box-ball system with carrying capacities...... 185 Figure A.4. The Carrier Algorithm...... 185 Figure A.5. Successive applications of the Carrier Algorithm with input se- quence from an advanced box-ball system...... 187 10
List of Figures—Continued
Figure A.6. Schensted insertion encoded in a unit carrying capacity advance box-ball evolution...... 189 11
Abstract
We study the relationship between the algorithm underlying the Robinson-Schensted- Knuth correspondence (Schensted insertion) and the Toda lattice, exploring this in the settings of discrete-time, ultradiscrete, and continuous-time dynamical systems.
Starting with the work of Noumi and Yamada [NY04] and their observation of a similarity between Hirota’s [Hi77] discrete-time Toda lattice and Kirillov’s [Ki00] ge- ometric lifting of the RSK (geometric RSK) equations for Schensted insertion, we derive solutions to the former in its unbounded setting and provide an explicit em- bedding of geometric RSK in the discrete-time Toda lattice.
Mimicking the ultradiscretisation of the discrete-time Toda lattice to the soliton cel- lular automaton, the box-ball system [To04], we produce an extension of the classical box-ball system for Schensted insertion, which we call the ghost-box-ball system. We study this new cellular automaton in relation to Schensted insertion, demonstrating their equivalence, both on their respective coordinatisation and also on the algorith- mic level.
O’Connell et al. ([O13], [BBO09], [COSZ14], [O12]) demonstrate an impressive treat- ment of the relation between a continuous version of geometric RSK and the Toda lattice. Through the introduction of dressing transformations and Painlev´eanaly- sis [EFH91], we reformulate some of these connections in a more integrable systems theoretic way. In this continuous setting, we also see the general Toda flows arise and present results on the Poisson geometry of the full Kostant-Toda lattice to lay the foundation for future probing of these exciting connections between algorithms, combinatorics. and dynamical systems theory. 12
Chapter 1 Introduction
Recently, a great deal of remarkable development in the literature has been made to bring together motivations and methods from the study of algorithms, combina- torics, integrable systems theory, and representation theory ([O13], [BBO09], [NY04], [Fu04], [Ga70], [Wo02], to list a few). A consistent theme is the translation of ideas and approaches from one area to another, serving to provide insight from new per- spectives. Of particular interest in this dissertation is the idea that an algorithm can effectively be thought of as being like a discrete dynamical system; continuum limits of the latter may provide insights into the analysis of the former.
The main focus of this dissertation will be an instance of this principle, studied in great detail and on multiple levels. We will see this principle in the context of Schen- sted insertion, the algorithm at the heart of the famous Robinson-Schensted-Knuth correspondence (cf. 2.3). Noumi and Yamada [NY04] observed a remarkable connec- tion between Schensted insertion (a purely algorithmic/combinatorial process) and the Toda lattice (a similarly well-known integrable dynamical system). The key piece of the bridge was provided by Kirillov [Ki00], in the form of tropicalisation (cf. 1.1). Through the exploration of this correspondence, other intriguing systems will be seen to emerge.
We now preview all of these ideas in the following sections of this introduction, starting with Maslov’s [LMRS11] elegant description of tropicalisation in Section 1.1. 13
1.1 The Tropical Semiring and Maslov Dequantisation
Tropical mathematics ([LMRS11], [Li07], [Vi01]) is the study of the max-plus semir- ing, which we will now define. In this section, we follow the presentation given by
Maslov [LMRS11]. The structure of the semiring (R≥0, +, ×) is carried over to the
set S = R ∪ {−∞} by a family of bijections D~, for ~ > 0, given by
ln x if x 6= 0 D (x) = ~ . (1.1.1) ~ −∞ if x = 0
This induces a family of semirings, parametrised by ~ > 0, (S, ⊕~, ⊗~) with operations given by
ln(ea/~ + eb/~) if a, b 6= −∞ a ⊕ b = D (D−1(a) + D−1(b)) = ~ (1.1.2) ~ ~ ~ ~ max(a, b) otherwise a ⊗ b = D (D−1(a)D−1(b)) = a + b. (1.1.3) ~ ~ ~ ~
In the limit, ~ → 0, Maslov ‘dequantises’ (R≥0, +, ×) to obtain the tropical semiring (R ∪ {−∞}, max, +), where its addition is the usual max operation and its multipli- cation operation is usual addition, hence the name “max-plus semiring”.
Maslov views this construction as an analogue of the correspondence principle from
quantum mechanics, with (R≥0, +, ×) as the quantum object and (R∪{−∞}, max, +) as its classical counterpart.
Tropicalisation takes one into the realm of the piecewise linear. This relatively new field of mathematics has seen applications in many areas of mathematics, including algebraic geometry, numerical analysis, cryptography, and, as we will see shortly, combinatorics and integrable systems. 14
1.2 The Robinson-Schensted-Knuth Correspondence
The celebrated Robinson-Schensted-Knuth (RSK) correspondence is a fundamental correspondence (see Section 2.3) between permutation groups and their representa- tion theory. This may be regarded as a kind of discrete nonlinear Fourier transform.
At the heart of the RSK correspondence is an insertion procedure called Schensted insertion which is a revamped version of patience sorting [AD99], which itself is a solitaire-like algorithm. Patience sorting takes a permutation (σ(1), . . . , σ(n)) of n numbers and forms a sequence of piles by taking each number in succession, placing it onto the left-most pile whose top number is greater than the number being placed. The process is performed by initialising with a single pile consisting of the first num- ber in the sequence.
For example, applying patience sorting to the permutation (1, 3, 6, 2, 4, 7, 5, 8) in S8, one obtains the following sequence of piles, initialised at the top-left, and terminating at the bottom-left:
3 6 2 2 1 1 3 1 3 6 1 3 6
4
2 4 5 8 2 4 5 5 2 4 7 2 4 1 3 6 7 8 1 3 6 7 1 3 6 7 1 3 6
Figure 1.1. Patience sorting a permutation in S8.
A key property of patience sorting is that the total number of piles is equal to the length of the longest increasing subsequence of the permutation. There may be more than one subsequence of this length (e.g. (1, 3, 6, 7, 8) and (1, 2, 4, 5, 8) are both of 15 length 5).
Although the build-up to Schensted insertion is rather involved, we point out that
n n Schensted insertion can be reduced to a discrete evolution on N0 × N0 , where N0 := N ∪ {0}. The foundation for Schensted insertion is a prescription for taking an input pair of sequences
(a, x) = ((a1, . . . , an), (x1, . . . , xn)), which encode words (weakly increasing sequences of positive integers) from an alpha- bet {1, . . . , n}, applying an extended version of patience sorting on the words, and transforming the input sequences into an output pair of sequences
(b, y) = ((b1, . . . , bn), (y1, . . . , yn)) which encode the the result of performing this extension of patience sorting. This encoding is introduced in full detail in Section 2.3.6.
The prescription, as we will see in Section 2.3, is given by recursively defining an n-tuple (η1, . . . , ηn) by
η1 = ξ1 + a1,
ηj = max{ηj−1, ξj} + aj, ∀ j = 2, . . . , n,
j P where ξj = xi for j = 1, . . . , n, and then solving for y and b via the following: i=1
y1 = η1,
yj = ηj − ηj−1, ∀ j = 2, . . . , n
b1 = 0,
bj = aj + xj − yj, ∀ j = 2, . . . , n.
The ξ and η variables above are auxiliary, providing these equations describing Schen- sted insertion. 16
1.3 Geometric RSK and the Discrete-Time Toda Lattice
Kirillov [Ki00] noticed that the RSK equations (cf. Corollary 2.20) were of a max- plus nature, and performed the reverse of dequantisation to obtain the quantised (or de-tropicalised) RSK equations. These equations have now come to be known as the geometric RSK (or gRSK) equations, the construction of which we very briefly provide. Treating the Schensted insertion equations as if they were obtained from tropicalisation/dequantisation, one can recover their quantum analogue by performing the following conversion on them (max, +, −, 0) 7→ (+, ×, ÷, 1). This calculation, due to Kirillov [Ki00], results in the following equations:
η1 = ξ1a1
ηj = (ηj−1 + ξj)aj ∀ j = 2, . . . , n,
where ξj = x1 ··· xj for j = 1, . . . , n, and then solving for y and b via the following:
y1 = η1
ηj yj = ∀ j = 2, . . . , n ηj−1
b1 = 1
xj ξjηj−1 bj = aj = aj ∀ j = 2, . . . , n. yj ξj−1ηj
Finally, one can eliminate (cf. Lemma 2.21) the supplementary variables ηi and ξi to obtain the following equations, commonly known as the geometric RSK (gRSK) 17
equations: b1 = 1 a1x1 = y1 ajxj = yjbj ∀ j = 2, . . . , n (1.3.1) 1 1 1 + = a x b 1 2 2 1 1 1 1 + = + ∀ j = 2, . . . , n. aj xj+1 yj bj+1 Remarkably, the system of equation in 1.3.1 can be written equivalently as the fol- lowing matrix factorisation [NY04]:
a¯1 1 x¯1 1 y¯1 1 1 0 a¯ 1 0 x¯ 1 0 y¯ 1 0 ¯b 1 0 2 2 2 2 . . . . . . . . .. .. .. .. = .. .. .. .. , (1.3.2) ¯ a¯n−1 1 x¯n−1 1 y¯n−1 1 bn−1 1 ¯ 0 a¯n 0 x¯n 0 y¯n 0 bn
1 where variables with bars are reciprocated, e.g.a ¯i = . ai
Given the left-hand side of Equation 1.3.2, the solution (the y’s and b’s), if it exists, is uniquely determined by virtue of the b-matrix, specifically by the 1 and 0 in the top-left.
Noumi and Yamada [NY04] then observed that, modulo certain boundary condi- tions, a change of coordinates could be performed on the gRSK equations to yield Hirota’s [Hi77] integrable discretisation of the famous Toda lattice, one of the most important examples of a completely integrable system, especially due to its natural generalisations to the setting of semisimple Lie algebras and their representations ([Ko78], [EFS93]). Noumi and Yamada’s change of coordinates [NY04], which forms 18
the starting point for our work, is the following:
t −1 t −1 t+1 −1 t+1 −1 ai = (Ii+1) , xi = (Vi ) , yi = (Vi ) , bi = (Ii ) . (1.3.3)
When performed on Hirota’s discretisation of the Toda lattice [Hi77]:
t+1 t t t+1 Ii = Ii + Vi − Vi−1 t t , (1.3.4) t+1 Ii+1Vi Vi = t+1 Ii the result is the following system of equations:
aixi = yibi (1.3.5) 1 1 1 1 + = + ai xi+1 yi bi+1 both for all i ∈ Z. This is clearly seen to be related to Equations 1.3.1, and this observation provides the initial point of contact between the worlds of RSK and the Toda lattice. The exact means by which we recover geometric RSK involves degenerate boundary conditions, and is the focus of Chapter3.
1.4 The Continuous-Time Toda Lattice and B¨acklund Transformations
The classical Toda lattice, introduced by Toda [To67], is a dynamical system in con- tinuous time. It models the mechanics of a chain of unit mass particles repelling each other with an exponential potential. It is a prototype for integrable systems theory, with generalisations from gln or sln to other semisimple Lie algebras ([Ko78] and [EFS93]). To solve the Toda lattice, one usually uses the factorisation method (cf. Theorem 2.5), which involves computing the LU decomposition or Gauss de- composition (cf. Definition 2.2). An equivalent formulation is in terms of B¨acklund 19
transformations, whose discrete analogues provide a convenient means of discretisa- tion to yield Hirota’s discrete-time Toda lattice.
Its initial formulation was in terms of relative positions and momenta, the j-th parti-
cle from the left has relative position qj and momentum/velocity pj. In the so called open-ended case, which is the only case we will treat in this dissertaion, there are n + 2 particles, labelled 0, 1, . . . , n + 1, with the 0-th pinned at negative infinity and the (n + 1)-st pinned at positive infinity.
Under Flaschka’s change of variables [Fl91], given by aj = −pj for j = 1, . . . , n, and
qj −qj+1 bj = e for j = 1, . . . , n−1, the phase space is presented conveniently as matrices of the following form:
a1 1 .. b1 a2 . X = . (1.4.1) . . .. .. 1 bn−1 an
A physical interpretation of the Toda lattice naturally gives rise to an understanding of the asymptotic behaviour of the Toda lattice that would be harder to access by analysis of the defining equations alone. The particles will repel each other until the system is sufficiently spread out so as to allow the particles to essentially move freely, no longer experiencing any significant repelling forces from their neighbours. Eventu- ally, as time goes to positive infinity, the particles willl sort themselves by ascending momenta, and the matrix in Equation 1.4.1 will have bj → 0 for each j, with the a’s taking on the values of the ordered momenta. This phenomenon of sorting by momenta is appropriately referred to as the sorting property of the Toda lattice, and we will see an analogue of this property follow through in the next section, for the 20
box-ball system.
In the Flaschka representation [Fl91], on symmetric matrices, this long-term sorting amounts to a diagonalisation of the matrix which is yet another algorithm. The ma- trix in 1.4.1 can by conjugated by the diagonal matrix
1 √ b1 √ D = b b (1.4.2) 1 2 . .. p b1 ··· bn−1
to obtain the symmetric form of the Toda lattice on Jacobi matrices:
√ a1 b1 √ .. b1 a2 . D−1XD = . (1.4.3) . . .. .. p bn−1 p bn−1 an
1.5 Box-Ball Systems
The box-ball system ([TS90], [TTS96], [To04]) is a famous example of a cellular automaton, another being Conway’s game of life [Ga70]. The box-ball system consists of a row of infinitely many boxes, filled with finitely many balls. A simple time evolution rule is given as follows:
(1) Take the left-most ball that has not been moved and move it to the left-most empty box to its right.
(2) Repeat (1) until all balls have been moved precisely once. 21
To illustrate this algorithm, below are six box-ball states (the top is the initial state, followed by its five subsequent time evolutions):
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
Figure 1.2. The box-ball time evolution (iterated five times).
The box-ball system exhibits a sorting property: chains of balls (consecutive strings of balls) travel as coherent masses (solitons) with velocity equal to the number of balls in a chain. When two solitons collide, the faster one comes out in front, and the two experience a phase shift (a deviation in position from where they would have been, barring the collision, cf. 2.2.4). Asymptotically, with enough iterations of the time evolution, the solitons will order themselves from slowest to fastest, travelling freely after being sorted.
The similarity to the behaviour of the Toda lattice is absolutely no accident: Toki- hiro [To04] shows that there is a coordinatisation of the box-ball system, with clear rules prescribing the evolution on these coordinates. Tokihiro also demonstrates the process of ultradiscretisation (cf. 2.2.6) on the discrete-time Toda lattice, and this is precisely the evolution rule on the box-ball system equations.
Although we will imminently repeat this, we would be remiss to not whet the reader’s appetite with our developments on the connection between Schensted insertion and the box-ball system. In Chapter4, we offer a multitude of new results on this con- nection. We mention the two main results now.
1. We show that by expanding the domain of the box-ball coordinate evolution, 22
one retrieves precisely the Schensted insertion rules.
2. Another rather exciting result is in the development of a new variant of the box-ball system that encodes Schensted insertion.
1.6 The Structure of this Dissertation
We provide the following table as a roadmap of the general topics in this dissertation. The numbers between two adjacent cells of the table act as a key for the more detailed outline below the table.
de-tropicalisation continuum limit
discrete space continuous space continuous space discrete time discrete time continuous time
RSK Continuous Algorithm 2 Geometric RSK 5 (Schensted Insertion) gRSK
. 1 4 7
(Ghost) (Hirota’s) Dynamics 3 6 Toda Lattice Box-Ball System Discrete Toda
Maslov tropicalisation stroboscope
Figure 1.3. Roadmap of the dissertation, with numbers corresponding to connec- tions between adjacent cells.
Chapter2 provides most of the background details concerning the above. In Section 2.1.6, we describe a relationship between the continuous-time Toda lattice and the discrete-time Toda lattice (6 in the table). Section 2.3.7 provides Kirillov’s geometric lifting of the RSK equations for Schensted insertion to the geometric RSK equations [Ki00](2 in the table). In Section 2.2.6, we present Tokihiro’s ultradiscretisation of 23 discrete-time Toda to yield the box-ball equations [To04](3 in the table). Section 6.2, whilst not in the Chapter2, is a presentation of background material, namely O’Connell presentation of a continuous time analogue of geometric RSK [O13](5 in the table). Finally, we include an analogy between a certain (spatial) extension of ge- ometric RSK and the discrete-time Toda lattice in what we have come to call “Noumi and Yamada’s observation” [NY04]. This last item is not quite 4 in the table, but it is what motivated our eventual description of 4 in the table.
Chapter3 concerns 4 in the table. In this chapter, we study the relationship between Schensted insertion and Toda on the discrete level, which is the original level to which Noumi and Yamada’s observation applies. We present the Hirota’s discretisation of Toda, which manifests as a stroboscopic dynamics of “factoring and flipping”, per- formed recursively. Central to this dynamics is the method of Gauss elimination/LU decomposition: factoring a matrix into a lower unipotent matrix, multiplied on the right by an upper triangular matrix. The original discrete-time Toda lattice concerns infinitely many particles, hence bi-infinite matrices (indexed by pairs of integers, with- out bound). By imposing boundary conditions, one obtains semi-infinite (indexed by pairs of integers, each bounded from one (the same) direction) and finite versions.
The first part of Chapter3 is an exploration into the solution of the discrete-time Toda lattice in the semi-infinite and bi-infinite cases only (the solution to the semi- infinite case truncates to the solution to the finite case). For semi-infinite discrete-time Toda, we provide an explicit description of the solution, when it exists, in terms of τ-functions (principal minor determinants). We then show how to piggyback on the semi-infinite solution to a parametrised family of solutions for the bi-infinite discrete- time Toda lattice. We also prove an extension of a result of Murphy [Mu18] on the factorisation of discrete-Schr¨odingeroperators, expressing the bi-infinite discrete-time Toda solutions in terms of generalised eigenfunctions. 24
Chapter4 concerns 1 in the table. In this chapter, we perform an operation on geo- metric Schensted insertion called ultradiscretisation. At its heart is Maslov’s tropical- isation. Indeed, we show that this recovers the original RSK equations for Schensted insertion. In the work of Tokihiro [To04], it is shown that this ultradiscretisation procedure, when applied to the discrete-time Toda lattice, yields the famous box-ball system equations. By mimicking Tokihiro’s discrete Toda ultradiscretisation for ge- ometric Schensted insertion, we present Schensted insertion in a form lending itself very naturally to comparison to the box-ball system. This natural view of Schensted insertion as a box-ball coordinate dynamics gives rise to one’s needing to make sense of zero length chains of balls and zero length chains of empty boxes, which is not inherently part of the original box-ball system but makes complete sense at the level of the equations describing the box-ball evolution. On this level, we view Schensted insertion as a degeneration of the box-ball equations. Furthermore, since the elegance of the original box-ball system lies in the simplicity of its description (a recipe for moving balls into empty boxes), we hoped to recapture the extended box-ball equa- tions in a similarly pictographic/procedural manner, which we accomplished in the creation of a new cellular automaton, the ghost-box-ball system. The remainder of Chapter4 is dedicated to studying the ghost-box-ball system in relation to Schensted insertion. We show that the ghost-box-ball system, at each stage of a single time step, encodes the stages of a single Schensted word insertion. We also show that the underlying evolution, modulo the ghosts, is precisely the original box-ball evolution.
Chapter5 concerns 7 in the table. In this chapter, we summarise the material in [BBO09] and [O13] pertaining to the connection between a continuous time analogue of geometric RSK and the classical (tridiagonal) Toda lattice. We begin with the construction of a partial right action of a unitary group on the space of continuous paths through a subgroup, starting with the 2 × 2 case, motivated by Sturm-Liouville 25 equations and dressing transformations. We also review the Lusztig parametrisa- tion, with Berenstein, Fomin and Zelevinsky’s [BFZ96] explicit specialisation of this parametrisation to a certain right-translation of the set of upper triangular matrices defined by the positivity of certain minor determinants.
We provide the background on the Painlev´eanalysis of the Toda lattice, using this and the geometry of flag manifolds to construct a novel proof of a main theorem (cor- responding to Theorem 5.9) of O’Connell’s paper on geometric RSK and the Toda lattice [O13].
Chapter6, we provide O’Connell’s [O13] description of relationship between the con- tinuous time geometric RSK equations and the Toda lattice, in terms of triangular arrays.
1. By relaxing a condition on triangular arrays, we provide a proof that the general Toda flows drive the dynamics on the triangular arrays.
2. We then provide a description of the Poisson structure and symplectic geometry underlying the full Kostant-Toda lattice through the study of the Arhangelskij normal form (6.6.11) and its associated invariants ([GS99], [Ar79]).
1.7 Main Results
We conclude the introduction with a discussion of the main results of the dissertation.
1.7.1 Realisation of the Quantisation of the RSK Algorithm as a Discrete Dynamical System
Our main result for Chapter3 is Theorem 3.7, providing a very explicit way of view- ing the quantisation of Schensted insertion (geometric RSK) as a time step of the discrete-time Toda lattice. Before our result, this was a problem of interest in the literature, but the closest solution was a result of Noumi and Yamada [NY04] showing 26 an equivalence of geometric RSK and the discrete-time Toda lattice in the bi-infinite setting, rather than the finite setting in which geometric RSK actually lives. Our result takes this relationship to the finite setting, resulting in witnessing geometric RSK as lying on the boundary of the discrete-time Toda lattice. This helps to finally make precise what had previously only existed as an observation/analogy in the lit- erature.
In addition to our main result, with Noumi and Yamada’s observation connecting to the bi-infinite discrete-time Toda lattice which is solved by bi-infinite matrix fac- torisations, we provide two results classifying the LU (lower-upper) decompositions of tridiagonal, bi-infinite, Hessenberg matrices. The first result, Theorem 3.2, offers a parametrisation of the factorisations by leveraging the already known solution for semi-infinite and finite matrices. In addition to this parametrised factorisation result, we offer Theorem 3.4 as a classification of these factorisations in terms of generalised eigenfunctions.
1.7.2 Interpretation of RSK in Terms of Solitonic Particle Dynamics
Motivated by our precise connection between geometric RSK and the discrete-time Toda lattice we searched for the analogous connection on the level of their respective dequantisations (tropicalisations or ultradiscretisations). The main result for this goal is Theorem 4.2 which shows how Schensted insertion is captured by the coordi- nate evolution on the box-ball system, a soliton particle dynamics that arises from dequantising the discrete-time Toda lattice. Theorem 4.2 and Corollary 4.3 achieve this goal by demonstrating that the RSK equations for Schensted insertion are given by the box-ball system equations. Key to these results was a rewriting of the RSK equations without which the relation to the box-bal system equations was obscured. 27
Prior to this work, a relationship between RSK and an advanced version of the box- ball system was known by Fukuda [Fu04]. However, this advanced box-ball system requires various extra features not automatically possessed by the dequantisation of the discrete-time Toda lattice. Our results work solely with what is obtained from the discrete-time Toda lattice, potentially opening up an avenue for more direct future connections between dynamical systems theory and the representation theory underlying the RSK correspondence.
1.7.3 The Ghost-Box-Ball System
Whilst the connection between Schensted insertion and the coordinate evolution of the box-ball system is quite satisfying, it relinquishes one of the appealing qualities of the box-ball system: its graphical representation as an asymmetric simple exclusion process on boxes and balls. The ghost-box-ball system is our answer to restoring a graphical nature to the extension that was necessary for the classical domain of the box-ball coordinates. This produced a completely new cellular automaton which comes equipped with its own motivation for study due to its creation from the already important algorithm of Schensted insertion. We show in Corollary 4.9 that this new cellular automaton does indeed capture the RSK equation. We further amplify this in Theorem 4.8 which establishes a one-to-one correspondence between each step of Schensted insertion and each step of the ghost-box-ball evolution.
Knowing some of the classical features of the box-ball system, we answer some of the natural questions on what persists in the modification leading to the ghost-box-ball system. The key to leveraging properties of the box-ball system is a transformation we call exorcism. Lemma 4.5 establishes that the ghost-box-ball evolution, under this transformation, is precisely the box-ball evolution. Using this, we show in Lemma 28
4.6 that the ghost-box-ball system possesses the classical solitonic behaviour of the box-ball system. Using this, we also prove in Corollary 4.7 the invariance of a com- binatorial signature (the shape) of the ghost-box-ball system. Aside from answering natural questions, these results show that the ghost-box-ball system possesses prop- erties that have historically attracted attention to the classical box-ball system.
Our final work presented in this dissertation on the ghost-box-ball system is an at- tempt at answering the question of whether the extended box-ball coordinate evolu- tion completely captures our ghost-box-ball evolution for ghost-box-ball systems not arising from RSK. We conjecture that it does (Conjecture 4.10), and prove a strati- fied (restricted to certain level sets of an invariant function of the box-ball coordinate evolution) version of the conjecture in Theorem 4.11.
1.7.4 Direct Integrable Systems Construction of Continuous and Discrete Time Geometric RSK
The (classical) Toda lattice is a continuous-time dynamical system. O’Connell [O13] studied a continuous-time version of geometric RSK, and related this to the Toda lattice. This work, whilst establishing this relationship, was rather involved, requir- ing one to pass through ideas and constructions from last passage percolation and continuum versions of the Gessel-Lindstr¨om-Viennottheorem, as well as continuum analogues of Gelfand-Tsetlin patterns. Additionally, this approach requires the in- volvement of semi-discrete random polymers, Whittaker vector and quantum Toda lattices. Our approach is much simpler in nature, and we demonstrate the simplicity in our proof of Theorem 5.9, which re-derives the key result of O’Connell, without needing to pass through such complicated structures. Additionally, Theorem 5.11 establishes a passage from continuous-time geometric RSK back to (discrete-time) geometric RSK, aiding in further increasing our mobility through our roadmap table 29
(Figure 1.3).
An advantage of our approach is that we bring in the geometry of flag manifolds, making available the machinery of [EFH91] and [EFS93] to our perspective. To aid in connecting to the geometry of flag manifolds, we derive explicit formulæ for describing the key components involved in mapping to the flag manifold. Lemma 6.8 provides an explicit description of the matrix used to conjugate a tridiagonal, Hessenberg matrix to its companion matrix, and Lemma 6.9 provides a relatively explicit description of the matrix that conjugates the companion matrix to a particular distinguished member of its coadjoint orbit. Finally, we write a slightly more explicit version of a result of [EFH91] for diagonalising the aforementioned distinguished matrix (when possible).
1.7.5 The Full Toda Lattice
In O’Connell’s work [O13], it is shown how classical (tridiagonal) Toda is obtained from continuous geometric RSK. We show very explicitly in Theorem 6.13 that re- laxing a certain condition, imposed by O’Connell, yields the full Toda lattice (not restricted to the triadiagonal case). We capture this in a commutative diagram in Theorem 6.15 which also contains a complete roadmap between O’Connell’s various components and the flag manifold and associated embeddings of the Toda flow. This opens up a potential path to generalising the constructions of this dissertation to the full Toda lattice. For example, we hope this will provide a way of find an analogue of the (ghost-)box-ball system for the full Toda lattice.
In light of this hopeful avenue to full Toda, we lay the groundwork for some of the relevant Poisson geometry. We present (between Lemma 6.18 and Corollary 6.19) a correction to Gekhtman and Shapiro’s formula [GS99] for the matrix conjugating 30 a matrix to the so-called Arhangelskij form and a proof that this correction works. Additionally, we offer Theorem 6.20 as a very explicit proof that the Arhangelskij form produces invariants of a matrix under the coadjoint orbit of certain parabolic subgroups. These results have been known in the literature (cf., for example, [GS99]). However, we provide more concrete details to help illuminate the path forward. 31
Chapter 2 General Background
2.1 The Toda Lattice
In this section, we present the relevant background on the Toda lattice: from its basic definition, to its solutions, geometry and discretisation.
2.1.1 Definition and Description of the Toda Lattice Dynamical System
The Toda lattice, due to Toda [To67], is a nearest neighbour interaction dynamical system on a collection of particles, in which two particles exert a repelling force on each other, given by the exponential of the distance between them.
Take n + 2 particles of unit mass, labelled j = 0, 1, . . . , n + 1. Let Particle j have
position qj (relative to its equilibrium position) and momentum pj. Because each particle has unit mass, one has
q˙j = pj (2.1.1)
for each j.
qj−1−qj qj −qj+1 Furthermore, taking the force as described above, one hasp ˙j = e − e for each j.
Lastly, boundary conditions of q0 = −∞ and qn+1 = ∞ are imposed, which, formally, result in eq0−q1 = eqn−qn+1 = 0. These boundary conditions are chosen to truncate the lattice to a finite system. Consequently, the Toda lattice is expressed as the following 32 system of differential equations:
q˙j = pj j = 1, . . . , n (2.1.2)
q1−q2 −e if j = 1 qj−1−qj qj −qj+1 p˙j = e − e if 1 < j < n (2.1.3) eqn−1−qn if j = n
2n When R , with coordinates (p1, . . . , pn, q1, . . . , qn), is equipped with the standard Poisson bracket, the Toda lattice is a Hamiltonian system with Hamiltonian function
n n−1 1 X X H(p , . . . , p , q , . . . , q ) = p2 + eqj −qj+1 . (2.1.4) 1 n 1 n 2 j j=1 j=1
Flaschka’s change of variables (p1, . . . , pn, q1, . . . , qn) 7→ (a1, . . . , an, b1, . . . , bn) is given
qj −qj+1 by by setting aj = −pj for j = 1, . . . , n, and bj = e for j = 1, . . . , n − 1.
In Flaschka variables, the Toda lattice assumes the following simple form:
˙ bj = (aj+1 − aj)bj, j = 1, . . . , n − 1 (2.1.5)
b1 if j = 1 a˙ j = bj − bj−1 if 1 < j < n . (2.1.6) −bn−1 if j = n
One can arrange the variables neatly into a matrix
a1 1 .. b1 a2 . . (2.1.7) .. .. . . 1 bn−1 an 33
Equations 2.1.5 and 2.1.6 amount to the following matrix differential equation:
b1 0 a1 1 . .. (a − a )b b − b .. d b1 a2 . 2 1 1 2 1 = . dt . . .. .. 1 . . .. .. 0 bn−1 an (an − an−1)bn−1 −bn−1 (2.1.8)
Remark 2.1. The boundary conditions chosen in Section 2.1.1 amount to setting b0 = bn = 0, which are precisely what enables this expression of the Toda lattice as a dynamical evolution on this finite dimensional matrix phase space.
2.1.2 Integrability of the Toda Lattice
Let X be the matrix
a1 1 .. b1 a2 . X = (2.1.9) . . .. .. 1 bn−1 an
Proposition 2.1. Equations 2.1.5 and 2.1.6 are equivalent to the following Lax equa- tion d X = [X, π (X)] (2.1.10) dt − where π−(X) is the projection of X onto its strictly lower triangular part, i.e. X with the a’s and 1’s replaced by zeroes. 34
Remark 2.2. If π+(X) is the projection of X onto its upper triangular part, then one could replace the right-hand side of Equation 2.1.10 with [π+(X),X], by the usual properties of the matrix commutator.
One can uniformly shift the particles in the Toda lattice, so we may assume that the average of the a’s is zero, i.e. a1 + ··· + an = 0. Under this assumption, X is a traceless matrix and is therefore an element of the Lie algebra sl(n, R). We will allow for complex numbers and make the following definitions.
Definition 2.1. Let g := sl(n, C) be the Lie algebra of traceless n × n matrices. Let n+ (n−) be the subalgebra of upper (respectively, lower) nilpotent matrices. Similarly, let b+ (b−) be the subalgebra of upper (respectively, lower) triangular matrices with trace zero.
Let π+ be the projection π+ : g → b+ and π− be the projection π− : g → n−.
We now define the Lie group analogues of the above.
Definition 2.2. Let G = SL(n, C) be the Lie group of determinant one n×n matrices.
Let N+ (N−) be the group of upper (respectively, lower) unipotent matrices. Let B+
(B−) be the group of upper (respectively, lower) triangular matrices with determinant
1. When g ∈ G is expressed as g = nb, where n ∈ N− and b ∈ B+, one calls nb the LU- decomposition (lower-upper decomposition). For g ∈ G, if g has an LU-decomposition, define Π− and Π+ by g = Π−(g)Π+(g), where Π−(g) ∈ N− and Π+(g) ∈ B+.
Remark 2.3. Not all matrices have an LU decomposition, i.e. G 6= N−B+. However,
N−B+ is dense in G. Furthermore, one can find a neighbourhood of the identity on which this factorisation always exists.
Definition 2.3. . Let εn be the matrix in g with 1’s on its superdiagonal, and 0’s 35
elsewhere. For example, when n = 3,
0 1 0 ε3 = 0 0 1 . 0 0 0
When it is unambiguous to do so, we drop the subscript and simply write ε.
With these definitions in place, we see that the phase space for the matrix form of the
Toda lattice is the subset of tridiagonal matrices in ε + b−, the so-called Hessenberg matrices.
Lemma 2.2. If X solves Equation 2.1.10, then
d Xk = [Xk, π (X)] (2.1.11) dt −
for all k ∈ N.
This is shown by induction and the product rule for matrix differentiation.
Proposition 2.3. The Toda flow is isospectral.
Proof. By Lemma 2.2, one has
d d trXk = tr Xk = tr[Xk, π (X)] = 0 (2.1.12) dt dt − since the commutators are traceless. With all trace powers having zero derivative, it follows that the characteristic polynomial, hence spectrum, of X is conserved by the Toda flow.
Moser [Mo75] showed that, generically, as time goes to +∞ or −∞, the repelling forces will cause the system to expand: the particles will move away from each other
asymptotically. This is exhibited in qj+1 − qj → ∞ for all j and as t → ±∞. There-
fore, asymptotically, each pj will limit to a constant. In the Flaschka variables, this 36
means that bj → 0 as t → ±∞, and aj will eventually become constant. Since the constant values for the aj’s will be eigenvalues for the asymptotically attained matrix, and the Toda lattice is isospectral, one must have that (a1, . . . , an) is some permuta- tion of the eigenvalues (λ1, . . . , λn) of X.
Furthermore, as the particles separate, the particles behave more like free particles with momenta given by the eigenvalues. A fast particle will move past a slower one, each undergoing a phase shift (cf. Section 2.1.3), and so at the end, the particles will arrange themselves from slowest to fastest, as t → ∞. Thus, asymptotically, if
λ1 < λ2 < ··· < λn, then
X(t) → diag(λ1, λ2, . . . , λn) + ε as + t → ∞. (2.1.13)
A similar argument holds in reverse time:
X(t) → diag(λn, λn−1, . . . , λ1) + ε as t → −∞. (2.1.14)
This property of the Toda lattice is often called the sorting property. In analogy to the box-ball system and its sorting property in Section 1.5, these eigenvalues are the ordered asymptotic speeds of the particles in the Toda lattice, like the speeds of the blocks of adjacent balls in the box-ball system.
2.1.3 The Phase-Shift Formulæ for the Toda Lattice
Returning to the original Hamiltonian variables for the Toda lattice (Equations 2.1.2 and 2.1.3),
q˙j = pj j = 1, . . . , n
q1−q2 −e if j = 1 qj−1−qj qj −qj+1 p˙j = e − e if 1 < j < n eqn−1−qn if j = n 37
+ n − n Definition 2.4. Define two sequences of numbers (αk )k=1 and (αk )k=1 by
+ − α = lim pk, α = lim pn−k+1. (2.1.15) k t→+∞ k t→+∞
In the notation of 2.1.13, since ai = −pi for all i, and ai → λi as t → +∞, we have
+ − αk = −λk, αk = −λn−k+1. (2.1.16)
The following is a result of Moser.
Theorem 2.4 ([Mo75]). The asymptotic behaviour of the solution to the (Hamilto- nian form of the) Toda lattice is given by
+ + −δt qk(t) = αk t + βk + O(e ) (2.1.17)
− − −δt qk(−t) = −αk t + βk + O(e ) (2.1.18) for t → +∞ with δ > 0.
Furthermore,
+ − X − βn−k+1 = βk + φjk(α ), (2.1.19) j6=k where
− − 2 log(αj − αk ) for j < k φjk(α) = − − 2 . (2.1.20) − log(αj − αk ) for j > k
Equation 2.1.19 is what we refer to as the phase shift: asymptotically, comparing the
+ − solution at −∞ and ∞, for λk, the difference is given by βk − βn−k+1. The quantity − − φjk represent the phase shift between two particls with velocities αj , αk at t = −∞. Moser [Mo75] interprets Equation 2.1.19 if the interactions take place two at a time, resulting in this overall scattering. 38
2.1.4 Explicit Solution of the Toda Lattice: Method of Factorisation
We now state, without proof, a method for solving the Toda lattice by the factorisation of matrices.
Theorem 2.5. (The Factorisation Theorem) To solve
d X = [X, π (X)],X(0) = X (2.1.21) dt − 0
X0t X0t X0t Factor e = Π−(e )Π+(e ), if possible. Then, the solution is given by
−1 X0t X0t X(t) = Π− (e )X0Π−(e ). (2.1.22)
We provide an example, in lieu of the proof, the latter of which is by direct com- putation. In fact, the main aspects of the proof are replicated in our later proof of Theorem 6.13. a 1 1 X0t Example 2.1. Let X0 = . Then, for t 6= − a , one has e = I + X0t −a2 −a 2 (since X0 = 0), hence
X t 1 + at t 1 0 1 + at t e 0 = = . (2.1.23) 2 −a2t 1 −ta 1 − at 1+at 1 0 1+at
The solution to the Toda lattice with initial condition X(0) = X0 is then given by
−1 a 1 0 a 1 1 0 1+at 1 X(t) = = . (2.1.24) −a2t 2 −a2t −a2 a 1+at 1 −a −a 1+at 1 (1+at)2 − 1+at 39
2.1.5 Geometry of the Solutions: Embeddings into the Flag Manifold Phase Space
In Example 2.1, the solution blows up at precisely the value(s) at which eX0t fails to have an LU-decomposition. We now present the method of [FH91], [EFH91] and
[EFS93], in which the Toda flows are mapped to the flag manifold G/B+, thus com- pactifying the flow and continuing the solution beyond its singularities. The key tool will be a theorem of Kostant [Ko78].
Definition 2.5. Let λ = (λ1, . . . , λn). Define the isospectral set of Hessenberg ma- trices with spectrum λ, denoted (ε + b)λ or Fλ, by
(ε + b−)λ = Fλ = {X ∈ ε + b− : σ(X) = λ}, (2.1.25) where σ(X) denotes the spectrum of X. We also define the subset of tridiagonal Hessenberg matrices with this spectrum:
Mλ = {X ∈ Fλ : X is tridiagonal}. (2.1.26)
Definition 2.6. Define two distinguished matrices in Fλ: λ1 1 .. λ2 . ελ = diag(λ) + ε = (2.1.27) .. . 1 λn and the companion matrix 0 1 . 0 .. .. cλ = . 1 , (2.1.28) 0 1 −c0 −c1 · · · −cn−2 −cn−1 40
n Q n Pn−1 i where (x − λi) = x + i=0 cix is the characteristic polynomial for λ. i=1
Remark 2.4. In sln, since the matrices are traceless, one has λ1 + ··· + λn = 0,
so that the bottom-right entry of cλ is zero. We provide the more general definition above for both convenience and for generalisability.
Theorem 2.6. [Ko78] For each X ∈ Fλ, there exists a unique lower unipotent L ∈
N−, such that −1 X = LcλL . (2.1.29)
The same statement holds (with a different L ∈ N−) when cλ is replaced by ελ.
Definition 2.7. The companion embedding is the map κλ : Fλ → G/B+ defined as −1 follows: for X ∈ Fλ, if X = LcλL , then
−1 κλ(X) = L mod B+. (2.1.30)
Proposition 2.7. [EFS93] Under the companion embedding, the Toda flow becomes linear, given by left multiplication by ecλt.
Proof. Suppose L0 is Kostant’s lower unipotent matrix for the initial condition X0, −1 i.e. X0 = L0cλL0 . Combining this with Equation 2.1.45, one finds that the Toda lattice solution is
−1 X0t −1 X0t X(t) = Π− (e )L0cλL0 Π−(e ). (2.1.31)
Therefore, by Kostant’s theorem, one has
−1 X0t κλ(X(t)) = L0 Π−(e ) mod B+. (2.1.32)
X0t We now compute Π−(e ) in terms of cλ:
−1 X0t L0cλL0 t cλt −1 cλt −1 Π−(e ) = Π−(e ) = Π−(L0e L0 ) = L0Π−(e L0 ). (2.1.33) 41
Thus,
cλt −1 cλt −1 κλ(X(t)) = Π−(e L0 ) mod B+ = e L0 mod B+. (2.1.34)
Returning to Example 2.1, we compute the companion embedding of the Toda flow: a 1 0 1 Example 2.2. For X0 = , one has cλ = and −a2 −a 0 0
a 1 1 0 0 1 1 0 = . (2.1.35) −a2 −a −a 1 0 0 a 1
Then, the companion embedding of the Toda flow is given by
1 t 1 0 1 + at t κλ(X(t)) = mod B+ = mod B+. (2.1.36) 0 1 a 1 a 1
Remark 2.5. The flows for the other integrals of Toda, the flows for the higher trace powers, are similarly linearised by mapping to the flag manifold.
2.1.6 Symes’s Discrete-Time Matrix Dynamics and the Discrete-Time Toda Lattice
In 1980, Symes [Sy80] proposed a new approach for Moser’s result (Theorem 2.4), based on matrix factorisations. In this section, the matrix factorisation is the so- called LU decomposition that arises from Gaussian elimination.
We inductively define a two-step discrete evolution on Hessenberg matrices. If at (discrete) time n we have a matrix X(n), we obtain X(n + 1) as follows: 42
1. Perform Gaussian elimination to factor X(n) = L(n)R(n), with L(n) lower unipotent and R(n) upper triangular.
2. Permute the factors to define X(n + 1) = R(n)L(n)
Remark 2.6. By construction, one has
X(n + 1) = R(n)L(n) = (L(n)−1X(n))L(n) = L(n)−1X(n)R(n). (2.1.37)
Thus, this discrete evolution is given by conjugating a matrix by its lower unipotent factor. Since the spectrum of a matrix is invariant under conjugation, it follows that the eigenvalues are constants of motion for this discrete evolution. Hence, if
X(n) ∈ (ε + b−)λ = Fλ, then so is X(n + 1).
Furthermore, if X(n) is tridiagonal, then one can show that L(n) is lower bi-diagonal with ones on its diagonal and R(n) is upper bi-diagonal with ones on its superdiag- onal. Therefore, the product X(n + 1) = R(n)L(n) is itself once again a tridiagonal
Hessenberg matrix. Thus, if X(n) ∈ Mλ, then so is X(n + 1).
Symes [Sy80] showed that this discrete evolution extends to a continuous evolution with Lax equation of the same form as the Toda lattice Lax equation (Equation
2.1.10), but with π−(X) replaced by π−(log X):
d X = [X, π (log X)]. (2.1.38) dt −
This is intimately connected to the work of Deift, Nanda and Tomei [DNT83] in which they describe a general framework associating to each real, injective function G(λ) on the spectrum of a a symmetric tridiagonal matrix X, a unique isospectral flow on the space of tridiagonal matrices convergent to a diagonal matrix as t → ±∞. When G(λ) = λ, this yields the classical Toda flow, and twhen G(λ) = log(λ), this 43 produces the so-called QR flow that is the symmetric analogue of Symes’s factorisa- tion evolution, with the factorisation given by the QR factorisation instead of the LU decomposition.
When working with Hessenberg matrices, Symes showed that his discrete LU evolu- tion extends to the continuous evolution on Hessenberg matrices with Lax equation 2.1.38.
To write the Symes discrete-time evolution out explicitly, let
t I1 1 1 . t . V t 1 I2 . 1 L(t) = , and R(t) = , (2.1.39) .. .. .. . 1 . . t t In Vn−1 1 then Symes’s discrete-time evolution produces what has come to be known as the finite discrete-time Toda lattice:
Definition 2.8. The finite discrete-time Toda lattice is the system
t+1 t t t+1 Ii = Ii + Vi − Vi−1 , i = 1, . . . , n t t t+1 Ii+1Vi Vi = t+1 , i = 1, . . . , n − 1 (2.1.40) Ii t t V0 = Vn = 0 which is expressible as
L(t + 1)R(t + 1) = R(t)L(t) (2.1.41)
This, in turn, extends to the (infinite) discrete-time Toda lattice:
Definition 2.9. The discrete-time Toda lattice, originally due to Hirota [Hi77][HT95], is the system 44
It+1 = It + V t − V t+1 i i i i−1 It V t (2.1.42) V t+1 = i+1 i i t+1 Ii for i, t ∈ Z. Alternatively, this can be represented as the bi-infinite matrix equation
L(t + 1)R(t + 1) = R(t)L(t) (2.1.43) where
X X t X t L(t) = (Ei,i + Vi Ei+1,i),R(t) = (Ei,i+1 + Ii Ei,i), (2.1.44) i∈Z i∈Z i∈Z
p q and Ei,j are the usual standard basis matrices, i.e. (Eij)pq = δi δj .
We conclude this coverage of the discrete-time Toda lattice with the discrete-time analogue of the Factorisation theorem (Theorem 2.5):
Lemma 2.8. [Su18] To solve the discrete-time Toda latitice with initial condition
t log X0 t t t X(0) = X0, factor e = X0 = Π−(X0)Π+(X0), if possible. Then, the solution is given by
−1 t t X(t) = Π− (X0)X0Π−(X0), (2.1.45) for all t ∈ N ∪ {0}. 45
2.2 Box-Ball Systems
A cellular automaton is a special type of discrete dynamical system with both discrete time steps and a discrete (in fact finite) number of states. Of particular interest is the box-ball system (BBS) which was introduced in 1990 by Takahashi and Satsuma [TS90]
The process of ultradiscretisation can be used to transform a discrete dynamical sys- tem with continuous variables into one with variables taking on a finite number of values. In [To04], it is shown that the box-ball system arises as a result of ultradis- cretisation of both the discrete KP equation and the discrete-time Toda lattice. In this chapter, we define the box-ball system and present a coordinatisation of box-ball systems and the equations these coordinates solve. We finish by following the process of ultradiscretisation of discrete-time Toda to yield the box-ball system equations. This section follows [To04] closely.
2.2.1 The Box-Ball Evolution
The (basic) box-ball system consists of a one-dimensional infinite array of boxes with a finite number of the boxes filled with balls, and no more than one ball in each box (single capacity). A simple evolution rule is provided for a box-ball system state:
(1) Take the left-most ball that has not been moved and move it to the left-most empty box to its right.
(2) Repeat (1) until all balls have been moved precisely once.
Since the algorithm requires one to know which balls have been moved, we can, with- out technically changing the algorithm, introduce a colour-coding based on whether balls have moved or not. Balls will be blue until they have moved, after which they will become red. When all balls are red, the colours should be reset to blue, ready 46 for the next time step. Or, equivalently, a 0-th step of colouring all balls blue should be prescribed. We will use the latter for a minor benefit in brevity. Below is an example of the evolution with this colour-coding, with each ball move separated into a sub-step:
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
Figure 2.1. A box-ball system time evolution (one time step).
Some authors use ones and zeroes to represent filled boxes and empty boxes, respec- tively. The numbered representation lends itself to the generalisation of the basic box-ball system to the advanced box-ball system, which we will define and utilise in AppendixA.
2.2.2 Soliton Behaviour and the Sorting Property
The box-ball system is sometimes referred to as the soliton cellular automaton. A chain of n consecutive balls travels with velocity n, so a chain (or soliton) of n parti- cles will have greater velocity than soliton of length m, if m < n. The solitons collide, and come out of the collision ordered with the longer chain ahead. There may be mixing phase, but the solitons come out ordered, with a phase shift. Here, by “phase shift”, we mean the difference between where the chain ends up after the collision and where the chain would have been if it were not for the collision. For now, we take this for granted, and provide a more detailed analysis shortly in Section 2.2.4, along 47 with a conjectured formula.
In the following figure, we demonstrate how the solitons become ordered after suffi- ciently many time evolutions. Once sorted, they travel with their respective velocities, never to collide again. ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
Figure 2.2. The sorting property of the box-ball system.
2.2.3 Invariants of the Box-Ball System
In [TTS96], a method is described for producing a pair of Young tableaux from a box- ball state via Dyck language, stackable permutations and the RSK correspondence. Not only are the shapes of the two Young tableaux the same, but this shape is also conserved under the box-ball dynamics. This invariant shape can be described in terms of balls and boxes, but it is slightly more convenient to represent the box-ball system as a sequence of 0’s (for empty boxes) and 1’s (for boxes with balls), which is made explicit in [To04]. The procedure goes as follows:
Let p1 be the number of 10’s in the sequence.
Eliminate all of these 10’s, and let p2 be the number of 10’s in the resulting sequence.
Repeat this process until no 10’s remain.
The sequence (p1, p2,...) is weakly decreasing, hence a partition of the number of balls. It can be represented by a Young diagram by taking the jth column to
have pj boxes. 48
For example, the associated shape to the sequence of box-ball states in Figure 2.2 is
Figure 2.3. The invariant shape of the box-ball system(s) in Figure 2.2
Remark 2.7. The row lengths then act as a signature for the system: if the columns
of the tableau are the pi’s, then the row lengths give the lengths of the solitons. One can see this heuristically by noting that as time goes to ±∞, the chains will be sufficiently separated by empty boxes so that each chain provides precisely one “10” for each particle comprising it. 2.2.4 The Box-Ball Phase Shift
We have seen how, as t → +∞, the blocks sort themselves by increasing length. The same holds in reverse time: as t → −∞, the blocks are ordered by decreasing lengths. The asymptotic sequence of lengths are revealed at any finite time using the invariant shape construction. The following example demonstrates why this is needed:
··· ··· ··· ··· ··· ··· ··· ··· ··· ···
Figure 2.4. A phase shift interaction between two colliding chains.
In the above example, we can discern the asymptotic ordering in the first, second, fourth and fifth rows, simply by counting the numbers of balls in each block of adja- cent balls. The middle row (the third) could be misleading, since it reveals a (2, 2) structure for the blocks. If two blocks are spaced far enough apart, then no such 49
obfuscation occurs.
Barring this intricacy (i.e. when there is enough space between adjacent blocks), one can take two blocks, evolve sufficiently many times according to the box-ball evolu- tion, and compare the position of the blocks to where they would have been if it had not have been for the collision.
In the figure below, we replicate Figure 2.2.4. However, we use green balls to keep track of where the block of three balls would have been without the collision, and magenta balls to keep track of where the block of one ball would have been.
··· ··· ··· ··· ··· ··· ··· ··· ··· ···
When the collision has concluded, we see that the three-block is two positions ahead of where it would have been, and the one-block is two positions behind where it would have been. Therefore, we say that the three-block experiences a +2 phase shift, and the one-block experiences a −2 phase shift.
We present here two conjectures on the phase shift formulæ for the box-ball system.
Conjecture 2.9. Take a box-ball system consististing of just two blocks of adjacent balls, subject to the following:
1. The left-most block has k balls.
2. The right-most block has l balls. 50
3. k > l
4. The two blocks are separated by at least l empty boxes.
After sufficiently many time steps of the box-ball evolution, after the blocks have collided and ordered themselves, the k-block will have experienced a phase shift of +2 min(k, l) = 2l, and the l-block will have experienced a phase shift of −2 min(k, l) = −2l.
This conjecture extends to the following conjectured formula for the phase shifts for box-ball systems with any number of blocks.
Conjecture 2.10. If one has a box-ball system with a total of n blocks, with Qk-many balls in the k-th block, and with blocks separated sufficiently so as to be able to identify
n the asymptotic soliton structure by simply ordering (Qk)k=1. After sufficiently many time-steps have passed (i.e. after the blocks have finished all collisions), the k-th block will have experienced a total phase shift of
X X 2 min(Qj,Qk) − 2 min(Qj,Qk). (2.2.1) j>k j
enced by the Qk-block as a result of colliding with slower chains in front, and the
second sum is the total (negative) phase shift experienced by the Qk-block as a result of colliding with faster chains initially behind it.
We draw the readers attention to the similarity between these conjectures and the phase shift formula for Section 2.1.3 for the Toda lattice: phase shifts propagate collision-by-collision. Whilst we defer the proof to our future direction, the next two sections help to shed some light on some of the analogies between the observed behaviour of the Toda lattice and the observed and conjectured behaviour of the box-ball system. 51
2.2.5 Coordinates on the Box-Ball System
t t t Suppose at time t, there are N sets of consecutive filled boxes. Let Q1, Q2, ..., QN t denote the lengths of these sets of filled boxes, taken from left to right. Let W1, t t W2, ..., WN−1 denote the lengths of the sets of empty boxes between the N sets of t t filled boxes, again taken from left to right. Lastly, let W0 and WN be formally defined to be ∞, reflecting the fact that the empty boxes continue infinitely in both directions.
The following theorem gives evolution equations for these coordinates. They can be found, for example, in [To04].
t t t t t Theorem 2.11. ([To04]) The coordinates (W0,Q1,W1,...,QN ,WN ) evolve under the box ball dynamics according to
t+1 t+1 W0 = WN = ∞ (2.2.2)
t+1 t t t+1 Wn = Qn+1 + Wn − Qn , n = 1,...,N − 1 (2.2.3) n n−1 ! t+1 t X t X t+1 Qn = min Wn, Qj − Qj , n = 1,...,N, (2.2.4) j=1 j=1
Example 2.3. Take the initial state in Figure 2.1:
··· ···
t t t t t t t t t W0 Q1 W1 Q2 W2 Q3 W3 Q4 W4 ··· ···
t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 t+1 W0 Q1 W1 Q2 W2 Q3 W3 Q4 W4
Figure 2.5. The box-ball coordinates on a box-ball system and its time evolution.
Under the time evolution, the coordinates evolve as
(∞, 3, 3, 1, 2, 2, 1, 1, ∞) 7→ (∞, 3, 1, 1, 3, 1, 1, 2, ∞) (2.2.5) 52
To distinguish between the box-ball evolution and the induced coordinate evolution, we will make the following definitions.
Definition 2.10. Let BBS be the set of states of the box-ball system (with at least
2n−1 one ball) and Bn = {∞} × N × {∞} the set of coordinates of BBS for a state with
n solitons. We define % : BBS → BBS to be the box-ball evolution and χn : Bn → Bn S to be the evolution on coordinates. We also define a map C : BBS → B := Bn n∈N taking a box-ball system state to its coordinates.
With this definition, we have an immediate corollary of Theorem 2.11.
Corollary 2.12. The following diagram commutes: % BBS BBS
C C χ B B
where χ : B → B is the map naturally induced by {χn : Bn → Bn}n∈N. 2.2.6 Ultradiscretisation of Discrete-Time Toda
Continuing to follow [To04], we perform the process of ultradiscretisation on the discrete-time Toda lattice to see that this results in the box-ball system. Later, in Chapter4, we present the full procedure for ultradiscretising geometry RSK. For this reason, we omit some of the detailed calculations in the ultradiscretisation of the discrete-time Toda lattice.
Recall the discrete-time Toda lattice:
t t V0 = Vn = 0 t+1 t t t+1 In = In + Vn − Vn−1 n = 1,...,N . (2.2.6) t+1 t+1 t t Vn In = In+1Vn n = 1,...,N − 1 53
It can be shown that the discrete-time Toda lattice is equivalent to the following system V t = V t = 0 0 n It ··· It It+1 = V t + n 1 n = 1,...,N . (2.2.7) n n It+1 ··· It+1 n−1 1 t+1 t+1 t t Vn In = In+1Vn n = 1,...,N − 1
Making the change of variables
1 t 1 t t − ε Qn(ε) t − ε Wn(ε) In = e ,Vn = e , (2.2.8) one obtains
t+1 t t t+1 Wn (ε) = Qn+1(ε) + Wn(ε) − Qn (ε) n = 1,...,N − 1 (2.2.9)
1 t − 1 Pn Qt (ε)−Pn−1 Qt+1(ε) t+1 − ε Wn(ε) ε ( j=1 j j=1 j ) Qn (ε) = −ε log e + e n = 1,...,N (2.2.10)
t t W0(ε) = WN (ε) = ∞. (2.2.11)
Finally, assuming the limits
t t t+1 t+1 Wn := lim Wn(ε),Wn := lim Wn (ε), ε→0+ ε→0+ (2.2.12) t t t+1 t+1 Qn := lim Qn(ε),Qn := lim Qn (ε), ε→0+ ε→0+ exist, one obtains
t+1 t t t+1 Wn = Qn+1 + Wn − Qn , n = 1,...,N − 1 (2.2.13) n n−1 ! t+1 t X t X t+1 Qn = min Wn, Qj − Qj , n = 1,...,N (2.2.14) j=1 j=1 t t W0 = WN = ∞ (2.2.15)
which are exactly the box-ball system equations. 54
2.3 The Robinson-Schensted-Knuth Correspondence
In this section, we provide the background and some basic motivation behind the Robinson-Schensted-Knuth correspondence and Schensted insertion. We begin with a review of some of the combinatorial objects of interest, the RSK equations describ- ing Schensted word insertion, and Kirillov’s geometric lifting of the (tropical) RSK equations to the geometric RSK (gRSK) equations. We will be following the papers [AD99] by Aldous and Diaconis and [NY04] by Noumi and Yamada, and the book [Ai07] by Aigner.
The coverage of this background is fairly detailed, and proofs and examples have been provided to aid in following the rather algorithmic constructions presented here. However, what is most pertinent to this dissertation are the equations for Schensted insertion (which are described in Corollary 2.20 or their equivalent presentation in Section 2.3.7) and the material thereafter (in Sections 2.3.7 to 2.3.9).
2.3.1 Young Tableaux
Definition 2.11. Let n ∈ N and let λ = (λ1, λ2, . . . , λk) ` n be a partition of n, i.e.
λi ∈ N for each i, and λ1 ≥ λ2 ≥ · · · ≥ λk. Associated to λ is the Young diagram (or
Ferrers diagram) of shape λ which is composed of λ1 boxes in the first row, λ2 boxes in the second row, ..., and λk boxes in the k-th row. The boxes are of equal size and aligned in a grid, justified to the left.
Example 2.4. The partition λ = (5, 3, 3, 2, 1), which partitions n = 14, has Young diagram: 55
Definition 2.12. A Young tableau is a Young diagram with its boxes labelled with n distinct numbers, where n is the number of boxes in the diagram, such that the numbers increase along the rows and down the columns. A standard tableau is a Young tableau in which the n distinct numbers are the numbers 1-through-n.
Example 2.5. For the example given above, here are some standard tableaux:
1 2 3 4 5 1 6 10 13 14 1 3 6 10 14 1 2 3 4 5 6 7 8 2 7 11 2 5 9 6 10 11 9 10 11 3 8 12 4 8 13 7 12 14 12 13 4 9 7 12 8 13 14 5 11 9
Definition 2.13. To each box (or cell) of a Young diagram, we can assign a quantity known as the hook length. The hook length of a box is the number of boxes (in its row) to the right of the box, plus the number of boxes (in its column) below the box, plus one (counting the box itself).
Example 2.6. To illustrate this, the diagram below has each of its cells filled in with their hook numbers (note that the diagram is shaded to avoid confusion with Young tableaux) 9 7 5 2 1 6 4 2 5 3 1 3 1 1 56
Before moving on, we make the following definition that generalises that of standard tableaux:
Definition 2.14. A semistandard tableau (SST) is a Young diagram filled with pos- itive integers that are weakly increasing along rows and strictly increasing down the columns. For ease of writing, let SST denote the set of semistandard tableaux.
Example 2.7. Below are three examples of semistandard tableaux:
1 3 3 5 6 6 8 1 1 1 4 4 4 6 7 2 2 2 5 7 7 3 3 3 9
2.3.2 The Length of the Longest Increasing Subsequence of a Permutation and Patience Sorting
Definition 2.15. The length of the longest increasing subsequence of a permutation
σ ∈ Sn, denoted `(σ), is defined to be the length of the longest increasing subsequence of the sequence (σ(1), σ(2), . . . , σ(n)). i.e. `(σ) is the longest increasing subsequence of the bottom row of σ when written in the 2-row permutation notation:
1 2 ··· n . σ(1) σ(2) ··· σ(n)
For ease of notation, we may write (σ(1), . . . , σ(n)) to represent σ ∈ Sn.
Example 2.8. The permutation σ = (1, 4, 7, 2, 3, 6, 8, 5, 9) has `(σ) = 6 corresponding to the subsequence (σ(1), σ(4), σ(5), σ(6), σ(7), σ(9)) = (1, 2, 3, 6, 8, 9).
Definition 2.16. Patience Sorting is an algorithm by which a sequence of distinct
numbers, say i1, . . . , in, can be sorted into ‘piles’: 57
(1) Set k = 1.
(2) Scan through the top number of each pile. If there are no piles, start a pile with
ik. Otherwise, either ik is larger than the top number of each pile or it is not. In
the case of the former, start a new pile to the right of all piles, and place ik in
this new pile. If ik is not larger than the top number of each pile, place ik on the
top of the left-most pile whose top number is larger than ik.
(3) If k < n, replace k 7→ k + 1 and return to (2). Otherwise, the algorithm ends.
Example 2.9. Return to the permutation σ = (1, 4, 7, 2, 3, 6, 8, 5, 9). Patience sorting yields the following piles: 2 3 5 1 4 7 6 8 9.
It is no coincidence that the number of piles is equal to the length of the longest increasing subsequence. Clearly, the number of piles is at least `(σ), since, if σ(j) > σ(i) for some j > i, then σ(j) must be put into a pile at least one to the right of the σ(i)’s pile. To see that `(σ) is achieved, we can construct a longest increasing sequence of the permutation by taking one number in each pile as the piles are being formed: whenever a number is placed in a pile other than the first pile, draw an arrow from that number to the top number of the pile immediately to the left of it. This results in a path from the last pile to the first, which picks out a longest increasing subsequence.
Example 2.10. Taking Example 2.9, and including the arrows, we obtain:
2 3 5
1 4 7 6 8 9
This gives the subsequence (1, 2, 3, 6, 8, 9). An arrow is placed at each step of the patience sorting algorithm: the order in which the arrows above were placed is given 58
by taking the permutation (1, 4, 7, 2, 3, 6, 8, 5, 9) without the first entry and taking the sequence of arrows with these numbers as their tails.
2.3.3 Schensted Insertion
In this section, we describe a procedure called Schensted insertion, which takes a permutation and incrementally grows the empty tableau into a standard tableau, passing through intermediated semistandard tableaux along the way. This procedure is essentially a more aesthetically pleasing version of patience sorting. In patience sorting, once a number was placed atop a pile, all numbers below it became irrelevant for future steps. Schensted insertion creates a first row out of the top numbers from patience sorting, and bumped numbers are then essentially patience sorted on the piles that result from stripping away the top numbers. We will now make this precise.
Suppose we have a (standard) Young tableau, T , and we want to insert a number a into T (where a is not a number that is already contained in T ). To do this, we perform a process called ‘bumping’, prescribing how a number is inserted into a row.
(1) To insert a number, a, into an empty row, we simply create a box in the row and place the number into it. If the row is not empty, move onto Step (2).
(2) If the row is not empty, search the row for the smallest number, m, which is larger than the number to be inserted, if there is such a number. Replace this number by a, and ‘bump’ (i.e. remove) m from the row.
(3) If no such number exists, create a new box at the end of the row and fill it with a.
The process of inserting a into a row of T either results in an extension of the row (with nothing being replaced) or a replacement, which ‘bumps’ a number out of the row into which the insertion was performed. 59
To generalise this row insertion to insertion into a Young tableau, we perform the following:
(1) Insert the number into the first row.
(2) If no numbers are bumped, then the insertion is complete.
(3) If a number is bumped, insert that number into the next row.
Notation 2.17. The result of inserting a into T is sometimes denoted T ← a.
This procedure clearly generalises to ‘words’. If one has a sequence of (distinct) num-
bers (a1, a2, . . . , ak) to insert into a tableau, T , not already containing the numbers,
simply insert the numbers recursively: ((((T ← a1) ← a2) ··· ) ← ak). We can also build Young tableaux from scratch by starting with the empty tableau, and a stan- dard tableau can be build by inserting the numbers 1-through-n in some particular order. Na¨ıvely, one might think it is sufficient to just build a standard tableau from the per- mutation (π(1), π(2), . . . , π(n)). But, what we will see is that two different permuta- tions may give rise to the same tableau. This next example serves to demonstrate the insertion procedure, as well as demonstrate why we need more than just one tableau.
Example 2.11. Take two permutations π = (4, 3, 1, 2) and σ = (1, 4, 3, 2). For each, we produce the sequence of (semi)standard tableaux:
4 3 1 1 2 π : ∅ , , , , . 4 3 3 4 4
1 1 4 1 3 1 2 σ : ∅ , , , , . 4 3 4 60
We see that the resulting tableaux are the same, but the transition through the shapes was different for the two permutations. Therefore, just knowing the tableau that one obtains from Schensted insertion on a permutation is not enough to recover the permutation uniquely. However, if one knows the shape transition and resulting tableau, then one can uniquely pin down the permutation. This is essentially the statement of the Robinson-Schensted correspondence, which we will now present and then prove.
2.3.4 The Robinson-Schensted Correspondence
Definition 2.18. For each λ ` n, let Dλ = {T : T is a standard tableau of shape λ} and dλ = |Dλ|.
Theorem 2.13. (The Robinson-Schensted Correspondence) There exists a one-to- S one correspondence between elements of Sn and Dλ × Dλ, i.e. the elements of Sn λ`n are in bijection with pairs of standard tableaux of the same shape λ ` n.
To prove Theorem 2.13, we need to define a procedure that extends Schensted inser- tion (of a word into a tableau) to a procedure that ultimately produces the same final tableau, but accompanied by another tableau that can be used to backtrack to the original permutation. That is, we will describe an algorithm which assigns a pair of standard tableaux (P,Q) to a permutation, and proceed to show that this algorithm has an inverse algorithm. The tableau P will be called the insertion tableau and Q will be called the recording tableau.
S First, we define the map Sn → Dλ × Dλ: λ`n
Fix a permutation σ ∈ Sn. As in the previous section, from (σ(1), σ(2), . . . , σ(n)) we build (n + 1) semistandard tableaux, say P0, P1, ..., Pn, where P0 = ∅ and
Pi = Pi−1 ← σ(i) for all 1 ≤ i ≤ n. Then Pσ := Pn is the (standard) Young tableau 61
associated to σ, as in the previous section.
Viewing σ as a 2-row array:
1 2 ··· n , σ(1) σ(2) ··· σ(n)
we can use the top row to build another standard tableau, but this will not be built
via the insertion algorithm. Instead, set Q0 = ∅, and define Qi to be the result of
adding a box to Qi−1 so that Qi and Pi have the same shape, and then insert i into
this new box. At the final stage, we have Qn, which we will denote by Qσ. S The map Sn → Dλ × Dλ is then defined as: λ`n [ Sn → Dλ × Dλ λ`n
σ 7→ (Pσ,Qσ).
S This map is invertible. To show this, we construct its inverse: Dλ × Dλ → Sn: λ`n
Given a pair of standard tableaux (P,Q) of the same shape λ ` n, we can obtain a
permutation σ(P,Q) ∈ Sn. This is achieved via the following algorithm:
(1) Identify the box in Q with the largest number, m. Delete this box from Q, and delete the corresponding box (positionally-speaking, that is) in P , making a note of the number, e, that was in this box. If Q was empty, then we are done.
(2) If e was in the first row, set σ(P,Q)(m) = e, and go back to Step (1). Otherwise, go onto Step (3).
(3) Search in the row above e’s row for the largest number, f, that is smaller than e. Replace f by e, and repeat Step (2) with f taking the role of e.
We will omit the proof that these maps are inverse to each other in favour of an example, which should be more enlightening in demonstrating two algorithms. 62
Example 2.12. Start with
1 2 6 7 1 4 6 7 P = ,Q = . 3 5 2 8 4 3 8 5
1 2 6 7 1 4 6 7 1 5 6 7 1 4 6 7 (1) (2) σ(P,Q)(8) = 2 3 5 2 8 3 2 4 3 4 3 8 5 8 5
1 5 6 1 4 6 1 5 1 4 (3) σ(P,Q)(7) = 7 (4) σ(P,Q)(6) = 6 3 2 3 2 4 3 4 3 8 5 8 5
3 5 1 4 3 1 (5) σ(P,Q)(5) = 1 (6) σ(P,Q)(4) = 5 4 2 4 2 8 3 8 3
4 1 8 1 (7) σ(P,Q)(3) = 3 (8) σ(P,Q)(2) = 4 8 2
1 2 3 4 5 6 7 8 leaving σ (1) = 8. Thus, the pair (P,Q) maps to . (P,Q) 8 4 3 5 1 6 7 2
1 2 3 4 5 6 7 8 Now, to go in the other direction, start with σ = , and set 8 4 3 5 1 6 7 2 (P0,Q0) = (∅, ∅). Then, we obtain the following sequence of pairs of Young tableaux, recalling that the sequence of Q tableaux is formed by mimicking the growing shape of the P -tableaux, and inserting successive numbers from the top row of σ into the newly formed boxes: 63
8 1 4 1 (P1,Q1): (P2,Q2): 8 2
3 1 3 5 1 4 (P3,Q3): (P4,Q4): 4 2 4 2 8 3 8 3
1 5 1 4 1 5 6 1 4 6 (P5,Q5): (P6,Q6): 3 2 3 2 4 3 4 3 8 5 8 5
1 5 6 7 1 4 6 7 1 2 6 7 1 4 6 7 (P7,Q7): (P8,Q8): 3 2 3 5 2 8 4 3 4 3 8 5 8 5
Which is the original pair of standard tableaux.
Theorem 2.14. ([NY04]) Given a permutation σ ∈ Sn, `(σ) is equal to the number
of columns in Pσ (or, equivalently, in Qσ).
Proof. This fact follows from the observation in Section 2.3.2 that patience sorting a sequence σ(1), σ(2), . . . , σ(n) yields `(σ) piles, paired with the fact that the first row of the tableau Pi contains precisely the top number of each pile in the patience sorting algorithm.
2.3.5 Semistandard Young Tableaux and the Robinson-Schensted-Knuth Correspondence
We have defined both (standard) Young tableaux and semistandard tableaux, yet the Robinson-Schensted correspondence only involves pairs of standard Young tableaux of the same shape. A natural question is the question of what happens to the Robinson- Schensted correspondence when the pairs of tableaux are allowed be semistandard. The answer is given by the celebrated Robinson-Schensted-Knuth correspondence. 64
Recall that permutations in Sn are in one-to-one correspondence with n × n permu- tation matrices. One can embed all such matrices, for all n, as semi-infinite matrices by extending by zeroes. The image is a subset of the set of non-negative arrays which are semi-infinite in two directions with entries zero for all but finitely many indices:
A := {A = (aij)1≤i,j<∞ : aij ∈ N0 ∀i, j and aij = 0 for all but finitely many i, j}.
I.e. if σ ∈ Sn, then σ ↔ Aσ, where 1 if 1 ≤ i ≤ n and j = σ(i) (A ) = . σ ij 0 otherwise
Lemma 2.15. ([NY04]) The elements of A are in one-to-one correspondence with ‘generalised permutations’
i i ··· i w = 1 2 m j1 j2 ··· jm
where ik ≤ ik+1 for all 1 ≤ k ≤ m − 1, and the j’s are weakly increasing in their i-blocks (i.e. if ik = ik+1, then jk ≤ jk+1).
Proof. Given w as in the statement of the lemma, let aij be equal to the number of i times the column appears in w. j i Conversely, given A ∈ A, arrange a copies of the column in a 2-by-||A|| ij j 1,1 array as prescribed by the conditions described above.
Theorem 2.16. ([NY04]) (The Robinson-Schensted-Knuth Correspondence) The set of non-negative integer matrices is in bijection with the set of pairs of semistandard Young tableaux of the same shape.
By identifying A with the set of generalised permutations, the previous algorithms establish the correspondence. We just need to make a few notes on how to generalise the procedures to generalised permutations and semistandard tableaux. 65
Generalised Permutations to Semistandard Tableaux:
The algorithm is essentially the same as before, with a couple of additional comments to be made on how row insertion works. In the following, we are inserting a:
(1) To insert the number into an empty row, we just create a box in the row and place the number into it. If the row is not empty, move onto Step (2).
(2) If the row is not empty, search the row for the smallest number which is strictly larger than the number to be inserted, if it exists, and call the found number m. If more than one such number exists, take m to be the left-most such number. Replace m by a, and ‘bump’ (i.e. remove) m from the row.
(3) If no such number exists, create a new box at the end of the row and fill it with a.
To insert a word into a (possibly empty) semistandard tableau, we cascade in the same manner as before. Insert the first ‘letter’ (as prescribed above). If a letter is bumped, then insert that letter into the next row. Otherwise, move onto the next letter in the word. Continue in this manner until no letters remain.
Lemma 2.17. ([NY04]) Word insertion, when performed on a semistandard tableau, yields another semistandard tableaux.
Proof. We check this by considering the case when the word is just one letter/number, w = a. The result will then follow by induction.
We need to check three things: we need insertion T ← a to have (weakly) increasing row lengths, to be (weakly) increasing along the rows, and to be (strictly) increasing down the columns.
(1) To fail to be weakly increasing in row lengths, one would have to encounter the situation of two adjacent rows having the same row lengths, with a bump number 66
from the first being appended to the second row. Thus, we need not consider the
full tableau, just two rows, R1 and R2 of the same length.
Suppose a bumps w from R1. Then, this implies that w > a, since w is the
smallest number in R1 which is strictly larger than a. Now, for w to be appended 0 0 to R2, one would have to have w ≥ w for all w in R2. However, consider the 00 number, w in R2, below the original position of w in R1. Since the tableau was semistandard before, we have w00 > w. But, this then gives w00 > w > a, which
tells us that w cannot possibly be appended to R2. Thus, row insertion must preserve the property of the row lengths being weakly decreasing.
(2) The fact that the weakly increasing property of rows is preserved by word insertion can be checked at the level of rows, since, if it is shown that insertion of a into a row, R, results in a weakly increasing row, then the result will follow inductively.
There are two possibilities: either a bumps a number in R, or a is appended to R. The latter case is trivial; a is only appended to R if a is greater than or equal to every number in R. Since R started off being weakly increasing, the result follows immediately.
Now, suppose a does bump a number, w, from R. Since w > a, we have no violations to the right of the bumped position. If w0 the number to the left of the bump, then w0 ≤ w. Since w was the smallest (and left-most) entry that was larger than a, we must have w0 ≤ a, which confirms that there are no issues to the left of the bump.
Therefore, we have that row insertion preserves the weakly increasing property of rows.
(3) Similarly, to see the preservation of column-strictness under word insertion, we only need to consider two rows again. The only thing to consider is whether a 67
row insertion can affect the relation of a row to the row above. The reason for this is that, if a bumps w, we have a < w, thus, if w0 was below w previously, then a < w < w0 gives us a < w0. If a is simply appended to a row, then, by virtue of the row lengths previously being weakly increasing, there cannot be anything below the new box, so there is nothing to check.
Now we turn our attention to the relation between a row and the row above. The key fact is that a bump must go to the left (not strictly) of the box below the
bump from the previous row. If we have two rows, R1 and R2, with R1 above R2,
then consider R1 ← a. If a bump occurs, then a bumps some number w. If w had no box below it, then this fact is trivially true. However, if there were a box below w, containing w0, then we have w0 > w, which means, when it comes to
0 inserting w into R2, it cannot go to the right of w (since we bump the left-most instance of the smallest number greater than w). Now, by (2), we know that a is
greater than or equal to everything to its left (in R1), and we know that a < w (again, since a bumped w), and so we have that w is inserted beneath a number strictly lesser than it.
Example 2.13. Take the matrix
0 1 0 0 1 A = (aij) = 0 0 2 0 0 . 1 0 0 0 0
The index pairs (i, j), counted with multiplicity aij, are
(2, 1), (1, 5), (2, 3), (2, 3), (3, 1).
The corresponding generalised permutation is then
1 1 2 2 3 w = . 2 5 3 3 1 68
Applying the RSK correspondence, one obtains the following sequences of semistan- dard tableaux, resulting in the pair (P,Q) of semistandard tableaux corresponding to A (or w).
2 2 5 2 3 2 3 3 1 3 3 = P 5 5 2 5
1 1 1 1 1 1 1 2 1 1 2 = Q. 2 2 2 3
Semistandard Tableaux to Generalised Permutations:
Given a pair of semistandard tableaux, (P,Q), of the same shape λ ` n, we obtain a generalised permutations via the following algorithm:
(1) Identify the box in Q with the largest number, m. If more than one box contains the largest number, pick the right-most instance of it. Delete this box from Q, and delete the corresponding box (positionally-speaking, that is) in P , making a note of the number, e, that was in this box. If Q was empty, then go to Step (4).
m (2) If e was in the first row, form the vector , and go back to Step (1). e Otherwise, go onto Step (3).
(3) Search in the row above e’s row for the largest number, f, that is strictly smaller than e. If more than one instance occurs, pick the right-most one. Replace f by e, and repeat Step (2) with f taking the role of e.
(4) Form all of the vectors into a 2 × n-array, according to the conditions imposed on generalised permutations. 69
Example 2.14. Take
1 1 2 4 2 2 3 3 P = and Q = 2 3 3 4 5 6 4 5
Which leads to the following series of pairs of tableaux and vectors:
1 1 2 4 2 2 3 3 1 1 3 4 2 2 3 3 6 , → , , 2 3 3 4 5 6 2 3 4 5 2 4 5 4 5 1 3 3 4 2 2 3 3 5 → , , 2 4 1 4 5 2 3 3 4 2 2 3 3 5 → , , 4 4 1 2 3 4 4 2 2 3 3 4 → , , 3 3 → 2 3 4 , 2 2 3 , 4 3 → 2 3 , 2 2 , 4 2 2 2 → , , 3 2 → ∅ , ∅ , . 2
Thus, this pair of semistandard tableaux yields the following generalised permutation:
2 2 3 3 4 5 5 6 . 2 3 4 4 3 1 1 2
2.3.6 The RSK Equations for Schensted Insertion
We now turn our attention back to Schensted insertion. We consider insertion of a word v into a tableau T , which produces a new tableau T 0. We will deal with the 70
a1 a2 an case of v being weakly increasing, i.e. v = 1 2 ··· n , where ai ∈ N0 for each 0 i ∈ [n] := {1, 2, . . . , n}. Thus, T is obtained from T by inserting a1 lots of 1 into
T , then a2 lots of 2, ..., finishing off by inserting an lots of n . The result is a new semi-standard tableau T 0 = T ← v.
Since the rows of a semistandard tableau are weakly increasing, we can identify T ∈
Dλ, where λ = (λ1, . . . , λm) ` n and m ≤ n, with a sequence of weakly-increasing
i i i xi xi+1 xn m i words (wi = i (i + 1) ··· n )i=1. Here, xj is the number of times j appears in Pn i row i of T . Thus, j=i xj = λi.
Note 2.8. Since semistandard tableaux are strictly increasing down their columns, this identification really does capture all of the information of T .
Example 2.15. Inserting the word v = 11203241 = 113241 into the tableau
1 1 2 4 T = 2 3 3 4 4 71 yields
T 0 = ((((T ← 1) ← 3) ← 3) ← 4) 1 1 2 4 = ← 1 ← 3 ← 3 ← 4 2 3 3 4 4 1 1 1 4 = ← 3 ← 3 ← 4 2 2 3 3 4 4 1 1 1 3 = ← 3 ← 4 2 2 3 4 3 4 4 1 1 1 3 3 = ← 4 2 2 3 4 3 4 4 1 1 1 3 3 4 = 2 2 3 4 3 4 4
The procedure T ← v yields a semistandard tableau T 0. If we can identify T 0 with 72
i i i 0 0 yi yi+1 yn m 0 its sequence of row words (wi = i (i + 1) ··· n )i=1, then finding T in general (given T and v) amounts to finding how the y’s arise from the x’s and a’s. Since word insertion is performed iteratively on each of the rows of the tableau, it suffices to understand the exponents at the level of rows.
Notation 2.19. We will use labelled arrows to denote word insertion. For word insertion into a tableau T 0 = T ← v, we will write T −−→v T 0. For word insertion into v a row, r, of a tableau, we write 0r r0 where r0 is the row after the word v has v0 been inserted and v0 is the word that has been bumped out of r.
v Once 0r r0 is understood, T −−→v T 0 is understood as v0
v v 0 1 0 0 2 0 w1 w1 , w2 w2 , ··· v2 v3
0 where wi are the words associated to the rows of T , wi are the words associated to 0 the rows of T , and v1 = v.
Explicit Formulæ for Dynamics:
Write w = 1x1 2x2 ··· nxn for the row into which we wish to insert the word v =
1a1 2a2 ··· nan , write w0 = 1y1 2y2 ··· nyn for the result, and write v0 = 1b1 2b2 ··· nbn for the word bumped by the process of the insertion w ← v. Thus, we want to know how
(y1, . . . , yn) and (b1, . . . , bn) arise from (x1, . . . , xn) and (a1, . . . , an).
In order to simplify the calculations, we introduce new variables by taking partial sums:
ξj = x1 + ··· + xj, ηj = y1 + ··· + yj (2.3.1)
for j = 1, . . . , n. These will serve to reformulate the information in a “max-plus” form (Lemma 2.18), which will be crucial for detropicalisation. 73
The y’s can then be recovered from the η’s as y1 = η1 and yj = ηj − ηj−1 for j > 1.
Clearly, the b’s can be obtained once the y’s are known. For example, bj represents the number of j’s bumped from w. Since w started with xj lots of j’s, and we introduced aj lots of j’s, and ended up with yj lots j’s, the number of bumped j’s must equal bj = xj + aj − yj = aj + ξj − ξj−1 − ηj + ηj−1 for j > 1. For j = 1, note that 1 cannot
be bumped, so y1 = x1 + a1, so that b1 = 0 always.
Lemma 2.18. ([NY04]) When v = ka and w = 1x1 2x2 ··· nxn , we get
ξj if j < k ηj = ξk + a if j = k . (2.3.2) max{ηk, ξj} if j > k
Proof. If ξj > ηk, then some of the j’s ‘survived’ the bumping. This means the bumping did not get past the last j, and so the number of boxes with numbers ≤ j is still given by ξj, hence ηj = ξj. If ξj = ηk, then the bumping got to the very last j, which still gives the same conclusion of ηj = ξj. However, if ξj < ηk, then this means that the bumping eradicated all instances of j (hence, by the nature of the insertion algorithm, no numbers k < l ≤ j remain) and so ηj = ηk, hence ηj = max{ηk, ξj} for all j > k.
Corollary 2.19. ([NY04]) Inserting v = 1a1 2a2 ··· nan into w = 1x1 2x2 ··· nxn , one has
ηj = max {x1 + ··· + xk + ak + ··· + aj} (2.3.3) 1≤k≤j for all j.
Proof. We apply Lemma 2.18 recursively:
1a1 (ξ1, ξ2, ξ3, . . . , ξj,...)−−−→ (ξ1 + a1 = η1, max{η1, ξ2}, max{η1, ξ3},..., max{η1, ξj},...)
2a2 −−−→ (η1, max{η1, ξ2} + a2 = η2, max{η1, η2, ξ3},..., max{η1, η2, ξj},...)
3a3 −−−→ (η1, η2, max{η1, η2, ξ3} + a3,..., max{η1, η2, η3, ξj},...). 74
Thus, ηj = max{η1, η2, . . . , ηj−1, ξj} + aj for all j > 1 and η1 = ξ1 + a1.
Since the η’s are weakly-increasing, we have
ηj = max{ηj−1, ξj} + aj = max{ηj−1 + aj, ξj + aj} (2.3.4) for all j > 1.
Unpacking this (to remove the ηl’s, for l < j), we get
ηj = max{max{ηj−2 + aj−1, ξj−1 + aj−1} + aj, ξj + aj} (2.3.5)
= max{ηj−2 + aj−1 + aj, ξj−1 + aj−1 + aj, ξj + aj} (2.3.6)
= ··· = max {ξk + ak + ak+1 + ... + aj} (2.3.7) 1≤k≤j
= max {x1 + ··· + xk + ak + ... + aj} (2.3.8) 1≤k≤j which also covers j = 1.
To summarise, we now have found
x1 + a1 if j = 1 yj = max1≤k≤j{x1 + ··· + xk + ak + ··· + aj} − max1≤k≤j−1{x1 + ··· + xk + ak + ··· + aj−1} if j > 1 and bj = xj + aj − yj for all j.
However, for the purpose of convenient calculation, dividing the computation into phases, using Equation 2.3.8, one has
Corollary 2.20. ([NY04]) Given input coordinates (a1, . . . , an) and (x1, . . . , xn), one obtains the output coordinates (b1, . . . , bn) and (y1, . . . , yn) as follows:
1. compute ξj = x1 + ··· + xj for j = 1, . . . , n,
2. compute ηj = max{ηj−1, ξj} + aj recursively for j = 1, . . . , n, initialising with
η1 = ξ1 + a1, 75
3. the y-coordinates are obtained by taking y1 = η1 and yj = ηj − ηj−1 for j = 2, . . . , n,
4. the b-coordinates are obtained by taking b1 = 0 and bj = aj + xj − yj for j = 2, . . . , n.
Example 2.16. Take w = 1223314052 = 12233152 and v = 11224151, i.e. x = (2, 3, 1, 0, 2) and a = (1, 2, 0, 1, 1), where emboldened letters denote the vectors of the corresponding variables. According to Equation 2.3.8, we should have
η1 = 3, η2 = max{5, 7} = 7, η3 = max{5, 7, 6} = 7,
η4 = max{6, 8, 7, 7} = 8, η5 = max{7, 9, 8, 8, 9} = 9.
Thus,
y1 = 3, y2 = 7 − 3 = 4, y3 = 7 − 7 = 0, y4 = 8 − 7 = 1, y5 = 9 − 8 = 1.
So,
b1 = 1 + 2 − 3 = 0, b2 = 2 + 3 − 4 = 1, b3 = 0 + 1 − 0 = 1,
b4 = 1 + 0 − 1 = 0, b5 = 1 + 2 − 1 = 2.
To check this, let us perform the word insertion:
1 1 2 2 2 3 5 5 ← 11224151 = 1 1 1 2 2 3 5 5 ← 224151 2 1 1 1 2 2 2 5 5 = ← 214151 2 3 1 1 1 2 2 2 2 5 = ← 4151 2 3 5 1 1 1 2 2 2 2 4 = ← 51 2 3 5 5
= 1 1 1 2 2 2 2 4 5 2 3 5 5 76
So, the word w0 = 13244151 is left, and v0 = 213152 is bumped. This gives y = (3, 4, 0, 1, 1) and b = (0, 1, 1, 0, 2) agreeing with the results of the formulæ.
2.3.7 Kirillov’s Geometric Lifting: gRSK
The formulæ in the previous section involve only the operations max and addition, hence the formulæ live in the tropical max-plus algebra. We will make the change of operations:
(max, +) → (+, ·) to the formulæ in the Corollary 2.20, making the necessary algebraic analogue for the ‘additive’ identities (0 → 1) to go from
ξj = x1 + ··· + xj ∀ j = 1, . . . , n
η1 = ξ1 + a1
ηj = max{ηj−1, ξj} + aj ∀ j = 2, . . . , n and
y1 = η1
yj = ηj − ηj−1 ∀ j = 2, . . . , n
b1 = 0
bj = aj + xj − yj = aj + ξj − ξj−1 − ηj + ηj−1 ∀ j = 2, . . . , n. to the (de)tropicalised analogue:
ξj = x1 ··· xj ∀ j = 1, . . . , n
η1 = ξ1a1
ηj = (ηj−1 + ξj)aj ∀ j = 2, . . . , n 77
and
y1 = η1
ηj yj = ∀ j = 2, . . . , n ηj−1
b1 = 1
xj ξjηj−1 bj = aj = aj ∀ j = 2, . . . , n. yj ξj−1ηj
Lemma 2.21. ([NY04]) Returning to the original variables of
x1, . . . , xn, a1, . . . , an, y1, . . . , yn, b1, . . . , bn,
the above formulæ reduce to the following system (x, a) 7→ (y, b):
b1 = 1 a1x1 = y1 ajxj = yjbj ∀ j = 2, . . . , n (2.3.9) 1 1 1 + = a1 x2 b2 1 1 1 1 + = + ∀ j = 2, . . . , n. aj xj+1 yj bj+1
Proof. The first two formulæ are by virtue of η1 = y1 and ξ1 = x1. For the other formulæ, take η1 = ξ1a1 and ηj = (ηj−1 + ξj)aj and rearrange to get
η 1 = 1 (∗) ξ1a1
ηj − ηj−1aj = 1 ∀ j = 2, . . . , n (∗j) ξjaj 78
Equating (∗) and (∗2) yields
η − η a η 2 1 2 = 1 ξ2a2 ξ1a1 y 1 1 ⇒ 2 − = x2a2 x2 a1 1 1 1 ⇒ − = b2 x2 a1 resulting in the third formula.
Equating (∗j+1) and (∗j) for j = 2, . . . , n yields
η − η a η − η a j+1 j j+1 = j j−1 j ξj+1aj+1 ξjaj η y 1 η 1 1 ⇒ j j+1 − = j − ξj xj+1aj+1 xj+1 ξj aj yj y 1 1 1 ⇒ j+1 + = + xj+1aj+1 yj aj xj+1 1 1 1 1 ⇒ + = + . bj+1 yj aj xj+1
2.3.8 A Matrix Representation of the Geometric RSK
Returning to the system of equations presented in Lemma 2.21, and letting bars
1 denote reciprocals (i.e.x ¯ := x ), one obtains the following equations:
a¯1x¯1 =y ¯1 ¯ a¯jx¯j =y ¯jbj ∀ j = 2, . . . , n ¯ a¯1 +x ¯2 = b2 ¯ a¯j +x ¯j+1 =y ¯j + bj+1 ∀ j = 2, . . . , n. 79
This can be represented in the following form:
a¯1 1 x¯1 1 y¯1 1 1 0 a¯ 1 0 x¯ 1 0 y¯ 1 0 ¯b 1 0 2 2 2 2 . . . . . . . . .. .. .. .. = .. .. .. .. . ¯ a¯n−1 1 x¯n−1 1 y¯n−1 1 bn−1 1 ¯ a¯n x¯n y¯n bn 0 0 0 0 (2.3.10)
As we stated before, the full Schensted insertion procedure can be reduced to single word insertions. Noumi and Yamada [NY04] provide a very explicit generalisation of Equation 2.3.10 to the problem of describing the output insertion tableau from the RSK correspondence. i.e. the tableau obtained by inserting a sequence of words into the empty tableau.
Let w1, w2, . . . , wm be a sequence of weakly increasing words, and suppose each is xi xi coordinatised as wi = 1 1 ··· n n for i = 1, . . . , m. As Schensted insertion prescribes:
1 1,1 1. One inserts w1 (coordinatised by x =: x ) into the empty word, which pro- duces a new first row, which we will say is coordinatised by y1,1 (the first row 1). Nothing is bumped.
2. One then inserts x2 =: x2,1 into the new first row (y1,1) which produces a new first row (its second form), which we denote by y2,1. A word may be bumped, and we call it x2,2.
Continuing on in this manner, keeping the indexing going, one can capture all of the above concisely in the following diagram: 80
x1 = x1,1 x2 = x2,1 x3 = x3,1
∅ y1,1 y2,1 ···
∅ x2,2 x3,2
∅ y2,2 ···
∅ x3,3
∅ ···
∅ . ..
Figure 2.6. Iterative Schensted word insertions building the RSK correspondence.
Definition 2.20. For x ∈ Rm, 1 6= m ≤ n, define
E(x) = diag(x) + εn (2.3.11)
where εn is as in Definition 2.3.
If x = (1, 1,..., 1, xk, xk+1, . . . , xn), then define
I 0 E (x) = k−1 (2.3.12) k 0 E(x0)
0 0 0 where x = (xk, xk+1, . . . , xn) and E is modified for the size of x , i.e. E(x ) = diag(xk, . . . , xn) + εn−k+1.
In [NY04], they prove a result that is equivalent to the following:
Theorem 2.22. ([NY04]) The geometric analogue of Figure 2.6, one obtains yk,m 81 for 1 ≤ m ≤ k ≤ n by solving the following factorisation problems recursively:
1 1,1 E(x¯ ) = E1(y¯ )
2 1 2,1 1,1 E(x¯ )E(x¯ ) = E(x¯ )E1(y¯ )
2,1 2,2 = E1(y¯ )E2(y¯ )
3 2 1 3,1 2,1 2,2 E(x¯ )E(x¯ )E(x¯ ) = E(x¯ )E1(y¯ )E2(y¯ )
3,1 3,2 2,2 = E1(y¯ )E2(x¯ )E2(y¯ )
3,1 3,2 3,3 = E1(y¯ )E2(y¯ )E3(y¯ ).
In general, at the k-th stage, one has
k k−1 2 1 k,1 k,2 k,k E(x )E(x ··· E(x )E(x )) = E1(y )E2(y ) ··· Ek(y ).
Remark 2.9. Noumi and Yamada’s proof of Theorem 2.22 in [NY04] is based on detailed path switching arguments that go by the name of Lindstr¨om-Gessel-Viennot formulæ [GV85], which are in turn based on the characterisation of totally positive matrices in terms of weighted path counting matrices. Although Theorem 2.22 is pre- sented as a factorisation into increasingly many matrices, each step involves only the permutation and factorisation of a pair of consecutive matrices.This suggests the idea that this full geometric RSK is built from a tower of sequential geometric Schensted insertions which are described by equations of the type seen in 2.3.10.
2.3.9 Noumi and Yamada’s Observation
In [NY04], the following observation was made:
Starting with the discrete-time Toda lattice (Definition 2.9):
t+1 t t t+1 Ii = Ii + Vi − Vi−1 t t (2.3.13) t+1 Ii+1Vi Vi = t+1 Ii 82
and performing the change of variables
t −1 t −1 t+1 −1 t+1 −1 ai = (Ii+1) , xi = (Vi ) , yi = (Vi ) , bi = (Ii ) , (2.3.14)
the discrete-time Toda lattice transforms into the following
1 1 1 1 aixi = yibi, + = + , i ∈ Z. (2.3.15) ai xi+1 yi bi+1 which bears a striking resemblance to the Equations 2.3.9. 83
Chapter 3 Geometric RSK and Toda: The Discrete Picture
In [NY04], Noumi and Yamada made a connection between geometric RSK and the discrete-time Toda lattice:
t+1 t+1 t t t+1 t+1 t t Ii Vi = Ii+1Vi ,Ii + Vi−1 = Ii + Vi (3.0.1)
for i, t ∈ Z. By setting
t −1 t −1 t+1 −1 t+1 −1 ai = (Ii+1) , xi = (Vi ) , yi = (Vi ) , bi = (Ii ) , (3.0.2)
in Equations 3.0.1, one obtains
1 1 1 1 aixi = yibi, + = + (3.0.3) ai xi+1 yi bi+1 for i ∈ Z.
Inspired by this observation, we proceed to explore the connections and direct conse- quences in this chapter. First, we delve into the bi-infinite discrete-time Toda lattice, exploring its solutions, and then we present a solution to the problem of preserving this connection under the restriction to the finite-dimensional setting, which is the natural home of the original geometric RSK correspondence.
3.1 The Factorisation Problem
In this section, we present the factorisation problem that underlies solving the discrete- time Toda lattice, first in the bi-infinite case, then in the semi-infinite case. 84
Definition 3.1. For a sequence x = (xi)i∈Z, define two operators, L(x) and U(x), to have the following matrix representations: X X L(x) := (Eii + xiEi+1,i), U(x) := (xiEii + Ei,i+1). (3.1.1) i∈Z i∈Z
Given input sequences a = (ai)i∈Z, x = (xi)i∈Z, the problem of finding sequences b = (bi)i∈Z and y = (yi)i∈Z such that
L(y)U(b) = U(a)L(x), (3.1.2)
t t t+1 t+1 when ai = Ii , xi = Vi , b = Ii and yi = Vi is the problem of solving the discrete- time Toda lattice for one time-step.
More generally speaking, this is the problem of factoring a bi-infinite, tridiagonal matrix, with 1’s on the super-diagonal (a tridiagonal Hessenberg operator) X H = (Ei,i+1 + giEii + hiEi+1,i) i∈Z into a product of a lower bidiagonal operator and an upper bidiagonal operator.
In general, this factorisation problem does not have a unique solution. In the next two sections (Section 3.2 and Section 3.3), we present two descriptions of the solution set to this problem: the first is a parametric description of the solution set and the second is a generalisation of a result developed by Murphy [Mu18] for one-dimensional discrete Schr¨odingeroperators.
3.2 Parametrised Factorisations
With the factorisation problem now established, we work our way to a new result on the LU-decomposition of bi-infinite Hessenberg operators by leveraging the simple formulation for semi-infinite Hessenberg operators. The first result of this section is a description of the semi-infinite LU-decomposition in terms of τ-functions. The following theorem provides a parametrisation of the LU-decompositions in the bi- infinite case. Ultimately, since the semi-infinite solution to the factorisation problem 85 truncates to the solution to the finite factorisation problem, this provides full coverage of the three cases (finite, semi-infinite and bi-infinite) of interest for the discrete-time Toda lattice.
Definition 3.2. For S a semi-infinite (half-line) operator and k ∈ N, define Sk to be the principal k × k submatrix of S, viewing S as a semi-infinite matrix (i.e. a matrix
(sij){(i,j)∈N: i≥a,j≥b}), and define τk(S) = det(Sk).
To state the next lemma, let us extend the definitions of L and U to both semi-infinite sequences and finite sequences.
Definition 3.3. Let x = (x1, x2,...) be a sequence, then
X X L(x) := (Eii + xiEi+1,i), U(x) := (xiEii + Ei,i+1). i∈N i∈N
If x = (x1, . . . , xm) is a finite sequence, then
m m X X L(x) := Im+1 + xiEi+1,i, U(x) := εm + xiEi,i, i=1 i=1 where Im+1 is the (m + 1) × (m + 1) identity matrix and εm is the m × m matrix with 1’s on its superdiagonal and 0’s elsewhere.
In the semi-infinite setting, the factorisation solution is unique for generic tridiagonal Hessenberg operators, if it exists. The following lemma does seem to be known in the literature, but for completeness, we include a proof here.
Lemma 3.1. Let H = P (E + g E + h E ), with h 6= 0 for all i, then i∈N i,i+1 i ii i i+1,i i either there exists unique sequences y = (y1, y2,...) and b = (b1, b2,...) such that
L(y)U(b) = H or no solution exists. Furthermore, if a solution exists, it is given by 86
τi(H) bi = i = 1, 2,... (3.2.1) τi−1(H)
hiτi−1(H) yi = i = 1, 2,... (3.2.2) τi(H)
where τ0(H) is defined to be 1.
Proof. We start by substituting in and multiplying to obtain
∞ ∞ X X X (Ei,i+1 + giEii + hiEi+1,i) = (Ei,i+1 + biEii + yibiEi+1,i) + yj−1Ejj (3.2.3) i∈N i=1 j=2
Comparing coefficients yields the following system of equations:
b1 = g1 (3.2.4)
biyi = hi i = 1, 2,... (3.2.5)
bi+1 + yi = gi+1 i = 1, 2,... (3.2.6)
Since hi 6= 0 for all i, it is clear that one can solve uniquely for the (b1, y1, b2, y2,...) in that order, if there is no obstruction to solving Equation 3.2.5.
Completion of the proof boils down to proving 3.2.1, since 3.2.2 follows immediately from this and Equation 3.2.5. We proceed by induction on i:
For i = 1, we have
τ1(H) b1 = g1 = τ1(H) = (3.2.7) τ0(H)
so that the base case holds trivially.
τi(H) Assume bi = for some i ∈ , then, by Equations 3.2.1 and 3.2.2, one has τi−1(H) N
hi gi+1bi − hi bi+1 = gi+1 − = . (3.2.8) bi bi 87
Using the induction hypothesis, one then obtains
gi+1τi(H) − hiτi−1H τi+1(H) bi+1 = = (3.2.9) τi(H) τi(H)
where the latter equality follows from a cofactor expansion of Hi+1 along its bottom row.
Definition 3.4. Let B be a bi-infinite operator, viewed as a bi-infinite matrix B = P b E . For (m, n) ∈ 2, define two semi-infinite operators from B: i,j∈Z ij ij Z
∞ ∞ + X − X Bm,n := bm+i−1,n+j−1Eij, Bm,n := bm−i+1,n−j+1Eji. (3.2.10) i,j=1 i,j=1
To elucidate this definition, suppose H is the bi-infinite Hessenberg operator H = P (E + g E + h E ). In matrix form, H is i∈N i,i+1 i ii i i+1,i
. . .. .. . .. g−1 1 H = (3.2.11) h−1 g0 1 . h g .. 0 1 . . .. .. where the 0-th column and 0-th rows are encased by bars for reference. The two
+ instances of interest are the semi-infinite, tridiagonal, Hessenberg operators H1,1 and − H−1,−1: g1 1 + . H = h g .. (3.2.12) 1,1 1 2 . . .. .. 88
g0 1 − . H = h g .. (3.2.13) 0,0 −1 −1 . . .. ..
We can now state the analogous result for bi-infinite operators.
Theorem 3.2. Let a = (ai)i∈Z and x = (xi)i∈Z be sequences and define H = U(a)L(x). For ρ 6= 0, define two operators:
h a x H+ = H+ − 0 E = H+ − 1 0 E ρ 1,1 ρ 11 1,1 ρ 11
and
− − Hρ = H0,0 − ρE11.
Then, there exists a unique pair of sequences (b, y) with b0 = ρ satisfying L(y)U(b) = ± U(a)L(x) if and only if τk(Hρ ) 6= 0 for all k ∈ N. If this solution exists, it is given by + + ai+1xiτi−1(H ) τ (H ) ρ i ≥ 1 i ρ + + i ≥ 1 τi(H ) τ (H ) ρ i−1 ρ a1x0 yi = i = 0 , and bi = ρ i = 0 . ρ − − τ−i(Hρ ) ai+1xiτ−i+1(Hρ ) i ≤ −1 i ≤ −1 − − τ−i+1(Hρ ) τ−i(Hρ )
Proof. Define P := L(y)U(b), then P = H is equivalent to the following simultane- ous system of equations:
y0b0 = a1x0 (3.2.14)
+ + P1,1 = H1,1 (3.2.15)
− − P0,0 = H0,0 (3.2.16) 89
a1x0 We start by setting ρ := b0, so that y0 = ρ . Then
+ X P1,1 = (Ei,i+1 + (bi + yi−1)Eii + yibiEi+1,i) (3.2.17) i∈N ! ∞ X X = y0E11 + (Ei,i+1 + biEii + yibiEi+1,i) + yj−1Ejj (3.2.18) i∈N j=2 a x = 1 0 E + L(y )U(b ), (3.2.19) ρ 11 + + where y+ := (y1, y2,...) and b+ := (b1, b2,...). Thus, Equation 3.2.15 is equivalent to a x L(y )U(b ) = H+ − 1 0 E = H+. (3.2.20) + + 1,1 ρ 11 ρ By Lemma 3.1, we have + + τi(Hρ ) ai+1xiτi−1(Hρ ) bi = + , yi = + (3.2.21) τi−1(Hρ ) τi(Hρ ) for i = 1, 2,....
Turning our attention to Equation 3.2.16:
− X P0,0 = (Ei,i+1 + (y−i + b−i+1)Eii + y−iib−iEi+1,i) (3.2.22) i∈N ! ∞ X X = b0E11 + (Ei,i+1 + y−iEii + y−ib−iEi+1,i) + b−j+1Ejj (3.2.23) i∈N j=2 ∞ ! ∞ ! X X = b0E11 + (Eii + b−iEi+1,i) (y−jEjj + Ej,j+1) (3.2.24) i=1 j=1
= ρE11 + L(b−)U(y−), (3.2.25)
where y− := (y−1, y−2,...) and b− := (b−1, b−2,...). Thus, Equation 3.2.16 is equiv- alent to
− − L(b−)U(y−) = H0,0 − ρE11 = Hρ . (3.2.26)
By Lemma 3.1, we have − − ai+1xiτ−i+1(Hρ ) τ−i(Hρ ) bi = − , yi = − (3.2.27) τ−i(Hρ ) τ−i+1(Hρ ) for i = −1, −2,.... 90
3.3 Factorisations by Generalised Eigenfunctions
Noumi and Yamada’s original observation (2.3.9) drew a connection between an bi- infinite analogue of geometric RSK and the bi-infinite discrete-time Toda lattice. Whilst we ultimately reduce this connection to the finite setting, exploring the orig- inal observation led us to studying solutions to the bi-infinite factorisation problem. In addition to the parametrised solution in Theorem 3.2, we now provide another so- lution extending the Murphy’s [Mu18] analogous result for one-dimensional discrete Schr¨odingeroperators.
Definition 3.5. We define the following three operators:
X X X dl = (Ejj − Ej+1,j), dr = (Ej,j+1 − Ejj), sr = Ej+1,j, j∈Z j∈Z j∈Z the first two are first-order difference operators, and the latter is a shift operator.
The main result of this section is a factorisation by generalised eigenfunction. The result was inspired by and generalised from the following result of Murphy [Mu18]
Proposition 3.3. [Mu18] Let L = ∆ + u be a discrete Schr¨odingeroperator, and
f = {fi} a solution to the equation Lf = λ0f for a fixed value λ = λ0, with fi 6= 0.
Define (with a slight abuse of notation) sequences µl and µr by
fi − fi−1 −1 fi+1 − fi −1 (µl)i = = f dlf, (µr)i = = f drf, (3.3.1) fi fi then
L − λ0 = (dr + µl)(dl − srµr).
Remark 3.1. Taking λ0 = 0 yields a family of UL-decompositions of the operator. Modulo the LU-vs-UL component, this is a result on the factorisation of a subclass of the operators in which we are interested: the discrete Schr¨odingeroperators, ma- tricially speaking, are tridiagonal operators both of whose super- and sub-diagonals
consists solely of 1’s (∆ = dr − dl and u is diagonal). 91
Theorem 3.4. Let H = U(a)L(x), and suppose H = L(y)U(b) has a solution (b, y).
fj+1 Let f = (fi)i∈ be the sequence defined via f0 = 1, f1 = −ρ and bj = − for all Z fj j ∈ Z. Then f is a generalised eigenfunction for H with eigenvalue 0: Hf = 0, and we have
L(y) = (dl + srµr) and U(b) = (dr + µl), (3.3.2)
where µr and µl are diagonal operators defined via
fj+1 − aj+1xjfj fj − fj+1 (µr)j = and (µl)j = . (3.3.3) fj+1 fj
Conversely, starting with a generalised 0-eigenfunction f = (fi)i∈Z such that fi 6= 0 for all i, and building µr and µl as prescribed yields an LU-decomposition of H. Therefore, the familiy of LU-decompositions of H are completely classified by the nonvanishing generalised 0-eigenfunctions.
Remark 3.2. The family of generalised eigenfunctions of H is (generically) two- dimensional. However, multiplication by a nonzero scalar does not change the result- ing factorisation, so the eigenfunctions can be normalised, without loss of generality, so that f0 = 1. This normalisation picks out a 1-dimensional subspace of eigenfunc- tions, which, in connection to our parametrised factorisations, is parametrised by our choice of f1, or, equivalently, −f1 = ρ.
Proof. By Theorem 3.2, we know the form of every solution, if it exists. With b0 = ρ
fixed, define the sequence of interest: f0 = 1, f1 = −ρ, and define fj for j 6= 0, 1
fj+1 recursively by forcing bj = − for all j. fj 92
We compute dl + srµr first: X fi+1 − ai+1xifi dl + srµr = Eii − Ei+1,i + Ei+1,i fi+1 i∈Z X ai+1xifi = Eii − Ei+1,i fi+1 i∈Z X ai+1xi = Eii + Ei+1,i bi i∈Z X = (Eii + yiEi+1,i) i∈Z = L(y).
The verification that U(b) = (dr + µl) is similarly straightforward, so is omitted.
Next, we prove that f = (fi)i∈Z is a generalised 0-eigenfunction for H. Recall that b and y satisfy the following system of equations biyi = ai+1xi , ∀ i ∈ Z. (3.3.4) bi + yi−1 = ai + xi
Since
(Hf)i = fi+1 + (ai + xi)fi + aixi−1fi−1 ∀ i (3.3.5)
(Hf)i and each fi 6= 0, it suffices to show that = 0 for all i ∈ . Indeed, fi Z
(Hf)i fi+1 fi−1 = + ai + xi + aixi−1 (3.3.6) fi fi fi aixi−1 = −bi + ai + xi − (3.3.7) bi−1 bi−1yi−1 = yi−1 − (3.3.8) bi−1 = 0. (3.3.9)
Hence, f is indeed a generalised 0-eigenfunction for H.
Finally, we establish that if fi+1 + (ai + xi)fi + aixi−1fi−1 = 0 and fi 6= 0 for all i, 93
then (dl + srµr)(dr + µl) = H: X ai+1xifi X fj+1 (dl + srµr)(dr + µl) = Eii − Ei+1,i Ej,j+1 − Ejj (3.3.10) fi+1 fj i∈Z j∈Z X fi+1 + aixi−1fi−1 = Ei,i+1 − Eii + ai+1xiEi+1,i (3.3.11) fi i∈Z X (ai + xi)fi = Ei,i+1 + Eii + ai+1xiEi+1,i (3.3.12) fi i∈Z = H. (3.3.13)
3.4 Geometric RSK as a Degeneration of the Discrete-Time Toda Lattice
Whilst the observation of Noumi and Yamada (2.3.9) provides a direct change of coordinates to transition between bi-infinite discrete-time Toda and the bi-infinite extension of geometric RSK, it does not provide a direct way to connect to the original (finite-dimensional) geometric RSK. In this section, our main result, Theorem 3.7, demonstrates how geometric RSK is attainable as a degeneration of discrete-time Toda.
Remark 3.3. From this point on, we shall omit the bars on the geometric RSK variables.
Lemma 3.5. Let M be a square matrix with block structure A B M = C D with A invertible. Then
det(M) = det(A) det(D − CA−1B). 94
This is a fairly standard result for block matrices. For a reference, see [EFS93].
Definition 3.6. For an n × n matrix M, and for a, b ∈ [n], let M(a),(b) be the (n − 1) × (n − 1) submatrix of M obtained by deleting the ath column and bth row of M.
Lemma 3.6. [NY04] Let S = U(a)U(x), for a = (a1, . . . , an) and x = (x1, . . . , xn),
and suppose τk(S(1),(n)) 6= 0 for k = 1, 2, . . . , n − 1, then solution to the geometric RSK equations with input (a, x) exists and is given by the unique pair of sequences
y = (y1, . . . , yn) and b = (b2, . . . , bn), where
τk−1(S(1),(n)) bk = , k = 2, . . . , n (3.4.1) τk−2(S(1),(n))
τk−2(S(1),(n)) yk = akxk , k = 1, . . . , n, (3.4.2) τk−1(S(1),(n)) and where, in addition to setting τ0 = 1, we also set τ−1 = 1.
To illustrate the lemma and definition, consider the case of n = 3. a1 1 0 x1 1 0 a1x1 a1 + x2 1 H = = (3.4.3) a2 1 x2 1 a2x2 a2 + x3 a3 x3 a3x3 then one has a1 + x2 1 H(1),(3) = (3.4.4) a2x2 a2 + x3
and the gRSK solution (y1, y2, y3, b2, b3) is given by
τ−1(H(1),(3)) y1 = a1x1 = a1x1 (3.4.5) τ0(H(1),(3))
τ1(H(1),(3)) b2 = a1 + x2 = (3.4.6) τ0(H(1),(3))
a2x2 τ0(H(1),(3)) y2 = = a2x2 (3.4.7) a1 + x2 τ1(H(1),(3))
a2x2 τ2(H(1),(3)) b3 = a2 + x3 − = (3.4.8) a1 + x2 τ1(H(1),(3))
a1 + x2 τ1(H(1),(3)) y3 = a3x3 = a3x3 . (3.4.9) (a1 + x2)(a2 + x3) − a2x2 τ2(H(1),(3)) 95
We now present our main result: the degeneration from the (finite) discrete-time Toda lattice to geometric RSK.
Theorem 3.7. Let (a, x) = ((a1, . . . , an), (x1, . . . , xn)), and define two (n+1)×(n+1)
matrices Uδ and L via
a U = U x 1 − 1 , a , a , . . . , a ,L = L(x , x , . . . , x ), (3.4.10) δ 1 δ 1 2 n 1 2 n
i.e. a1 x1 − 1 1 1 δ .. a1 . x1 1 U = ,L = , (3.4.11) δ . . . .. .. .. 1 an xn 1
˜ ˜ ˜ δ δ then the LU decomposition problem LδUδ = UδL, where Uδ = U(c1, . . . , cn+1) and ˜ δ δ Lδ = L(z1, . . . , zn) satisfies:
δ δ y1 = c1z1, (3.4.12)
bk = lim ck, k = 2, . . . , n, (3.4.13) δ→0
yk = lim zk, k = 2, . . . , n (3.4.14) δ→0 where ((1, b2, . . . , bn), (y1, y2, . . . , yn)) is the geometric RSK solution with initial data
((a1, . . . , an), (x1, . . . , xn)).
Proof. Let Hδ = UδL, then by Lemma 3.1, one has
δ τk(Hδ) ck = (3.4.15) τk−1(Hδ)
δ τk−1(Hδ) zk = akxk (3.4.16) τk(Hδ)
a1 for k = 1, 2, . . . , n, and where a0 := x1 δ − 1 . 96
Now, for S := U(a)U(x), one has:
a + x 1 1 2 . a x a + x .. 2 2 2 3 S(1),(n) = . (3.4.17) . . .. .. 1 an−1xn−1 an−1 + xn
By Lemma 3.6, the true gRSK solution is given by
τk−1(S(1),(n)) bk = , k = 2, . . . , n (3.4.18) τk−2(S(1),(n))
τk−2(S(1),(n)) yk = akxk , k = 1, . . . , n. (3.4.19) τk−1(S(1),(n))
δ a1x1 δ By direct calculation, one finds y1 = x1a1, c1 = δ , and z1 = δ, so Equation 3.4.12 obviously holds.
It therefore remains to show
τ (H ) τ (S ) k δ → k−1 (1),(n) (3.4.20) τk−1(Hδ) τk−2(S(1),(n)) as δ → 0, for k = 2, . . . , n. To see this, first note that only the first n principal minors are needed from Hδ, the the principal n × n submatrix of Hδ has the following block structure: a1x1 δ 1 0 ··· 0 a1x1 a1 + x2 1 .. (Hδ)(n+1),(n+1) = 0 a x a + x . . (3.4.21) 2 2 2 3 . . .. .. . . . 1 0 an−1xn−1 an−1 + xn 97
Thus, for k = 2, . . . , n, one has
a1x1 T δ e1 τk(Hδ) = det (3.4.22) a1x1e1 (S(1),(n))k−1 where e1 is the first column of the (k − 1) × (k − 1) identity matrix and (S(1),(n))k−1 is the principal (k − 1) × (k − 1) submatrix of S(1),(n).
By Lemma 3.5, one has a1x1 δ T τk(Hδ) = det (S(1),(n))k−1 − a1x1e1 e1 (3.4.23) ε a1x1 δ 0 ··· 0 a x 0 0 ··· 0 = 1 1 det (S ) − . (3.4.24) (1),(n) k−1 . . . δ . . ··· . 0 0 ··· 0 Therefore, δ 0 ··· 0 0 0 ··· 0 a1x1 det (S ) − δ (1),(n) k−1 . . . . . ··· . τk(Hδ) 0 0 ··· 0 lim = lim (3.4.25) δ→0 τk−1(Hδ) δ→0 δ 0 ··· 0 0 0 ··· 0 a1x1 det (S ) − δ (1),(n) k−2 . . . . . ··· . 0 0 ··· 0 δ 0 ··· 0 0 0 ··· 0 det (S ) − (1),(n) k−1 . . . . . ··· . 0 0 ··· 0 = lim (3.4.26) δ→0 δ 0 ··· 0 0 0 ··· 0 det (S ) − (1),(n) k−2 . . . . . ··· . 0 0 ··· 0 τ (S ) = k−1 (1),(n) (3.4.27) τk−2(S(1),(n)) 98
δ by continuity of determinants and by the algebra of limits. Thus, ck → bk for k =
δ δ akxk akxk 2, . . . , n, and also zk → yk, since zk = δ and yk = . ck bk
We conclude this chapter by remarking that the embedding of geometric RSK into the discrete-time Toda lattice falls into the realm of the Deift-Nanda-Tomei scheme [DNT83] described back in Section 2.1.6. That is to say, the embedding produces ge- ometric RSK as a stroboscope of the logarithmic continuous-time flow. This requires some justification on how to translate from our Hessenberg setting to their symmetric setting, as well as the well-definedness of the logarithm required for their flow:
d X = [X, π (log X)]. (3.4.28) dt −
Translating from the Hessenberg setting to the symmetric setting is well-known in the literature: if the sub-diagonal of a tridiagonal, Hessenberg matrix is nowhere zero, then one can carry the Hessenberg Toda flow over to the symmetric Toda flow by conjugation.
Lemma 3.8. [EMP08] Let
a1 1 .. b1 a2 . X = (3.4.29) . . .. .. 1 bn−1 an
with a1, . . . , an, b1, . . . , bn−1 ∈ R with bi > 0 for each i, then 99
√ a1 b1 √ .. b1 a2 . C−1XC = (3.4.30) . . .. .. 1 p bn−1 an where
p p p C = diag(1, b1, b1b2,..., b1 ··· bn−1). (3.4.31)
Recall that our Hessenberg matrices that arise from gRSK (in the sense of Theorem 3.7) are of the following form:
a1x1 1 δ a x a + x 1 1 1 1 2 .. a2x2 a2 + x3 . Hδ = . (3.4.32) .. .. . . 1 a x a + x 1 n−1 n−1 n−1 n anxn an
Recall also that tropicalisation takes one from (R≥0, +, ×) to (R ∪ {−∞}, max, +),
with 0 ∈ R≥0 corresponding precisely to −∞ ∈ R ∪ {−∞}. Therefore, the variables
a1, . . . , an, x1, . . . , xn in gRSK should all be (strictly) positive. In this case, Lemma
3.8 applies to Hδ, and we can carry the Hessenberg flow for gRSK to the symmetric analogue.
We also have the question of computing the logarithm of the resulting matrix. To
answer this, we first present a result on the nature of the τ-functions of Hδ. 100
Proposition 3.9. For each k ∈ {1, . . . , n + 1}, one has τk(Hδ) > 0 for δ > 0 sufficiently small.
a1x1 a1x1 Proof. For k = 1, τ1(Hδ) = δ . Since a1, x1 > 0 and δ > 0, we see that δ > 0.
a1x1 If k > 1, factoring out δ from the first column and performing a cofactor expansion down the first column, one obtains:
a x τ (H ) = 1 1 τ ((H ) ) − δτ ((H ) ) (3.4.33) k δ δ k−1 δ (1),(1) k−2 δ (1,2),(1,2) where is defined by the natural extension of Definition 3.6: remove the first two rows and columns of Hδ.
If τk−1((Hδ)(1),(1)) > 0 for all 1 < k ≤ n + 1, i.e. all principal minors of (Hδ)(1),(1) are positive, then the result will follow from Equation 3.4.33. We prove inductively that
τm((Hδ)(1),(1)) > 0 for each m ∈ {1, 2, . . . , n. For m = 1, we have
τ1((Hδ)(1),(1)) = a1 + x2 > 0. (3.4.34)
For m > 1, we can perform a cofactor expansion along the bottom row of the principal m × m submatrix of (Hδ)(1),(1) to obtain
τm((Hδ)(1),(1)) = (am + xm+1)τm−1((Hδ)(1),(1)) + amxmτm−2((Hδ)(1),(1)). (3.4.35)
By writing
zm = τm((Hδ)(1),(1)) − xm+1τm−1((Hδ)(1),(1)) (3.4.36) for all m, then Equation 3.4.35 can be rewritten as
zm = amzm−1. (3.4.37)
Observe:
z1 = τ1((Hδ)(1),(1)) − x2 = (a1 + x2) − x2 = a1. (3.4.38) 101
Therefore, the solution to Equation 3.4.37 is given by
m Y zm = ai. (3.4.39) i=1 With this, Equation 3.4.36 yields
m Y τm((Hδ)(1),(1)) = xm+1τm−1((Hδ)(1),(1)) + ai. (3.4.40) i=1
By induction, it follows that τm((Hδ)(1),(1)) > 0 for all m ∈ {1, . . . , n}.
By Equation 3.4.33, it now follows that for δ > 0 sufficiently small, τk(Hδ) > 0 for all k ∈ {1, . . . , n + 1}.
Corollary 3.10. The result of symmetrising Hδ using Lemma 3.8 is a matrix with positive eigenvalues. Furthermore, Hδ is diagonalisable to a diagonal matrix with positive diagonal.
√ √ Proof. To symmetrise Hδ, conjugate it by diag(1, a1x1,..., a1x1 ··· an−1xn−1). Since we are conjugating by a diagonal matrix with only positive entries on its diag- onal, we do not affect the positivity of the principal minors of Hδ.
By Sylvester’s criterion, since a real, symmetric matrix is positive defininite if and only if all of its principal minors are positive. Thus, the symmetrisation of Hδ is pos- itive definite. It then follows from positive definiteness that this matrix has positive eigenvalues. To conclude, note that the symmetrisation of Hδ is diagonalisable, by the spectral theorem for real, symmetric matrices, and, since Hδ is conjugate to this symmetric matrix, Hδ is itself diagonalisable, with the same (positive) eigenvalues as its symmetrisation.
We now have our means for computing the logarithm: ordinarily, one could use the Jordan-Chevalley decomposition to compute the matrix logarithm in terms of the 102
semisimple and nilpotent parts of the matrix. Since Hδ and its symmetrisation are both diagonalisable, they are both semisimple. This simplifies taking the logarithm of these matrices: we diagonalise the matrices, then apply the usual real-valued log- arithm to the (positive) diagonal entries. We now conclude by remarking that the embedding of geometric RSK into the discrete-time Toda lattice (Theorem 3.7) can now be viewed as recasting geometric RSK as a discrete-time evolution, which, by the work of Deift-Nanda-Tomei [DNT83] and Symes [Sy80], is a discretisation (a stroboscope) of the following flow:
d X = [X, π (log X)]. (3.4.41) dt −
Furthermore, by the work of Deift-Nanda-Tomei and Symes, this continuous-time flow commutes with the classical Toda lattice. 103
Chapter 4 RSK and BBS: The Ultradiscrete Picture
In this chapter, we relate Schensted insertion and the Toda lattice on the ultradiscrete level. We begin by showing that, as one would hope, the ultradiscretisation of the geometric RSK equations indeed results in the RSK equations. The purpose of this exercise, however, goes far beyond simply recovering the RSK equations: by replicat- ing the ultradiscretisation method of discrete-time Toda, we obtain a representation of the RSK equations in a form that lends itself to comparison to the box-ball system.
We find that our comparison results in the need to be able to interpret the box-ball system when some of its coordinates are zero. This leads us to a new cellular au- tomaton, extending the box-ball system by two new types of object, and we call this cellular automaton the ghost-box-ball system (GBBS). We prove some key results about our ghost-box-ball system, including its reduction to the original box-ball sys- tem under an operation we call exorcism. The operation of exorcism, along with the properties of the GBBS, allows us to extend a the classical Young diagram conserved quantity of the box-ball system to the ghost-box-ball system.
We note that there are some other results in the literature, one of which allows for a description of Schensted insertion in terms of an advanced box-ball system. This description, due to Fukuda [Fu04], is provided in AppendixA. In the final section of this chapter, Section 4.7, we compare and contrast our new results to Fukuda’s, highlighting the key differences and the appeal of our approach. 104
4.1 Ultradiscretisation of Geometric RSK
We begin with a demonstration of the ultradiscretisation process on the geometric RSK equations. Since we are just (re)tropicalising the detropicalised RSK equations, one should not be surprised to recover RSK. However, the process rewrites the RSK equations in a way that is paramount to seeing the connection between RSK and the box-ball system.
Lemma 4.1. The ultradiscretisation of the geometric RSK equations (Equations 2.3.9) results in the (tropical) RSK equations for Schensted insertion.
Proof. We begin with the geometric RSK equations:
y1 = a1x1 (4.1.1)
yibi = aixi i = 2, . . . , n (4.1.2)
b2 = a1 + x2 (4.1.3)
yi + bi+1 = ai + xi+1 i = 2, . . . , n − 1. (4.1.4)
Using Equations 4.1.2 and 4.1.4, one obtains
bi+1 = xi+1 + ai − yi (4.1.5)
ai = xi+1 + (bi − xi) (4.1.6) bi ai = xi+1 + (ai−1 − yi−1) (4.1.7) bi Qi aj = ··· = x + j=1 (4.1.8) i+1 Qi j=2 bj for i = 2, . . . , n − 1. 105
Taking b1 = 1, as it should be, the geometric RSK can be expressed as
b1 = 1 (4.1.9)
aixi yi = i = 1, . . . , n (4.1.10) bi Qi aj b = x + j=1 i = 1, . . . , n − 1 (4.1.11) i+1 i+1 Qi j=2 bj by following the convention of taking the empty product to be 1.
With the evolution expressed in a subtraction-free form, we begin ultradiscretisation, first by changing variables
−ai(ε)/ε −xi(ε)/ε −yi(ε)/ε −bi(ε)/ε ai → e , xi → e , yi → e , bi → e .
The change of variables, applied to Equations 4.1.9- 4.1.11, yields
− 1 b (ε) e ε 1 = 1 (4.1.12)
− 1 y (ε) − 1 (a (ε)+x (ε)−b (ε)) e ε i = e ε i i i i = 1, . . . , n (4.1.13)
− 1 b (ε) − 1 x (ε) − 1 Pi a (ε)−Pi b (ε) e ε i+1 = e ε i+1 + e ε ( j=1 j j=2 j ) i = 1, . . . , n − 1, (4.1.14) where the empty sum is taken to be zero.
Solving for the exponentiated variables on the left-hand side, we get the following
b1(ε) = 0 (4.1.15)
yi(ε) = ai(ε) + xi(ε) − bi(ε) i = 1, . . . , n (4.1.16)
1 1 Pi Pi − xi+1(ε) − ( aj (ε)− bj (ε)) bi+1(ε) = −ε log e ε + e ε j=1 j=2 i = 1, . . . , n − 1. (4.1.17)
The final step in ultradiscretisation is taking the limit as ε → 0+. If we abuse notation by recycling the orignal RSK variables in the ultradiscrete equations by letting, for example, bi = lim bi(ε), we obtain the following ultradiscretisation of the geometric ε→0+ 106
RSK equations
b1 = 0 (4.1.18)
yi = ai + xi − bi i = 1, . . . , n (4.1.19) i i !! X X bi+1 = min xi+1, aj − bj i = 1, . . . , n − 1. (4.1.20) j=1 j=2
It remains to show that the solution to this system of equations solves the RSK equations. Equations 4.2.1 and 4.2.2 are already part of the RSK equations. What is left of the RSK equations is for the following to hold
η1 = ξ1 + a1 (4.1.21)
ηj = max{ηj−1, ξj} + aj, j = 2, . . . , n (4.1.22)
where ηi = y1 + ··· + yi and ξi = x1 + ··· + xi for i = 1, . . . , n. Since both the RSK equations and the ultradiscrete geometric RSK equations have a unique solution, this will complete the proof. We proceed directly:
Equation 4.1.21 is equivalent to y1 = x1 + a1, which clearly holds.
For Equation 4.1.22, we use the defining equations for the ultradiscrete geometric 107
RSK equations in the following computation for i ≥ 1:
yi+1 = ai+1 + xi+1 − bi+1 i i !! X X = ai+1 + xi+1 − min xi+1, aj − bj j=1 j=2 i i !! X X = ai+1 + xi+1 − min xi+1, aj − (aj + xj − yj) j=1 j=2 i !! X = ai+1 + xi+1 − min xi+1, a1 + (yj − xj) j=2 i !! X = ai+1 + xi+1 − min xi+1, a1 − y1 + x1 + (yj − xj) j=1
= ai+1 + xi+1 − min (xi+1, (b1 + ηi − ξi))
= ai+1 + xi+1 + max (−xi+1, ξi − ηi)
= ai+1 + max (0, xi+1 + ξi − ηi)
= ai+1 + max (0, ξi+1 − ηi)
= ai+1 + max (ηi, ξi+1) − ηi.
Thus,
ηi+1 = yi+1 + ηi = ai+1 + max (ηi, ξi+1) , (4.1.23) which completes the proof.
4.2 RSK Insertion and the Box-Ball Coordinates
The key point of the work in Section 4.1 is that we have now obtained the RSK equations in the following form:
b1 = 0 (4.2.1)
yi = ai + xi − bi i = 1, . . . , n (4.2.2) i i !! X X bi+1 = min xi+1, aj − bj i = 1, . . . , n − 1. (4.2.3) j=1 j=2 108 which lends itself to comparison to the box-ball system equations with n + 1 solitons:
t+1 t t t+1 Wi = Qi+1 + Wi − Qi i = 1, . . . , n (4.2.4) i i−1 ! t+1 t X t X t+1 Qi = min Wi , Qj − Qj i = 1, . . . , n + 1. (4.2.5) j=1 j=1 In the box-ball system equations (Equations 4.2.4 and 4.2.5), we perform the following change of variables:
t t t+1 t+1 Qi+1 = ai,Wi = xi,Wi = yi,Qi = bi, (4.2.6) producing the following
yi = ai + xi − bi i = 1, . . . , n (4.2.7) i i−1 ! X X bi = min xi, aj−1 − bj i = 1, . . . , n + 1. (4.2.8) j=1 j=1
t t Since b1 = min(x1, a0) = min(W1,Q1), to obtain the RSK condition b1 = 0, we take t a0 = Q1 = 0. Under this condition, the box-ball equations now take the form
b1 = 0 (4.2.9)
yi = ai + xi − bi i = 1, . . . , n (4.2.10) i i ! X X bi+1 = min xi+1, aj − bj i = 1, . . . , n. (4.2.11) j=0 j=1
Finally, we make sense of bn+1 in the above system:
n n ! X X bn+1 = min xn+1, aj − bj = a1 + ··· + an − b2 − · · · − bn (4.2.12) j=0 j=1
t since xn+1 = Wn+1 = ∞.
Although the box-ball coordinate evolution was defined on Bn (as defined in Definition 2.10), in which all coordinates are positive integers, these equations naturally extend to coordinates which may contain zeroes. Furthermore, since this box-ball coordinate
t+1 t evolution has Q1 = 0 if Q1 = 0, we modify Bn in the following way: 109
0 2n 0 S 0 n 2 0 Definition 4.1. Let Bn = {0} × N0 × {∞}, B = Bn, and Rn = (N0 ) . On Bn, n∈N0 we have the dynamics induced by ξ : B → B (append ∞ on the left, then evolve the coordinates, and finally strip off the ∞), and this set of maps extend naturally to all
0 of B , we conflate these maps and call the result ξ as well. We define a map RSKn :
Rn → Rn by taking the first (respectively, last) n coordinates of (a, x) ∈ Rn to be
insertion (respectively, initial) word in the RSK algorithm, and letting RSKn(a, x) = S (b, y). Define R = Rn, and let RSK : R → R be defined naturally. n∈N 0 Remark 4.1. The identification used to induce a map on Bn by ξ : B → B is really 0 between Bn and Bn+1. The reason for not shifting one of the definitions here is that Bn, in the box-ball setting, is indexed by the number of solitons, whereas in this chapter, the index aligns with the index for RSK.
n Definition 4.2. For a pair of sequences a = (a1, . . . , an), x = (x1, . . . , xn) ∈ N0 , n 0 define a map φRSK→BBS : R → Bn by
n φRSK→BBS(a, x) = (0, x1, a1, x2, a2, . . . , xn, an, ∞). (4.2.13)
0 0 n Conversely, for a sequence z ∈ Bn, define a map φBBS→RSK : Bn → R by
n φBBS→RSK(0, z1, . . . , z2n, ∞) = ((0, z2, . . . , z2(n−1)), (z1, z3, . . . , z2n−1)). (4.2.14)
0 0 Let φRSK→BBS : R → B and φBBS→RSK : B → R be their natural extensions.
We summarise the above calculations in the following theorem:
Theorem 4.2. Under the box-ball evolution ξn+1 : Bn+1 → Bn+1, RSK insertion is captured as the following
ξn+1(∞, 0, x1, a1, x2, . . . , an−1, xn, an, ∞) = (∞, b1, y1, b2, y2, . . . , bn, yn, bn+1, ∞) . (4.2.15) 110
Corollary 4.3. One has
RSK = φBBS→RSK ◦ ξ ◦ φRSK→BBS. (4.2.16)
4.3 The Ghost-Box-Ball System
We now introduce the ghost-box-ball system which is designed to be governed by the extension of the box-ball equations. This amounts to modifying the original box- ball system to reflect the zeroes that we are allowing into the box-ball coordinates. Ultimately, what we want is a modified box-ball system into which one can encode an RSK pair, and from whose evolution one can read the RSK output.
Definition 4.3. A ghost-box-ball system (GBBS) is a finite-tailed infinite collection of boxes for which a finite number of boxes are designated precisely one of the following three states (the rest of the boxes are empty):
1. filled (with a ball),
2. filled ghost,
3. empty ghost, and subject to the following constraints:
1. the first box is a filled ghost,
2. a filled ghost may not be adjacent to another filled ghost, nor to a filled box, and
3. an empty ghost may not be adjacent to another empty ghost, nor to an empty box.
Whenever the term “filled box” (respectively “empty box”) is used, we will not be referring to ghosts. However, a single filled ghost or a block of filled boxes will both be referred to as a filled block, and similarly for empty. An important consequence 111 is that the constraints force filled and empty blocks to alternate, with the first block being a filled one, consisting of a single filled ghost, and the last being an empty block, consisting of infinitely many empty boxes. Under this observation, as with the box-ball system, we can coordinatise ghost-box-ball systems, the set of which will be denoted GBBS.
Definition 4.4. We define a coordinatisation C : GBBS → B0 by mapping a ghost- box-ball system to a tuple (F1,E1,F2,E2,...), where
0 if the i-th filled block is a filled ghost F = i #length of the i-th filled block otherwise 0 if the i-th empty block is an empty ghost E = i #length of the i-th empty block otherwise
0 For each ghost-box-ball system state, its coordinates lie in Bn for some n ∈ N0. Therefore, the full set of ghost-box-ball states is identified with B0. For a graphical representation of the GBBS’s, we employ the following key:
1. An empty box shall be represented by
2. A filled box shall be represented by
3. A filled ghost shall be represented by
4. An empty ghost shall be represented by
We now define the following evolution rule on GBBS.
Definition 4.5. (The Ghost-Box-Ball Algorithm)
1. Move each ball exactly once.
2. Move the leftmost unmoved ball to its nearest right empty box.
3. If a ball’s new position has a filled ghost to its immediate right, materialise (create) an empty ghost between them. 112
4. If a ball’s new position has a filled ghost to its immediate left, exorcise (delete) the ghost.
5. If a ball is moved from a position with an empty ghost right-adjacent of it, insert a filled ghost between the box vacated by the ball and the empty ghost.
6. If a ball is moved from a position with an empty ghost left-adjacent of it, exorcise that ghost.
7. Repeat steps (2)-(6) until all balls have been moved.
Lemma 4.4. The result of performing this algorithm on a ghost-box-ball system is again a ghost-box-ball system.
Proof. This is just a consequence of the construction: steps (3)-(6) serve the purpose of keeping constraints (2) and (3) satisfied.
Definition 4.6. The map %ˆ : GBBS → GBBS will be defined to be the result of applying this algorithm.
Example 4.1. We demonstrate the ghost-box-ball evolution, step-by-step below:
Figure 4.1. A single time-step of the ghost-box-ball evolution, split into its the subroutine that defines it. 113
Therefore, one time-step of the ghost box ball evolution is given by the following:
Figure 4.2. A single time-step of the ghost-box-ball evolution (without the inter- mediate steps).
4.4 Exorcism, Soliton Behaviour and the Invariant Shape
In the ghost-box-ball algorithm, ghosts can be exorcised as a result of the movement of balls (either by moving a ball to the right of an empty ghost or from the right of an empty ghost). There is a natural map z : GBBS → BBS given by exorcising all ghosts in a ghost-box-ball system, shifting the remaining balls and boxes to fill in the newly created voids (i.e., create a sequence of empty and filled boxes, scanning the ghost-box-ball system from left to right, ignoring all ghosts). Since the original box-ball system was bi-infinite, there is an issue to deal with here. We can do this by taking the ghost-box-ball systems to also be bi-infinite (by having an infinite number of empty boxes to the left of the initial filled ghost). We will assume this is the case, without explicitly drawing the empty boxes to the left of the filled ghost. This map shall be referred to as global exorcism.
Lemma 4.5. The following diagram commutes
%ˆ GBBS GBBS
z z % BBS BBS 114
Proof. The ghost-box-ball algorithm and the box-ball algorithm only differ in how the ghosts are materialised and exorcised: the new position of a moving ball in the ghost-box-ball algorithm agrees with that of the box-ball algorithm, relative to just the empty boxes and balls. Therefore, the result of globally exorcising the ghosts and then evolving according to the box-ball dynamics coincides with evolving according to the ghost-box-ball dynamics and then globally exorcising.
In light of this result, it also makes sense to ask about how the soliton structure of box-ball-system translates to the ghost-box-ball system. By conflating all ghosts (i.e. identifying empty and filled ghosts), any chain of ghosts can be thought of as constituting a single zero length soliton. It is not hard to see that the algorithm preserves the existence and location of such chains (a ghost cannot materialise unless it does so next to another ghost, and a ghost cannot be exorcised without a neighbouring ghost). Along with the previous lemma, this observation leads us to the following.
Lemma 4.6. The ghost-box-ball dynamics exhibits the same soliton behaviour as the box-ball dynamics, subject to the following modification to the traditional notion of a soliton in the box-ball system: a soliton is any consecutive chain of filled balls uninterrupted by empty boxes (i.e. ghosts do not split solitons up), or any consecutive chain of ghosts (a ghost soliton). A ghost soliton does not move (it has zero velocity), all other solitons travel with speed equal to the number of balls comprising it. The ghost-box-ball system therefore also exhibits the sorting property of the box-ball system.
Our final immediate analogue of the classical box-ball system is its conserved shape.
Corollary 4.7. If G ∈ GBBS, define B = z(G). Representing B as a sequence of 1’s and 0’s,
let p1 be the number of 10’s in the sequence.
Eliminate all of these 10’s, and let p2 be the number of 10’s in the resulting sequence. 115
Repeat this process until no 10’s remain.
We associate the weakly decreasing sequence (p1, p2,...), or, equivalently, the Young th diagram whose j column has pj boxes is the shape associated to G. This Young diagram is the same for %ˆkG for every k ∈ N.
Proof. This follows from Lemma 4.5 and Section 2.2.3. It is already known that
k k Young diagram is conserved for all % (B) = % ◦ z(G). Since % ◦ z = z ◦ %ˆ, it follows that
k k k z(ˆ% (G)) = % ◦ z(G) = % (B).
Example 4.2. Returning to the ghost-box ball system in Example 4.1, below are the initial state and the subsequence three evolutions.
··· ··· ··· ···
Figure 4.3. Three iterations of the ghost-box-ball algorithm.
Applying global exorcism to these four ghost-box-ball states yields
··· ···
··· ···
··· ···
··· ···
Figure 4.4. Three iterations of the ghost-box-ball evolution (after global exorcism). 116
where the blue lines indicate the locations previously occupied by ghosts. The invariant shape for this sequence (and all future states) is
Figure 4.5. The invariant shape of the ghost-box-ball system(s) in Figure 4.3
4.5 The Ghost-Box-Ball System and Schensted Insertion
0 0 In Corollary 4.3, we constructed maps φRSK→BBS : R → B and φBBS→RSK : B → R to represent Schensted insertion in terms of the natural extension of the coordina- tised box-ball evolution to the setting in which some coordinates may vanish. In this section, we demonstrate how to lift this to the ghost-box-ball evolution, using the coordinate map on GBBS.
The main tool will be the coordinate map C : GBBS → B0 in Definition 4.4. One can encode an RSK pair (a, x) ∈ R in a ghost-box-ball system by composing the maps
0 −1 0 φRSK→BBS : R → B and C : B → GBBS.
The main result of this section will be the following
−1 RSK = φBBS→RSK ◦ C ◦ %ˆ ◦ C ◦ φRSK→BBS. (4.5.1)
However, we will prove this by showing something stronger: if an RSK input pair is encoded in a ghost-box-ball system, the number of steps in the RSK insertion is equal to the number of steps in the ghost-box-ball evolution, and the data from each stage of the RSK insertion is fully recoverable from the corresponding stage of the ghost-box-ball evolution. 117
Definition 4.7. For a pair (a, x) ∈ Rn, define a sequence of triples
i i i i n 3 r (a, x) := (a , x , b ) ∈ (N0 ) (4.5.2)
n P for i = 0, 1,..., ak, where k=1 1. r0(a, x) = (a, x, (0, 0,..., 0)).
i i+1 2. If j = min{ak 6= 0}, then x is the first row of the semistandard Young k tableau obtained by (Schensted) inserting j into xi, ai+1 is obtained from ai by subtracting 1 from the j-th entry, and bi+1 is the result of adding 1 to the k-th entry of bi if a k is bumped from xi to obtain xi+1, or bi+1 = bi if nothing is bumped.
We write ri for ri(a, x) if it unambiguous to do so.
This construction clearly encodes the steps of Schensted insertion. In particular, if n P m RSK(a, x) = (b, y) and m = ak, then one has r = ((0, 0,..., 0), y, b). k=1
4.5.1 The RSK Walls in the Ghost-Box-Ball System
We now introduce a bookkeeping device for the ghost-box-ball system that arise from an RSK input pair. The key observation is that, in the steps of the RSK insertion,
i i i the total number of instances of a given number is conserved: aj +xj +bj is a function of just j (it is constant in i). In particular, taking i = 0, this quantity is
0 0 0 aj + xj + bj = aj + xj =: wj which is expressed solely in terms of the input pair. This quantity will be called the j-th width.
We now construct our walls using the wj quantities. 118
n −1 Definition 4.8. Let (a, x) ∈ R and G = C ◦ φRSK→BBS(a, x) be the associated ghost-box-ball system. The walls of G, which we will represent by red zigzags, will be placed as follows:
1. Take the initial filled ghost and the following w1 boxes of G, and separate them
from the rest of the subsequent boxes by a wall (a red zigzag). Note: If x1 = 0
(and/or a1 = 0), include the corresponding empty ghost (resp. and/or the corresponding filled ghost).
2. Take the next w2 boxes of G and place a wall at the end of them. Again, if
a2 = 0, include the corresponding filled ghost before the zigzag.
3. Continue in this manner until the boxes are separated into n+1 regions (n finite and one infinite).
Example 4.3. Take the RSK input to be a = (3, 0, 2, 1, 0) and x = (2, 0, 0, 2, 3), so that the associated ghost-box-ball system has coordinates
φRSK→BBS(a, x) = (0, 2, 3, 0, 0, 0, 2, 2, 1, 3, 0, ∞).
We put a wall before the first filled ghost, enclose the two subsequent empty boxes and three filled boxes by another wall (this is now “Region 1”), and so on. Here is the resulting picture:
···
1 2 3 4 5
Figure 4.6. The initial ghost-box-ball system with its finite regions labelled.
Next, we describe how the walls interact with the steps of the ghost-box-ball evolution.
1. When a ghost is exorcised: if it is bordered by a wall, the wall then borders the ghost’s other neighbour. 119
2. When a filled ghost materialises by the evacuation of a box (implying the pres- ence of a wall to the right of the evacuated box), the filled ghost that materialises does so to the left of the wall.
3. When an empty ghost materialises by the appearance of a ball, the empty ghost is created in the same region as the ball’s new location.
The key point is that the position of the walls, relative to the non-ghost boxes, does not change with the steps of the ghost-box-ball evolution. The walls keep a notion of the regions throughout the evolution, and it is precisely the numbers of moved balls, unmoved balls and empty boxes in the regions that we show encodes the correspond- ing RSK steps. For ease of visualisation, we once again employ a colouring of balls to keep track of balls that have moved and those that have not, colouring unmoved balls blue and moved balls red.
Before presenting the main theorem of this section, we provide an illustrative example showing the locations of the walls throughout the evolution of the ghost-box-ball system in Figure 4.6.
Example 4.4. Below is the evolution of the ghost-box-ball system in Figure 4.6, with the walls included at each step. 120
···
···
···
···
···
···
···
Figure 4.7. The evolution of a ghost-box-ball system with walls.
The initial ghost-box-ball system came from (a, x) = ((3, 0, 2, 1, 0), (2, 0, 0, 2, 3)). The result of RSK insertion is
a = (3, 0, 2, 1, 0)
x = (2, 0, 0, 2, 3) y = (5, 0, 2, 1, 0)
b = (0, 0, 0, 2, 3)
Figure 4.8. The Schensted evolution encoded by Figure 4.7
In the final stage of the above ghost-box-ball evolution, we can construct two se- quences: the number of empty boxes in each region and the number of (red) balls in each region. These two sequences are (5, 0, 2, 1, 0) and (0, 0, 0, 2, 3), which are y and b, respectively. We will prove that this holds in general, by proving a stronger result.
n −1 i Theorem 4.8. Let (a, x) ∈ R and let G = C ◦ φRSK→BBS(a, x). Let (r )i be as defined in Definition 4.7. At the i-th stage of the ghost-box-ball evolution of G, the following holds for each j ∈ [n]: 121
i (1) aj is equal to the number of blue (unmoved) balls in the j-th region,
i (2) bj is equal to the number of red (moved) balls in the j-th region,
i (3) xj is equal to the number of empty boxes in the j-th region.
Proof. We prove this by induction on i.
The base case (i = 0) is satisfied by construction: the initial ghost-box-ball state and wall structure were built out of the RSK input variables so that (1) and (3) are
0 satisfied, and (2) holds trivially because nothing has been bumped yet (so bj = 0 for each j) and no balls have been moved yet (so all balls, if any, are blue).
Let us now suppose that, at some stage, say the k-th stage, we have for each j ∈ [n]:
k (1) aj is equal to the number of blue (unmoved) balls in the j-th region of the k-th step in the GBBS algorithm,
k (2) bj is equal to the number of red (moved) balls in the j-th region of the k-th step in the GBBS algorithm,
k (3) xj is equal to the number of empty boxes in the j-th region of the k-th step in the GBBS algorithm.
We now consider the (k + 1)-st stage of the ghost-box-ball evolution. We need to show that the equalities hold for the region the ball was in and the region the ball moves to (unless it moves to the infinite region).
Suppose the ball we moved was in Region j. We first make the observation that the ball cannot move within Region j: the walls were placed so that each region either has no blue balls or the chain of blue balls is precisely the last part of the region (if there are blue balls, the wall comes right after the last of them). Therefore, the ball must move to Region l, for some l > j (which may be the infinite region). 122
With the observation, we can immediately check the counts for Region j after the the ball is moved: the number of red balls has not changed (by the observation), the number of blue balls has decreased by 1, and the number of empty boxes has increased by 1. We therefore need to show:
k+1 k aj = aj − 1
k+1 k bj = bj
k+1 k xj = xj + 1.
By the induction hypothesis, for us to be able to move a ball from Region j, we must
k k have had aj ≥ 1 and, since the ball was the left-most unmoved ball, aj0 = 0 for all j0 < j. Thus, in terms of RSK, this means we are inserting a j into the current row
k k k k for x . This reduces aj by one (since the j is inserted into x ), increases xj by one k (because the j finds a place in the row, and does not alter bj , since a number can only bump a number greater than itself). We see, therefore, the validity of the counts (in terms of the GBBS and the RSK insertion) agree for Region j.
Now we split into two cases (based on the destination of the ball):
(Case 1): The ball lands in some finite region, say Region l, where n ≥ l > k
(Case 2): The ball lands in the infinite region.
(Case 1): Since the ball moves to the left-most empty box to its right, all spaces between the ball’s origin and its new box must be full or ghosts. In particular, at the k-th stage, there were no empty boxes in the regions between Region j and Region
k k l. By the induction hypothesis, xm = 0 for all j < m < l and xl ≥ 1 (i.e. the j to be inserted bumps an l). In terms of Region l, we lose one empty box and gain a red 123
ball (there is no change to the number of blue balls here). We therefore need to show:
k+1 k al = al
k+1 k bl = bl + 1
k+1 k xl = xl − 1.
This is clearly the case, since a j is bumping an l. (Case 2): By the same reasoning in Case 1, there must be no empty boxes in any
k of the finite regions beyond the j-th, so xm = 0 for all j < m ≤ n. By the induction hypothesis, there are no numbers in the row that are strictly greater than j. Therefore, in this RSK insertion step, we do not bump anything. Instead, we extend the row by a box and fill it with the j. Since the theorem does not contain any conditions relating the current RSK step and the infinite region of the ghost-box-ball system, we have nothing more to check here. Simply by having the only changes be
k+1 k aj = aj − 1
k+1 k bj = bj
k+1 k xj = xj + 1. establishes that nothing has been bumped and that the row has been extended by a j; only the variables/counts for Region j are affected in this case.
We now have the following immediate corollary:
Corollary 4.9. One has
−1 RSK = φBBS→RSK ◦ C ◦ %ˆ ◦ C ◦ φRSK→BBS.
Proof. Since, at each stage, the RSK triple ri is encoded in the i-th stage of the ghost-box-ball evolution, and no blue balls remain at the end of the ghost-box-ball evolution, only empty boxes, empty ghosts, filled ghosts and red balls are left. One can simply apply the coordinate mapping C : GBBS → G and read off the RSK variables
(using φBBS→RSK) to find the output pair (b, y) for the RSK insertion a → x. 124
4.6 The Ghost-Box-Ball Evolution and the Extended Box-Ball Coordinate Dynamics
By exploiting the connection between geometric RSK and discrete-time Toda, we were able to obtain a direct interpretation of Schensted insertion as what essentially amounts to a box-ball evolution: literally the box-ball evolution on the coordinate level, or an extension of the box-ball evolution (the ghost-box-ball system). Whilst we have attained the desired connection between RSK and the Toda lattice (at the ultradiscrete level), it is natural to ask if the box-ball coordinate dynamics still de- scribes the ghost-box-ball evolution on ghost-box-ball systems that do not arise from an RSK pair, i.e. those not in the image of φRSK→BBS.
In this section, we provide a partial answer to this question simply by utilising the results of the previous sections.
From Corollary 4.3, we have
RSK = φBBS→RSK ◦ ξ ◦ φRSK→BBS (4.6.1) and Corollary 4.9 establishes
−1 RSK = φBBS→RSK ◦ C ◦ %ˆ ◦ C ◦ φRSK→BBS. (4.6.2)
Since φRSK→BBS is bijective, one obtains from the above
−1 φBBS→RSK ◦ ξ = φBBS→RSK ◦ C ◦ %ˆ ◦ C . (4.6.3)
Conjecture 4.10. The equality
ξ = C ◦ %ˆ ◦ C−1 (4.6.4) holds in general. 125
Remark 4.2. Purely by leveraging the relation to RSK of each of the maps ξ and %ˆ, the closest we get is the upcoming theorem (Theorem 4.11).
0 Definition 4.9. For z = (Q1 = 0,W1,Q2,W2,...,Qn,Wn,Qn+1, ∞) ∈ Bn, define 0 ζn : Bn → N0 by n+1 X ζn(z) = Qj. (4.6.5) j=1
0 Extend this to ζ : B → N0 in the natural way.
Since this counts the number of balls in the system with coordinates z, this should
t+1 be conserved. To see that this is the case, consider the Qn+1:
n+1 n ! t+1 t X t X t+1 Qn+1 = min Wn+1, Qj − Qj (4.6.6) j=1 j=1 n+1 n ! X t X t+1 = min ∞, Qj − Qj (4.6.7) j=1 j=1 n+1 n X t X t+1 = Qj − Qj , (4.6.8) j=1 j=1
Rearranging this shows that ζ is invariant under the coordinate dynamics on B0.
0,m −1 Theorem 4.11. For each m ∈ N0, let B = ζ (m). On this level set of ζ,
φBBS→RSK is injective, and one has
ξ = C ◦ %ˆ ◦ C−1 (4.6.9) when restricted to B0,m.
Proof. For z = (Q1 = 0,W1,Q2,W2,...,Qn,Wn,Qn+1, ∞), one has
φBBS→RSK(z) = ((0,Q2,...,Qn−1), (W1,...,Wn)). (4.6.10) 126
0,m Clearly the obstruction to injectivity is in losing the data of Qn+1. However, on B , one has n+1 n X X Qn+1 = m − m + Qn+1 = m − Qj + Qn+1 = m − Qj. (4.6.11) j=1 j=1
0,m Thus, when restricted to B , φBBS→RSK is injective and therefore has a left inverse. We post compose Equation 4.6.3 by this left inverse to complete the proof.
4.7 Fukuda: Remarks and Distinctions
We conclude this chapter by drawing the reader’s attention to AppendixA, which follows the work of Fukuda [Fu04]. In AppendixA, we discuss how Schensted inser- tion can be encoded in an advanced box-ball system with carrying capacities and ball colours. We draw attention to this work to address what already exists in the liter- ature, and to highlight the key differences in our work. In particular, we emphasise the appeal of having captured Schensted insertion in what is essentially only a slight deviation from the original box-ball system (in the sense that it is governed by the original box-ball coordinate evolution), rather than having to add the complexity of box labels and capacities to the box-ball system.
Both the ghost-box-ball and advanced box-ball (with uniform carrying capacity 1) systems reproduce Schensted/RSK insertion, but these two systems are fundamen- tally different. On the one hand, the ghost-box-ball system exhibits what we call zero solitons: particles with velocity zero, whereas no such zero soliton exists in the advanced box-ball system. Conversely, prioritisation of certain balls over others (via colouring/labelling) in the advanced box-ball system is not something currently in the ghost-box-ball system.
It would be interesting to study the applications of a hybrid of the two systems. As far as Schensted insertion alone is concerned, there is an obvious appeal to the ghost-box- 127
ball system over the advanced box-ball system: the former is simply a manifestation of the original (unlabelled, carrying capacity one) box-ball-system, originally intro- duced by Takahashi and Satsuma (at least this is true for ghost-box-ball systems created from an RSK input pair and conjectured for general ghost-box-ball systems). The advanced box-ball system is not governed by such simple equations. Therefore, purely from the standpoint of simplicity, the attraction of the ghost-box-ball system is clear.
Moving beyond just Schensted insertion, we have not investigated potential other applications of the ghost-box-ball system. In [Fu04], it is shown that there is a bijection between advanced box-ball systems and generalised permutations, and so, via the RSK correspondence, a pair of semistandard Young tableaux can be associated to an advanced box-ball system. It is showns that the Q tableau, under the time evolution of the advanced box-ball algorithm, is precisely what is known as the box- label algorithm, and that the P tableau is actually a conserved quantity of the time evolution. 128
Chapter 5 Geometric RSK and the Toda Lattice: The Continuous-Time Picture
In this chapter we build the foundations for describing the full geometric RSK al- gorithm as a kind of tower of stroboscopes associated to a sequence of continuous Toda lattices of increasing dimension. These stroboscopes are again seen to be given by a type of B¨acklund transformation known as a dressing transformation. The fact that the succession of stages corresponds to the tridiagonal Toda lattices is an abso- lutely crucial element for this description. We give a novel proof of this fact that is self-contained and based on Painlev´eanalysis and the geometry of flag manifolds.
5.1 Geometric Lifting and Path Operators
In this section, we will describe how the Toda flow and certain generalisations (from the tridiagonal Hessenberg Toda lattice, to full Hessenberg (general) Toda lattice and randomisations) can be described as a flow on upper triangular matrices.
Using the Lie algebra decomposition g = n− + a + n+, where a is the Cartan algebra
of diagonal matrices and g and n± are as defined back in Definition 2.1, one has the
Gauss decomposition g = [g]−[g]0[g]+ with [g]± ∈ N± and [g]0 ∈ A, which is defined on an open dense subset of G.
−1 Lemma 5.1. [BBO09] Let n ∈ n+, then one has [h nh]+ = n for all h ∈ N− if and L only if n belongs to the vector space Cei generated by ei = Ei,i+1. i In what follows, let η = η(t) be a path in the Cartan algebra a and let b = b(t) be a 129 solution to the differential equation
d d b = η + n b (5.1.1) dt dt L where n ∈ Cei. i Proposition 5.2. [BBO09] Let g ∈ G, and assume that bg has a Gauss decomposi- tion, then the upper part [bg]0+ := [bg]0[bg]+ in the Gauss decomposition of bg satisfies the equation d d [bg] = T η + n [bg] (5.1.2) dt 0+ dt g 0+ where Tgη(t) is a path in the Cartan algebra.
Proof. To see this, begin with a tautological rewriting of Equation 6.1.3:
d d ([bg] [bg] ) = η + n [bg] [bg] (5.1.3) dt − 0+ dt − 0+ and rewrite it in the form
d d d [bg] [bg] + [bg] [bg] = η + n [bg] [bg] . (5.1.4) dt − 0+ − dt 0+ dt − 0+
−1 −1 Left multiply Equation 5.1.4 by [bg]− and right the result by [bg]0+:
d d d [bg]−1 [bg] + [bg] [bg]−1 = [bg]−1 η + n [bg] . (5.1.5) − dt − dt 0+ 0+ − dt −
Projecting into n+ and applying Lemma 5.1, we see that
d −1 −1 d [bg]0+ [bg]0+ = [bg]− η + n [bg]− = n. (5.1.6) dt + dt + 130
Therefore, d −1 −1 d [bg]0+ [bg]0+ = [bg]− η + n [bg]− + n. (5.1.7) dt dt 0 R t −1 d Setting Tgη = 0 [bg]− η + n [bg]− ds yields ds 0
d d [bg] = T η + n [bg] (5.1.8) dt 0+ dt g 0+
From now on, assume n = ε and g ∈ N−. Performing manipulations similar to those in the proof above yields
d d d [g−1bg]−1 [g−1bg] + [g−1bg] [g−1bg]−1 = [g−1bg]−1g−1 η + ε g[g−1bg] . − dt − dt 0+ 0+ − dt − (5.1.9)
If we also assume that η(t) = λt is the linear path, then as g ranges over all of N−,
d g−1 η + ε g = g−1ε g (5.1.10) dt λ
ranges over all of (ε + b−)λ by Kostant’s theorem (Theorem 2.6).
ελt −1 We also recall that b(t) = e . Thus, if X0 := g ελg, then
g−1bg = g−1eελtg = eX0t. (5.1.11)
Thus, the right-hand side of Equation 5.1.9 is then
X0t −1 −1 X0t X0t −1 X0t [e ]− g ελg[e ]− = [e ]− X0[e ]−. (5.1.12) 131
So, by Theorem 2.5, the right-hand side of Equation 5.1.9 represents the solution of all
Toda equations on (ε + b−)λ, in general (i.e. for tridiagonal Toda, full Kostant-Toda, or anything in between).
5.2 Lusztig Parameters and Total Positivity 5.2.1 Lusztig Parameters
Definition 5.1. The set of totally positive lower unipotent matrices, denoted (N−)>0,
is the set of all matrices in N− such that each minor that is not automatically zero, as a result of the matrix being lower unipotent, is positive. If this is relaxed to the condition that all minors are nonnegative, then the resulting set, denoted N≥0, is called the set of totally nonnegative lower unipotent matrices.
Remark 5.1. There are many nice ways of defining these objects, one of which is due to Fomin and Zelevinsky [FZ00] in terms of weighted planar networks. However, the following parametrisation proposition [O13], parametrising totally positive lower unipotent matrices in terms of so-called Lusztig parameters, will be key in the defini- tions to follow. This parametrisation is given in terms of the matrices in the following definition.
Definition 5.2. For 1 ≤ i < n, let li : R → SL(n) be defined by li(a) = In +
aEi+1,i. The matrices li(a) are known as the elementary lower uni-triangular Ja- m cobi matrices. For u = (u1, . . . , um) ∈ R , with 1 ≤ m < n, define Lm(u) =
lm(um)lm−1(um−1) ··· l1(u1).
Proposition 5.3. [O13] Each L ∈ (N−)>0 can be written uniquely as a product
1 2 n−1 L = L1(u )L2(u ) ··· Ln−1(u ) (5.2.1)
m m where each u ∈ (R>0) for each m 132
λ 5.2.2 The Flow Rt on P
Definition 5.3. Let P be the set
m P = {b ∈ B+ : ∆k (b) > 0, 1 ≤ k ≤ m ≤ n} (5.2.2)
where
m ∆k (b) = det[bij]1≤i≤k, m−k+1≤j≤m. (5.2.3)
m where, by convention, ∆0 (b) = 1. I.e. the set of upper triangular matrices with the property that every connected subma- trix, with first entry in the first row of the original matrix, has positive determinant.
m (∆k (b) is the determinant of the connected k ×k submatrix of b whose top-right entry is b(1, m))
Remark 5.2. One also has that P can be defined equivalently as the totally positive part of the double Bruhat cell B+ ∩ B−w¯0B−.
Definition 5.4. Let φi : SL(2) → G be the embedding given by inserting g ∈ SL(2) into the ((i, i), (i, i + 1), (i + 1, i), (i + 1, i + 1)) block of the n × n identity matrix. For 0 −1 an adjacent transposition s = (i i + 1) ∈ S , let s¯ = φ . Such a matrix i n i i 1 0 is called a fundamental reflection.
Definition 5.5. Let w = si1 ··· sir be a reduced decomposition of w ∈ Sn into ad- jacent transpositions, then define w¯ :=s ¯i1 ··· s¯ir . Denote the longest Weyl group element by w0, which is expressible in terms of adjacent transpositions as w0 = s1(s2s1)(s3s2s1) ··· (snsn−1 ··· s3s2s1), and let w¯0 be the corresponding matrix for this reduced decomposition.
Berenstein, Fomin and Zelevinsky [BFZ96] work out not only the Lusztig parameters, but also formulæ completely describing the LDU-decomposition (the factorisation
corresponding to N−AN+) for the right-translate of P byw ¯0, Pw¯0. 133
Proposition 5.4. [BFZ96] Let b ∈ P. Then bw¯0 has a Gauss (LDU) decomposition
bw¯0 = LDU where n ∆i (b) Dii = n , 1 ≤ i ≤ n (5.2.4) ∆i−1(b) and L ∈ (N−)>0 is given by
1 2 n−1 L = L1(u )L2(u ) ··· Ln−1(u ) (5.2.5)
m m+1 m ∆i−1(b)∆i+1 (b) where ui = m m+1 for 1 ≤ i ≤ m < n. ∆i (b)∆i (b)
O’Connell shows that the flow given by Equation 6.1.3 in Section 5.1 maps reduces
m to a flow on P, i.e. the positivity of the minor determinants ∆k (b) is preserved. In what follows, we only consider the flow given by the linear path η(t) = λt, for
n λ = (λ1, . . . , λn) ∈ R . The flow is then the solution to the differential equation d b = ε b, (5.2.6) dt λ where we recall that ελ = diag(λ1, . . . , λn) + ε, as in Definition 2.6.
Then, since ελ is constant, if one allows an arbitrary initial condition b(0) = b0 ∈ P, then one obtains the solution
ελt b(t) = e b0. (5.2.7)
The following notation is introduced for this flow on P:
n λ Definition 5.6. For λ ∈ R , define the map Rt : P → P by
λ ελt Rt (b) = e b. (5.2.8)
5.3 Tw0 and the Linear Path η(t) = λt
Recall the differential equation in Equation 5.3.1 of Proposition 5.2: 134
d d [bg] = T η + n [bg] (5.3.1) dt 0+ dt g 0+
If one takes the linear path η(t) = λt and g = w0 the longest element, with fixed
reduced decomposition w0 = (s1)(s2s1) ··· (snsn−1 ··· s1), and n = ε, one obtains the
path Tw0 λt. Recalling how Tw0 η was defined, we first solve d b(t) = ε b(t) (5.3.2) dt λ with b(0) = Id.
This flow b(t), due to its relation to the Pitman operator Pw0 , is of major interest, and is in fact the focus of the final theorem (Theorem 8.5) of [O13]. They show that the flow defined by Equation 5.3.2, under certain conditions on b ∈ B+, can be transported over to Mλ, resulting in the classical (tridiagonal) Toda flow. In
Chapter6, we show in Theorem 6.13 that this generalises to a larger subset of B+ to yield general Toda lattice on Fλ, i.e. a flow that is not simply on tridiagonal matrices.
A significant part of the statement of Theorem 8.5 in [O13] is that the flow induced on (ε + b−)λ by Equation 5.3.2, with initial condition b(0) = Id, is a classical (tridiag- onal) Toda flow. We offer an alternative and much simpler, self-contained, integrable systems theoretic proof in this section, using the results on Painlev´ebalances in [EFH91]. 135
5.3.1 Painlev´eBalances
Recall the Toda lattice in Flaschka variables:
˙ bj = (aj+1 − aj)bj, j = 1, . . . , n − 1 (5.3.3)
b1 if j = 1 a˙ j = bj − bj−1 if 1 < j < n . (5.3.4) −bn−1 if j = n which one can rewrite equivalently as ˙ bj = (aj+1 − aj)bj, , j = 1, . . . , n, (5.3.5) a˙ j = bj − bj−1, by adopting the convention that b0 = bn = an+1 = 0.
Proposition 5.5. [EFH91] Let Θ ⊆ {1, . . . , n − 1} be nonempty. There is a Laurent series solution of Equations 5.3.5 having the form σΘ b (t) = − j + (Taylor) near t = 0, if j ∈ Θ, (5.3.6) j t2
bj(t) = (Taylor) near t = 0, if j 6∈ Θ. (5.3.7)
This solution depends on 2(n − 1) − |Θ| free parameters.
Each type of series in the proposition is called a Painlev´ebalance Θ of dimension equal to the number of free parameters.
Corollary 5.6. [EFH91] When Θ = {1, . . . , n−1}, which is called the lowest balance, its dimension is 2(n − 1) − (n − 1) = n − 1. 5.3.2 The Birkhoff Decomposition
Definition 5.7. The Birkhoff decomposition of the flag manifold G/B+ is the decom- position [ G/B+ = N−wB+/B+, (5.3.8) w∈W 136
where W is the Weyl group of G.
Remark 5.3. This decomposes the flag manifold into Birkhoff cells, N−wB+/B+, with dimension n(n − 1) dim(N wB /B ) = − `(w) (5.3.9) − + + 2
and where `(w) is the length, in the usual sense for the symmetric group, of the permutation in Sn to which w corresponds, i.e. the minimal number of transpositions needed to write the permutation as a product of adjacent transpositions.
The two extreme cases are for w = 1 and w = w0:
For w = 1, one obtains the “big cell”: n(n − 1) dim(N B /B ) = (5.3.10) − + + 2
which is in fact an open, dense subset of G/B+.
For w = w0 (the longest word), one obtains the “smallest cell”: n(n − 1) n(n − 1) dim(N w B /B ) = − = 0 (5.3.11) − 0 + + 2 2
which is just a point in G/B+. 5.3.3 Passage to General Toda: The Crystal Embedding
In what follows, we fix the spectrum λ = (λ1, . . . , λn).
In Section 6.5, we will present in full detail how one obtains the general Toda flows on ε + b− using O’Connell’s dynamics on the Richardson cell P, providing a proof in Theorem 6.13. For now, we will just state this passage to the general Toda lattice:
λ One begins with an initial matrix b0 ∈ P and defines the flow b(t) = Rt b0. For this simple flow, corresponding to the linear path η(t) = λt, we write this explicitly as
ελt b(t) = e b0. (5.3.12) 137
Defining a flow on N− via
L(t) = [b(t)w ¯0]−, (5.3.13) we map b(t) to a flow on the isospectral set of Hessenberg matrices with spectrum λ:
−1 b(t) 7→ L(t) ελL(t) ∈ (ε + b−)λ. (5.3.14)
We show in Theorem 6.13 that this flow is a general Toda flow.
We now define the crystal embedding, a modification of Definition 2.7, which is the embedding into the flag manifold utilised in [EFH91].
Definition 5.8. For X ∈ (ε + b−)λ, let L ∈ N− be the unique lower unipotent matrix such that
−1 X = LελL . (5.3.15)
The crystal embedding is defined to be the map
jλ :(ε + b−)λ → G/B+
−1 X 7→ L mod B+.
Given the flow b(t) on P, it is natural to ask what the image of the corresponding crystal embedded Toda flow is in terms of b(t). The answer is simple, given Equation 5.3.14 and Definition 5.8:
Lemma 5.7. Under the crystal embedding, the flow on the flag manifold arising from b(t) is
b(t)w ¯0 mod B+. (5.3.16) 138
Proof. Comparing Equation 5.3.14 and Definition 5.8, one sees that the
−1 jλ(L(t) ελL(t)) = L(t) mod B+ (5.3.17)
= [b(t)w ¯0]− mod B+ (5.3.18)
= [b(t)w ¯0]−[b(t)w ¯0]0+ mod B+ (5.3.19)
= b(t)w ¯0 mod B+ (5.3.20)
where 5.3.19 is due to [b(t)w ¯0]0+ ∈ B+.
We conclude this treatment of the crystal embedding with the following result of [EFH91]:
Lemma 5.8. ([EFH91]) The lowest balance Θ = {1, . . . , n − 1} corresponds under jλ to the smallest Birkhoff cell.
5.3.4 Tridiagonality of the Flow Associated to b(t) = eελt
We now prove part of Theorem 8.5, the part asserting the tridiagonality of the general
Toda flow induced by b(t) = eελt, using an integrable systems-theoretic proof.
n ελt −1 Theorem 5.9. Let λ ∈ R , b(t) = e , and let X(t) = L(t) ελL(t) for t > 0 be the induced general Toda flow on (ε + b−)λ, where L(t) = [b(t)w ¯0]−. Then X(t) is a classical (tridiagonal) Toda flow, which is singular at t = 0.
Proof. The singularity of the flow at t = 0 is immediate: when t = 0, one has b(t) = b0 = Id. Thus, b0w¯0 =w ¯0, which has no LU-decomposition (for n > 1, at least).
To elucidate this, we observe that with λ fixed, the flows on the flag manifold are extended for all time, without singularities ([EFS93], [FH91]). The image of X(0), which is singular in (ε + b−)λ, in the flag manifold, is, by Lemma 5.7, given by b(0)w ¯0 mod B+ =w ¯0 mod B+ which is in the smallest cell of the Birkhoff decomposition of
G/B+. 139
Therefore, by existence and uniqueness, since the smallest Birkhoff cell is a point, the flow on the flag manifold, given by embedding X(t), is the unique flow passing through this point in the flag manifold.
We conclude the proof by recalling Corollary 5.6, which states that the dimension of the lowest balance, to which the smallest Birkhoff cell corresponds (by Lemma 5.8), is n − 1. After fixing the spectrum, losing one degree of freedom to the trace zero condition, this fixes all n − 1 of the free parameters. Thus, this unique Laurent series solution to the tridiagonal Toda Equations 5.3.5, with lowest balance and spectrum λ, is necessarily the Toda flow given by X(t).
5.4 The gRSK Stroboscope and a Nesting of Toda Lattices
It is natural to expect there to be a continuous-time analogue of Noumi and Yamada’s hierarchical description of geometric RSK (Theorem 2.22). This is indeed the case and was developed in [O13], Prop. 3.1, using a continuous-time analogue of the Lind- str¨om-Gessel-Viennotformula [GV85] supplemented by the Karlin-McGregor formula [KM59]. However, based on what has been developed in this chapter, the result may be stated completely independently of those technicalities and will be presented in that way here. We conjecture that the proof, in fact, may be alternatively given in terms of the dressing transformation approach developed in this chapter at least for the case when η(t) is linear. We mention also that the result stated in this section underlies and is the starting point for what will be presented in Sections 6.1- 6.3 of the next chapter.
(k) Definition 5.9. Let 1 ≤ k ≤ n. For b ∈ B+, define b to be the principal k × k (k) submatrix of b (i.e. the top-left k × k submatrix of b). For the matrix w¯0, define w¯0 to be the bottom-left k × k submatrix of w¯0. 140
Lemma 5.10. For b(t) solving Equation 5.3.2 with b(0) = I, one has
(k) ε(k)t b (t) = e λ . (5.4.1)
(k) Proof. Write ελ in the 2 × 2 block structure with its first block equal to ελ : ε(k) ∗ λ ελ = . (5.4.2) 0 ∗
It follows that (k) j (ελ ) ∗ εj = . (5.4.3) λ 0 ∗ for all j ∈ N ∪ {0}. It then follows immediately, by substituting this into the Taylor series for the exponential function, that
(k) ε t (k) ε(k)t b (t) = (e λ ) = e λ . (5.4.4)
We can now state the main result.
Theorem 5.11. Let b(t) solve Equation 5.2.6. For each 1 ≤ k ≤ n, the vector
(k) k,1 k,k (k) y = (y , . . . , y ) := log[b w¯0]0 satisfies a gradient flow that is equivalent to the Toda flow. This final output, for k = n, is y(n) which is the output of geometric RSK (a discrete time process), initialised with input word λ, and hence y(n) = (yn,1, . . . , yn,n) corresponds to the geometric P -tableau.
Furthermore, the lower unipotent matrix L(t) = [bw¯0]− is the image of the crystal embedded Toda flow corresponding to y(n), and, by Lemma 5.10, this holds at all
(k) (k) (k) levels, so that L (t) := [b w¯0 ]− gives the crystal embedded Toda at each level, with each described by its corresponding Lusztig parameters in Proposition 5.4.
−1 Lastly, the process above yields the Hessenberg matrix X(t) = L(t) ελL(t) which is 141
a known as a dressing transformation of ελ and gives a solution to the classical Toda lattice.
Recall that the discrete-time Toda lattice is a time discretisation of the Deift-Nanda- ˙ Tomei [DNT83] / Symes [Sy80] log flow (i.e. for X = [X, π− log(X)]), and we showed in Section 3.4 how one can view geometric RSK in terms of this discrete flow.
We also recall that RSK is a series of successive Schensted insertions, and so this lifts to the geometric RSK setting. The Figure 2.6 from earlier helps us to visualise this:
x1 = x1,1 x2 = x2,1 x3 = x3,1
∅ y1,1 y2,1 ···
∅ x2,2 x3,2
∅ y2,2 ···
∅ x3,3
∅ ···
∅
. ..
Figure 5.1. Iterative Schensted word insertions. In the geometric setting, the y’s here are precisely the y’s in Theorem 5.11.
The coincidence of this continuous-time (gRSK) insertion diagram with that of the discrete-time (RSK) insertion diagram, together with the results alluded to at the end of 3.4, motivates us to make the following conjecture:
Conjecture 5.12. A discrete-time stroboscope of the chopped flows in Theorem 5.2.6 reproduces (full, as in the above figure) geometric RSK. 142
Chapter 6 The Full Kostant-Toda Lattice
In this chapter we start off (Sections 6.1- 6.3) reviewing O’Connell’s approach to establishing a relation between gRSK and the Toda lattice, for which our work, de- scribed in Chapter5, provides an alternate treatment.
One element appearing here that we have not seen yet are Gelfand-Tsetlin triangles. Such objects have played a significant role in the representation theory of compact Lie groups from the perspective of orbit theory [GS83]. This has potentially interesting interpretations related to Poisson and symplectic geometry though this is not brought out in [O13]. In Sections 6.4- 6.6, we are able to extend O’Connell’s picture to the setting of the full Kostant-Toda lattice and in particular enhance the significance of the Gelfand-Tsetlin triangles from an integrable systems perspective related to non- compact Lie groups.
In section 6.6 we start to lay the groundwork for the aforementioned connections to Poisson geometry.
6.1 Triangular Arrays and the Gelfand-Tsetlin Parametrisation
Definition 6.1. There is a bijection f : P → T , which is given by defining T =
m m m m m (ti ) ∈ T via t1 + ··· tk = log ∆k (b), for b ∈ P, where ∆k (b) are certain mi- nor determinants of b. The map f gives what is referred to as the Gelfand-Tsetlin parametrisation of P by triangles T .
Recalling Proposition 5.4: 143
Proposition 6.1. [BFZ96] Let b ∈ P. Then bw¯0 has a Gauss (LDU) decomposition
bw¯0 = LDU where n ∆i (b) Dii = n , 1 ≤ i ≤ n (6.1.1) ∆i−1(b) and L ∈ (N−)>0 is given by
1 2 n−1 L = L1(u )L2(u ) ··· Ln−1(u ) (6.1.2)
m m+1 m ∆i−1(b)∆i+1 (b) where ui = m m+1 for 1 ≤ i ≤ m < n. ∆i (b)∆i (b)
n m+1 m ti m ti+1 −ti If b = f(T ), then Dii = e for 1 ≤ i ≤ n and ui = e for 1 ≤ i ≤ m < n. Recall from Section 5.1 that a path in the Cartan algebra, say η(t), can be geomet- rically lifted to a path on B+ by solving the following differential equation:
d d b = η + ε b (6.1.3) dt dt
P where we now take ni = 1 for all i in n = i niei, to yield n = ε. We will also allow λ b0 = b(0) to take arbitrary initial values in P. This generalises the flow Rt in Section η 5.1 to a flow Rt on P.
η Definition 6.2. The dynamics Rt on P, under the bijection f : P → T , induces a η dynamics on T , which O’Connell [O13] denotes St . This is defined as
η η −1 St = f ◦ Rt ◦ f . (6.1.4)
m Proposition 6.2. ([O13]) Under this dynamics, the triangle T = (ti ) evolves as t˙1 =η ˙ , 1 1 m−1 m m−1 ˙m ˙ t2 −t1 t1 = t1 + e , 2 ≤ m ≤ n (6.1.5) tm−tm−1 t˙m =η ˙ − e m m−1 , 2 ≤ m ≤ n m m m−1 m m−1 m m−1 ˙m ˙ ti+1−ti ti −ti−1 ti = ti + e − e , 1 < i < m ≤ n. 144
Definition 6.3. The path η(t) in the Cartan algebra defines a path Π: η → (T (t), t > η 0) in the set of triangles via Rt and f:
Π(η) = (f(b(t)), t > 0), (6.1.6) i.e. T (t) = f(b(t)).
Remark 6.1. Despite this generalisation, the type of path that will ultimately be of interest is still the linear path η(t) = λt.
6.2 Continuous-Time gRSK and Dynamics on T and P
We continue to follow [O13] here, but with results and definitions restricted to our setting, the deterministic one.
The mapping Π : η 7→ (T (t), t > 0), where η : [0, ∞) → Rn is a continuous path with η(0) = 0 is defined by taking T (t) = f(b(t)), where b(t) solving 6.1.3. In [O13], the mapping Π : η → (T (t), t > 0) is thought of as a continuous analogue of gRSK, where for each t > 0, one has the following analogies:
the path (η(s), 0 ≤ s ≤ t) is thought of as the input word,
m the triangle T (t) = (ti (t), 1 ≤ i ≤ m ≤ n) is thought of as the P -tableau,
the vector tn(t) is thought of as the shape of P ,
the path (tn(s), 0 < s ≤ t) is thought of as the Q-tableau. Note: this set contains the history of bottom rows (the shapes) of the triangles up to, and including, the current triangle, T (t).
O’Connell also defines a mapping Πξ : η 7→ (T (t), t > 0), which, under the same analogies, is thought of as inserting η into an initial triangle ξ ∈ T . O’Connell
η η ξ constructs a flow on triangles, St : T → T , where St ξ = Π η. 145
With the original full generality, O’Connell was able to extend the results that will follow through to geometric RSK with random input, ultimately making the connec- tion to the quantum Toda lattice. For our purposes, we will focus on the deterministic model, but will return to the stochastic setting when discussing future directions.
n m Definition 6.4. For η : [0, ∞) → R smooth, let T = (ti ) ∈ T evolve according to
˙1 t1 =η ˙1 (6.2.1)
m m−1 ˙m ˙m−1 t2 −t1 t1 = t1 + e (6.2.2)
m m−1 ˙m tm−tm−1 tm =η ˙m − e , 2 ≤ m ≤ n, (6.2.3)
m m−1 m m−1 ˙m ˙m−1 ti+1−ti ti −ti−1 ti = ti + e − e , 1 < i < m ≤ n, (6.2.4)
η η and denote this flow on T by St . i.e., if T (0) = ξ, then T (t) = St ξ.
We immediately restrict our focus to paths of the form η(t) = λt, where λ ∈ Rn, λ and, in this case, one write St for the flow defined by the path η(t) = λt. This is the type of path that will ultimately connect to the Toda lattice. Additionally, for this connection to be for the sl(n) Toda lattice, we assume λ1 + ··· + λn = 0. Recall also λ that, in this linear path case, the flow Rt is simply given by left-multiplication by eελt.
6.3 The Set Tλ
m Definition 6.5. Viewing a triangle T = (ti ) ∈ T as a directed graph (quiver) 146
1 t1
2 2 t2 t1
3 3 3 t3 t2 t1 ......
n−1 n−1 n−1 n−1 tn−1 tn−2 t2 t1
n n ∗ ∗ n n tn tn−1 t∗ ··· t∗ t2 t1
Figure 6.1. The quiver structure on triangular arrays. and then as a weighted directed graph by assigning a weight of ea−b to an arrow a → b,
m m m m define a set of quantities li = li (T ) and ri = ri (T ) for 1 ≤ i ≤ m < n as the m m sum of the weights into the node ti and the sum of the weights out from the node ti . Explicitly, for 1 ≤ i ≤ m < n
( tm+1−tm tm−1−tm m e i+1 i + e i i i < m li = tm+1−tm , (6.3.1) e m+1 m i = m
( tm−tm+1 tm−tm−1 m e i i + e i i−1 i > 1 ri = tm−tm+1 . (6.3.2) e 1 1 i = 1
n Definition 6.6. For λ ∈ R , define the subset Tλ of T to be
m m m Tλ = {T = (ti ) ∈ T : λm + li (T ) = λm+1 + ri (T ), 1 ≤ i ≤ m < n} . (6.3.3)
−1 Also, define Pλ = f (Tλ).
λ Proposition 6.3. [O13] The flows St on T restrict to a flow on Tλ.
With these preliminary definitions and results, we can now proceed onto O’Connell’s connection to Toda. The key lies in a theorem of Kostant [Ko78]. 147
6.3.1 The Connection to the Toda Lattice
Theorem 2.6 plays the main role in O’Connell’s connection to the Toda lattice. To map from P and T to the isospectral set of tridiagonal Hessenberg matrices, denoted
Mλ, O’Connell constructs a map gλ : T → Mλ to carry the flow on Tλ to a flow on
Mλ. Surprisingly, the result is the Toda lattice.
Definition 6.7. Let Fλ = (ε + b−)λ, M the set of tridiagonal Hessenberg matrices,
and Mλ = M ∩ Fλ the isospectral submanifold of tridiagonal matrices.
Definition 6.8. Define a map h : T → (N−)>0 by setting
1 2 n−1 h(T ) = L1(u )L2(u ) ··· Ln−1(u ) (6.3.4)
where
m+1 m m ti+1 −ti ui = e , 1 ≤ i ≤ m < n. (6.3.5)
Definition 6.9. Define a map gλ : T → M by setting p1 1 .. −q1 p2 . g (T ) = (6.3.6) λ . . .. .. 1 −qn−1 pn where
tm −tm qi = e i+1 i , 1 ≤ i ≤ n − 1, (6.3.7)
n m and pi = pi where pi are defined recursively by
m m−1 m m−1 t2 −t1 p1 = p1 + e (6.3.8)
n n−1 m tn−tn−1 pm = λm − e (6.3.9)
m m−1 m m−1 ti+1−ti pi = pi + e , 1 < i < m. (6.3.10) 148
Proposition 6.4. [O13] If T ∈ Tλ, then
−1 gλ(T ) = L ελL, (6.3.11) where L = h(T ).
−1 Remark 6.2. A key observation made by O’Connell is that h(T ) ελh(T ) ∈ Mλ if and only if T ∈ Tλ. Therefore, it should be understood that the defining condition for −1 Tλ is exactly the requirement for h(T ) ελh(T ) to be tridiagonal.
−1 Proposition 6.5. [O13] Let T ∈ T and b = f (T ). By Proposition 5.4, bw¯0 has an
LDU decomposition, bw¯0 = LDU, and this L is in fact h(T ).
λ We are finally set up to state O’Connell’s result connecting the flow St , restricted to
Tλ, to the Toda flow on Mλ.
n m λ Theorem 6.6. [O13] Let λ ∈ R , T (0) ∈ Tλ and T (t) = (ti (t)) = St T (0). Let −1 M(t) = gλ(T (t)), b(t) = f (T (t)), and
1 2 n−1 L = L1(u )L2(u ) ··· Ln−1(u ) (6.3.12) where
m+1 m m ti+1 −ti ui = e , 1 ≤ i ≤ m < n. (6.3.13)
Let Q = Π−(M) and P = M − Q. Then, for all t ≥ 0,
−1 M(t) = L(t) ελL(t), (6.3.14) and we have Gauss decompositions
tM(0) b(t)w ¯0 = L(t)R(t), e = n(t)r(t), (6.3.15)
−1 −1 where n(t) = L(0) L(t), R(t) ∈ B+, r(t) = R(t)R(0) and these satisfy
L˙ = LQ, R˙ = PR, n˙ = nQ, r˙ = P r. (6.3.16)
In particular, M(t) defines a solution to the toda flow on Mλ. 149
We will prove in Section 6.5 that one can extend Theorem 6.6 to the full Kostant-Toda lattice, by extension to all of T . Before that, however, we will present some results on
Kostant’s theorem (Theorem 2.6), and its ελ version, for use in relating O’Connell’s
ελ-driven dynamics to the companion version used by [EFS93] and [KS18].
6.4 Flag Manifolds: The Crystal Embedding and the Companion Embedding
We begin by recalling some definitions and results from Section 2.1, starting with
Definition 2.6 of the companion matrix cλ in Fλ = (ε + b−)λ (the set of Hessenberg matrices with spectrum λ): 0 1 . 0 .. .. cλ = . 1 , (6.4.1) 0 1 −c0 −c1 · · · −cn−2 −cn−1
n Q n Pn−1 i where (x − λi) = x + i=0 cix is the characteristic polynomial for λ. i=1
In the work of O’Connell [O13], the focus is on ελ, whereas the version used by [EFS93] and [KS18] is cλ. The latter has the advantage of providing a unique representative that is independent of the choice of ordering on λ.
We also recall Kostant’s theorem:
Theorem 6.7. [Ko78] For each X ∈ Fλ, there exists a unique lower unipotent L ∈
N−, such that −1 X = LcλL . (6.4.2) 150
The same statement holds (with a different L ∈ N−) when cλ is replaced by ελ (with a specific ordering of the λ’s).
Finally, we recall the companion embedding of the Fλ into the flag manifold, κλ :
Fλ → G/B+.
Definition 6.10. The companion embedding is the map κλ : Fλ → G/B+ defined as −1 follows: for X ∈ Fλ, if X = LcλL , then
−1 κλ(X) = L mod B+. (6.4.3)
Remark 6.3. An analogous embedding can be performed using ελ representation of
Fλ. We call this the crystal embedding, due to its direct relation to continuous crystals.
We now turn our attention to the L’s in both versions of Kostant’s theorem, finding explicit formulæ where possible and offering a means of translation between the two by expressing the relationship between the L’s corresponding to ελ and to cλ.
Lemma 6.8. For each n ∈ N, if X is a tridiagonal n × n Hessenberg matrix, and for 1 < k ≤ n, k−1 (k−1) k−1 X i−1 det(xIk−1 − X ) = x + lkix i=1 the n × n lower unipotent matrix L = (lij)i,j defined by the above n − 1 polynomials is the unique such matrix satisfying
−1 L XL = cX
where cX is the companion matrix of X (or cλ, where λ = Spec(X)).
Proof. We prove this by induction:
The base case of n = 1 is trivial: X = [a1], L = [1] and cX = [a1] clearly satisfies
XL = LcX . 151
Let us suppose the result holds for some n ∈ N. To proceed, suppose X is an (n + 1) × (n + 1) tridiagonal Hessenberg matrix. We make the key observation that if L is the conjugating matrix for X, then L(n) is the conjugating matrix for X(n), which follows immediately from the definition of the lki’s. Thus, the induction hypothesis asserts
(n) (n) (n) X L = L cX(n) . (6.4.4)
We impose a block structure on X:
(n) X en X = T bnen an+1 where en is the last column of the n × n identity matrix.
We impose the analogous block structure on L: L(n) 0 L = vT 1
where v = [ln+1,1 ln+1,2 ··· ln+1,n].
A p Let c = L−1XL with block structure c = . Since XL = Lc, one obtains qT r the following equation from the top-left block:
(n) (n) T (n) X L + env = L A, (6.4.5) with A an n × n matrix.
By the invertibility of L(n), there can be only one A satisfying this equation.
Claim. A = εn, where εn is the n × n matrix with 1’s on the superdiagonal and
zeroes elsewhere. This εn is not to be mistaken with ελ. If dλ = diag(λ1, . . . , λn), 152
then ελ = dλ + εn.
Proof of Claim. Using the induction hypothesis, and plugging in A = εn, Equation 6.4.5 becomes
(n) T (n) L cX(n) + env = L εn, (6.4.6) or, equivalently,
(n) T L (εn − cX(n) ) = env . (6.4.7)
Evaluating both sides, one obtains 0 0 ··· 0 0 0 ··· 0 . . . . . . . . = (6.4.8) 0 0 ··· 0 0 0 ··· 0 ln+1,1 ln+1,2 ··· ln+1,n p0 p1 ··· pn−1 where n−1 n n−1 (n) n X k n X i−1 n X i det(xIn − X ) = x + pkx = x + ln+1,ix = x + ln+1,i+1x . k=0 i=1 i=0
Hence, pk = ln+1,k+1 for k = 0, . . . , n − 1. Thus, the ansatz of A = εn was consistent, which proves the claim.
Returning to Equation 6.4.4, we turn our attention to the top-right block:
(n) en = L p. (6.4.9)
(n) −1 (n) −1 Since (L ) is lower unipotent, p = (L ) en = en, since the en is also the last column of (L(n))−1.
One can conclude therefore that this matrix c, given by L−1XL is a companion matrix. Since c conjugate to X, and the characteristic polynomial is invariant under matrix conjugation, one must have that c is indeed the companion matrix for X. This completes the induction step, proving the theorem. 153
−1 This gives a means of computing L in the ελ version of Kostant’s theorem.
Lemma 6.9. Let X ∈ (ε + b−)λ, and let L1 be defined as in Lemma 6.8, and let L2 be the lower unipotent matrix such that for i > j
i+j (L2)ij = (−1) ej−i(λ1, . . . , λi−1) (6.4.10)
where ej−1 is the (j − i)-th elementary symmetric polynomial
X ej−i(λ1, . . . , λi−1) = λk1 λk2 ··· λkj−i , (6.4.11)
1≤k1 −1 −1 then L = L2L1 satisfies X = L ελL. −1 Proof. This is a consequence of Lemma 6.8. One has X = L1cλL1 , and I claim −1 −1 −1 −1 −1 −1 that cλ = L2 ελL2. Thus, X = L1L2 ελL2L1 , and so L = (L1L2 ) = L2L1 . The claim itself is simply an application of Lemma 6.8 since k−1 k X k+i τk(xIn − ελ) = x + (−1) ek−i(λ1, . . . , λk). (6.4.12) i=0 When λi 6= λj for all i 6= j, one can of course diagonalise any matrix in Fλ. The following result, which is an explicit form of Lemma 7 in [EFH91], describes a diago- nalisation of ελ. −1 Lemma 6.10. If λ1, . . . , λn are distinct, then one has ελ = UdλU , where U = (uij) is the upper triangular matrix given by i−1 Y uij = (λj − λk), 1 ≤ i ≤ j ≤ n k=1 and dλ = ελ − ε = diag(λ1, . . . , λn). 154 Proof. The matrix U is clearly invertible if and only if λi 6= λj since n i−1 Y Y Y det(U) = (λi − λk) = (λi − λk). i=1 k=1 1≤k It just remains to show that ελU = Udλ. Let uj = (uij)1≤i≤n be the j-th column of U, then for i < n: n X (ελuj)i = (ελ)ikukj k=1 = λiuij + ui+1,j i−1 i Y Y = λi (λj − λk) + (λj − λk) k=1 k=1 i−1 Y = (λi + λj − λi) (λj − λk) k=1 i−1 Y = λj (λj − λk) k=1 = λj(uj)i. For i = n, we simply have (ελuj)n = λn(uj)n 0 j < n = = λj(uj)n. λj(uj)n j = n Thus, ελuj = λjuj for each j. 6.5 Extension to the Full Kostant-Toda Lattice In O’Connell’s work, everything was restricted to the set Tλ ⊂ T . In this section, we show that the nice forms the objects (h, gλ, etc.) took when restricted to Tλ can na¨ıvely be extended to all of T . When this extension is performed, one finds that (ε + b−)λ takes the place of Mλ, and one recovers the full Kostant-Toda lattice on (ε+b−)λ. We raise some natural questions about this extension, and provide answers for some of them. 155 6.5.1 The Map gˆλ The first task is to extend gλ|Tλ : Tλ → M to a mapg ˆλ : T → Fλ. We define this the na¨ıve way. Definition 6.11. Let gˆλ : T → Fλ be defined by −1 gˆλ(T ) = h(T ) ελh(T ), (6.5.1) where h : T → (N−)>0 is the same as in Definition 6.8. First, we define two Rn actions, one on T and another on P. n Definition 6.12. For r = (r1, . . . , rn) ∈ R , define αr : T → T by adding ri to k+i−1 m m the i-th diagonal, (tk )1≤k≤n−i+1. I.e. T = (ti ) ∈ T , then (si ) = S = αr(T ) is defined by m m si = rm−i+1 + ti (6.5.2) for 1 ≤ i ≤ m ≤ n. Definition 6.13. Let µr : P → P denote right multiplication by the diagonal matrix diag(er1 , . . . , ern ). Remark 6.4. Since exp(diag(r1, . . . , rn)) ∈ B+ ∩ B−, one clearly has µr(b) ∈ B+ ∩ B−w¯0B− for each b ∈ P. Furthermore, since m rm+rm−1+···rm−k+1 m ∆k (µr(b)) = e ∆k (b) (6.5.3) for each 1 ≤ k ≤ m ≤ n, it is clear that total positivity is preserved. Lemma 6.11. For each r ∈ Rn, the following diagram commutes 156 αr T T f f P P µr m Proof. Let b ∈ P and let T = (ti ) ∈ T be the unique triangular array given by solving m m m t1 + ··· + tk = log ∆k (b) (6.5.4) m for 1 ≤ k ≤ m ≤ n, i.e. T = f(b). Applying αr to T yields the triangle αr(T ) = (si ) given by m m m−i+1 si = ti + r . (6.5.5) m Thus, αr(f(b)) = (si ). Traversing the diagram the other way, one has er1 er2 µr(b) = b . . (6.5.6) .. ern r1 rn m Since µr multiplies the columns of b correspondingly by (e , . . . , e ), ∆k (b), which involves precisely columns m − k + 1, m − k + 2, . . . , m, is given by m rm+rm−1+···+rm−k+1 m ∆k (µr(b)) = e ∆k (b) (6.5.7) for each 1 ≤ k ≤ m ≤ n. 157 Now we observe that m m m m m m−k+1 s1 + ··· + sk = t1 + ··· + tk + r + ··· + r (6.5.8) m rm+···+rm−k+1 = log ∆k (b) + log(e ) (6.5.9) m = log ∆k (µr(b)) (6.5.10) for all 1 ≤ k ≤ m ≤ n, which proves that f ◦ µr = αr ◦ f. m m Lemma 6.12. Let T = (ti ) and S = (si ) be triangular arrays in T , then gˆλ(T ) = n gˆλ(S) if and only if S = αr(T ) for some r ∈ R . Proof. Supposeg ˆλ(T ) =g ˆλ(S), i.e. −1 −1 h(T ) ελh(T ) = h(S) ελh(S). (6.5.11) By Theorem 2.6, since these are both Hessenberg matrices with the same spectrum, there is a unique matrix for each that conjugates them to ελ. As this is exactly what is described above, we see that h(T ) = h(S). This reduces us to studying h : T → (N−)>0. m+1 m m+1 m m ti+1 −ti m si+1 −si Let ui = e and vi = e for 1 ≤ i ≤ m < n, then 1 2 n−1 1 2 n−1 L1(u )L2(u ) ··· Ln−1(u ) = L1(v )L2(v ) ··· Ln−1(v ). (6.5.12) By Proposition 5.3, each side is in (N−)>0 and the factorisation is unique, hence we m m conclude that ui = vi for all 1 ≤ i ≤ m < n: m+1 m m+1 m ti+1 −ti si+1 −si m+1 m m+1 m e = e ⇔ ti+1 − ti = si+1 − si . (6.5.13) m+1 m+1 m m Thus, ti+1 − si+1 = ti − si is constant for all 1 ≤ i ≤ m < n. For each 1 ≤ d ≤ m, d d let rd = t1 − s1, then T = α(r1,...,rn)(S). Therefore, two triangles have the same image n in Fλ if and only if they are in the same R orbit, as defined by αr. 158 Theorem 6.13. If T0 ∈ T , then λ λ −1 X(t) :=g ˆλ(St (T0)) =g ˆλ(f(Rt (f (T0)))) satisfies the following: (a) Initial Condition: X(0) =g ˆλ(T0) ˜ ˜ ˜ (b) Well-Defined: If T0, T0 ∈ T satisfy gˆλ(T0) =g ˆλ(T0), then X(t) = X(t). d (c) Lax Equation: dt X(t) = [(X(t))≥0,X(t)] Showing these three points will establish that, if X0 ∈ gˆλ(T ) ⊂ Fλ, then the flow induced on Fλ with initial conditiong ˆλ(T0) by the flow on T is exactly the full Kostant-Toda flow with this particular initial condition. Proof. λ (a) This is clear: S0 = idT , so λ X(0) =g ˆλ(S0 (T0)) =g ˆλ(T0) = X0. (6.5.14) ˜ ˜ (b) Suppose T0, T0 ∈ T have the same image underg ˆλ. By Lemma 6.12, T0 = αr(T0) for some r ∈ Rn. We now have λ ˜ λ −1 ˜ St (T0) = (f ◦ Rt ◦ f )(T0) (6.5.15) λ −1 = (f ◦ Rt ◦ f )(αr(T0)) (6.5.16) λ −1 −1 = (f ◦ Rt ◦ f )((f ◦ µr ◦ f )(T0)) (6.5.17) λ −1 = (f ◦ Rt ◦ µr ◦ f )(T0) (6.5.18) λ −1 = (f ◦ µr ◦ Rt ◦ f )(T0) (6.5.19) λ −1 = (αr ◦ f ◦ Rt ◦ f )(T0) (6.5.20) λ = αr(St (T0)), (6.5.21) 159 λ where 6.5.17 and 6.5.20 are by Lemma 6.11, and 6.5.19 is due to Rt being left multi- plication and µr being right multiplication, which commute. Since gλ ◦ αr = gλ, this proves the part (b) of the theorem. (c) The proof of (c) is based on the analogous proof in [O13], given for the restricted (tridiagonal) setting. λ Let T0 ∈ T and let ht := h(St (T0)), then λ −1 X(t) =g ˆλ(St (T0)) = ht ελht. (6.5.22) Differentiating this yields d X(t) = −h−1h˙ h−1ε h + h−1ε h˙ (6.5.23) dt t t t λ t t λ t −1 ˙ −1 ˙ = −ht htX(t) + X(t)ht ht (6.5.24) −1 ˙ = [X(t), ht ht]. (6.5.25) To finish the proof, we follow the proof of Theorem 8.4 in [O13]: Since, by Proposition 5.4, each matrix in bw¯0 ∈ Pw¯0 admits a Gaussian decomposition bw¯0 = LR with L ∈ N− and R ∈ B+, and also, by Proposition 6.5, h(f(b(t))) = L(t), where b(t)w ¯0 = L(t)R(t). λ ελt Now, since b(t) = Rt (b0) = e b0, one has ελt ελt L(t)R(t) = b(t)w ¯0 = e b0w¯0 = e L0R0, (6.5.26) where L0 := L(0) and R0 := R(0). −1 Since X(t) = L(t) ελL(t), one has tX0 −1 ελt −1 −1 e = L0 e L0 = L0 L(t)R(t)R0 . (6.5.27) 160 Differentiation then yields tX0 −1 ˙ ˙ −1 X0e = L0 (L(t)R(t) + L(t)R(t))R0 (6.5.28) hence ˙ ˙ tX0 L(t)R(t) + L(t)R(t) = L0X0e R0 (6.5.29) tX0 = ελL0e R0 (6.5.30) = ελL(t)R(t) (6.5.31) = L(t)X(t)R(t). (6.5.32) Solving for X(t), one obtains −1 ˙ ˙ −1 −1 ˙ ˙ −1 X(t) = L(t) L(t) + R(t)R(t) = ht ht + R(t)R(t) . (6.5.33) ˙ −1 ˙ Finally, observe that since ht is unipotent, ht is strictly lower triangular, and so ht ht is strictly lower triangular. Since R(t) is upper Borel, R˙ (t) is also upper Borel, and ˙ −1 −1 ˙ so R(t)R(t) is upper Borel. We therefore, must have ht ht = (X(t))<0, so that d X(t) = [X(t), (X(t)) ] = [(X(t)) ,X(t)], (6.5.34) dt <0 ≥0 which concludes the proof of Part (c) of the theorem. For the purpose of elucidation, we provide an example demonstrating these objects. Example 6.1. Take an initial triangle (with no conditions imposed on it being in TΛ): log(2) T0 = log(2) 0 ∈ T . log(2) 0 0 Additionally, fix a spectrum λ = (0, 1, −1). 161 Step 0: Computing gˆλ(T0) m+1 m m ti+1 −ti Since ui = e for 1 ≤ i ≤ m < n, we have t3−t2 0−0 1 log 2−log 2 2 e 2 1 e 1 u = [e ] = [1], u = t3−t2 = log 2−log 2 = . e 3 2 e 1 We then compute h(T0) using these u vectors and conjugate ελ to obtain X0. 1 0 0 1 0 0 1 0 0 1 0 0 1 2 1 2 h(T0) = L1(u )L2(u ) = u1 1 0 0 1 0 u1 1 0 = 2 1 0 . 2 0 0 1 0 u2 1 0 0 1 1 1 1 Then, 1 0 0 −1 0 1 0 1 0 0 2 1 0 X0 :=g ˆλ(T0) := 2 1 0 0 1 1 2 1 0 = −1 0 1 . 1 1 1 0 0 −1 1 1 1 −2 −2 −2 −1 Step 1: Computing b0 = f (T0) From m m m t1 + ··· + tk = log ∆k (b0), 1 ≤ k ≤ m ≤ n one obtains b11 b12 b13 2 1 1 −1 b0 := f (T0) = 0 b22 b23 = 0 1 2 . 0 0 b33 0 0 1 Step 2: Flow on P Since 1 1 1 0 0 0 1 1 1 −1 ελ = 0 1 −1 0 1 0 0 1 −1 , 0 0 2 0 0 −1 0 0 2 162 we have Λ b(t) = Rt (b0) (6.5.35) 1 1 1 1 0 0 1 1 1 −1 2 1 1 = 0 1 −1 0 et 0 0 1 −1 0 1 2 (6.5.36) 0 0 2 0 0 e−t 0 0 2 0 0 1 t 1 −t 5 t 2 e 2 e + 2 e − 2 = t 5 t 1 −t (6.5.37) 0 e 2 e − 2 e 0 0 e−t Step 3: Find T (t) = f(b(t)) For the avoidance of repetition, we will simply state that solving m m m t1 (t) + ··· + tk (t) = log ∆k (b(t)), 1 ≤ k ≤ m ≤ n yields the triangle T (t) = f(b(t)), given by log 2 log(2) t . t 1 t 1 −t 5 t 1 −t 5 t − log e − 2 log(2e − 1) − log 2 e + 2 e − 2 log 2 e + 2 e − 2 Step 4: Compute X(t) =g ˆΛ(T (t)) As before, we find h(T (t)), then conjugate ελ. Again, omitting repeating the calcula- tions in Step 0, we compute 1 0 0 L(t) := h(T (t)) = 5e2t−1 5e2t−4et+1 1 0 2 1 5e2t−4et+1 2et−1 1 then 5e2t−1 1 0 5e2t−4et+1 X(t) = L(t)−1ε L(t) = −4et(5e2t−5et+1) 1 5e2t−1 Λ 2t t 2 t − 2t t + 1 1 (5e −4e +1) 2e −1 5e −4e +1 −4et 2et 1 (2et−1)(5e2t−5et+1) − (2et−1)2 1−2et − 1 163 d One can indeed check that dt (X(t)) = [X(t)≥0,X(t)], and so we have obtained the full Kostant-Toda flow with initial condition X0. With this connection between O’Connell’s dynamics on Tλ, we move to answer some natural questions: 1. How would one modify the picture to have a dynamics driven by cλ? 2. If X0 ∈ gˆλ(T ), then the full Kostant-Toda flow with initial condition X0 is attainable. What isg ˆλ(T ), and is there a nice parametrisation of this set? The first question has a simple solution: Lemma 6.14. Let L = (lij) be the lower unipotent matrix defined (as L2) in Lemma −1 ˜ −1 λ ˜ ˜ 6.9, so that cλ = L ελL. Define P := L PL. Then, one has a flow Kt : P → P λ ˜ cλt ˜ defined by Kt (b) = e (b). ˜ −1 Proof. Let b0 = L b0L, where b0 ∈ P, then λ ˜ cλt −1 Kt (b0) = e L b0L (6.5.38) −1 L ελtL −1 = e L b0L (6.5.39) −1 ελt −1 = L e LL b0L (6.5.40) −1 ελt = L e b0L (6.5.41) −1 λ = L Rt (b0)L, (6.5.42) ˜ λ which is again in P since Rt is a flow on P. By construction, one has the following commutative square: λ Kt P˜ P˜ ConjL ConjL P P λ Rt 164 ˜ ˜ −1 where ConjL(b) = LbL . Before moving onto the second question, we take stock of all of the objects and flows, compiling them in the following theorem. Theorem 6.15. The following diagram commutes: Companion Toda Flow= MulteCλt G/B+ G/B+ κλ κλ f-KT Flow Fλ Fλ Toda Flow Mλ Mλ gˆλ gλ gλ gˆλ λ St Tλ Tλ f|P λ f|P λ St λ T T Pλ Pλ λ f Rt |Pλ f P P λ Rt = Multeελt ConjL ConjL λ Kt = MulteCλt P˜ P˜ 165 where the diagonal arrows are inclusions. If λ1, . . . , λn are distinct, then the top commutative square can be replaced by the torus embedding with torus action. Proof. The proof is an amalgamation of the results of [O13] and the results of this chapter. Now we move onto the question of the image ofg ˆλ. Lemma 6.16. The map h : T → (N−)>0 is onto. Proof. This follows immediately from Proposition 5.3. If L ∈ (N−)>0, then there m m exists unique u ∈ (R>0) for each 1 ≤ m < n such that 1 2 n−1 L = L1(u )L2(u ) ··· Ln−1(u ) (6.5.43) m+1 m m ti+1 −ti where, specifically, ui = e for 1 ≤ i ≤ m < n. Furthermore, since h : T → m (N−)>0 is invariant under precomposition with αr : T → T , one can assume t1 = 0 for 1 ≤ m < n. Solving m+1 m m ti+1 = ti + log(ui ) (6.5.44) m with t1 = 0 for all m yields i−1 ! m Y m−k ti = log ui−k . (6.5.45) k=1 m Then T = (ti ) satisfies h(T ) = L. Corollary 6.17. Since h(T ) = (N−)>0, we have −1 gˆλ(T ) = L ελL : L ∈ (N−)>0 . (6.5.46) Proposition 5.3 provides a nice parametrisation of (N−)>0, hence a concrete answer to the question of determining the image ofg ˆλ. 166 6.6 The Poisson Structure and Symplectic Geometry of Full Kostant-Toda In this second part of Chapter6, in light of the relation between continuous gRSK and full Kostant-Toda, we present some results pertaining to the full Kostant-Toda lattice, namely on the Arhangelskij form and parabolic Casimirs. 6.6.1 The Arhangelskij Normal Form and Parabolic Casimirs The following is a result from [GS99], based on [Ar79]. However, the statement was erroneous in its description of the entry γ1n, as well as having a small additional typographic error. These are corrected below, with a proof provided for the corrected formulæ. Lemma 6.18. [GS99],[Ar79] Let X = (xij) ∈ sln and let γ11 γ12 ········· γ1n . 0 1 0 ··· 0 . ...... ...... Γ = , (6.6.1) .. . 0 0 . 0 . 0 0 0 ··· 1 γn−1,n −1 0 0 0 ··· 0 γ11 where 1/2 (1) γ11 = xn1 , (2) γin = −xi1/xn1 for i = 2, . . . , n − 1, √ (3) γ1i = xn,i/ xn,i for i = 2, . . . , n − 1, and γ1,1 Pn−1 (4) γ1n = 2 xn,1xn,n − i=1 xn,ixi,1 , 2xn,1 167 then κ1 ∗ ∗ −1 ΓXΓ = 0 φ (X) ∗ . (6.6.2) 1 1 0 κ1 Proof. Let us impose a block structure on X and Γ: T x1 X2 x3 γ11 ~γ γ1n 1k X = X X X , Γ = 0 I ~γ . (6.6.3) 4 5 6 n−2 kn −1 x7 X8 x9 0 0 γ11 −1 where x1 and x9 are 1 × 1 matrices. One computes Γ as −1 1 T T γ − ~γ ~γ ~γkn − γ1n 11 γ11 1k 1k −1 Γ = . (6.6.4) 0 In−2 −γ11~γkn 0 0 γ11 Multiplying out ΓXΓ−1 yields T ~γ1kX4+γ1nx7 x1 + ∗ ∗ γ11 T T X4+x7~γkn X4~γ1k+x7~γkn~γ1k X5 + ~γknX8 − ∗ γ11 γ11 x7 1 x7 T x7 T 2 X8 − ~γ1k x9 − X8~γkn + (~γ1k~γkn − γ1n) γ11 γ11 γ11 γ11 (6.6.5) Equating this to Equation 6.6.2, one obtains the following six equations: 1 (i) 2 x7 = 1, γ11 1 (ii) (X4 + x7~γkn) = 0, γ11 168 1 x7 T (iii) X8 − ~γ = 0, γ11 γ11 1k 1 T T (iv) X5 + ~γknX8 − (X4~γ + x7~γkn~γ ) = φ1(X), and γ11 1k 1k 1 T x7 T (v) x1 + (~γ X4 + γ1nx7) = x9 − X8~γkn + (~γ ~γkn − γ1n) γ11 1k γ11 1k 1/2 1/2 Solving (i) yields γ11 = x7 = xn1 , which gives us (1). T 1 x21 xn−1,1 Solving (ii) yields ~γkn = − X4 = − ,..., − , which gives us (2). x7 xn1 xn1 T γ11 1 Solving (iii) yields ~γ = X8 = √ (xn2, . . . , xn,n−1), which gives us (3). 1k x7 xn1 By (ii), the left-hand side of (iv) is X5 + ~γknX8. By (ii), this is then equal to 1 −1 X5 − X4X8, which is indeed φ(X) = X5 − X4x X8. x7 7 (v) is equivalent to x7 1 T x7 T 2γ1n = x9 − x1 − ~γ1kX4 − X8 − ~γ1k ~γkn (6.6.6) γ11 γ11 γ11 1 T = x9 − x1 − ~γ1kX4 (6.6.7) γ11 n−1 1 X = x − x − x x (6.6.8) nn 11 x ni i1 n1 i=2 n−1 ! 1 X = x x − x x . (6.6.9) x n1 nn ni i1 n1 i=1 Thus, n−1 ! √ n−1 ! 1 X xn1 X γ = x x − x x = x x − x x , (6.6.10) 1n 3/2 n1 nn ni i1 2x2 n1 nn ni i1 2xn1 i=1 n1 i=1 verifying (4). Definition 6.14. Define a map Γn : sl(n) → B+ by setting Γ(X) to be the Γ in Lemma 6.6.1. 169 n−1 Definition 6.15. For X ∈ sl(n) and 0 ≤ k ≤ 2 , let [X]k denote the submatrix of X obtained by deleting the first and last k columns and k rows. That is, [X]k is the central (n − 2k) × (n − 2k) submatrix. Since the index on Γn is implicit in the input, we simply write Γ(X) for Γn(X). Corollary 6.19. [Ar79] Generically, each matrix X can be conjugated by an element of the upper triangular group to the following form κ1 ∗ ∗ · · · · · · ∗ 0 κ2 ∗ · · · · · · ∗ ...... ...... Ad X = (6.6.11) bX .. .. . 0 0 1 . . . .. 0 . 0 ··· κ2 ∗ 1 0 0 ··· 0 κ1 Proof. This is a recursive application of Lemma 6.6.1. Define two sequences of matrices: (Nk)0≤k≤n, (Γk)1≤k≤n recursively by 1. N0 = X, 2.Γ k = diag(Ik−1, Γ([Nk−1]k−1),Ik−1), −1 3. Nk = ΓkNk−1Γk , where the last two are defined for 1 ≤ k ≤ n. 170 We prove by induction that Nk has the following form for each k: κ1 .. A B . k k κk Nk = 0 Ck Dk . k κ .. Ik 0 . κ1 Let Mk = Γ([Nk−1]k−1) for each k, so that Ik 0 0 Γk = 0 M 0 . (6.6.12) k 0 0 Ik N0 has the desired form trivially (one simply has N0 = C0 = X) and N1 has the desired form by construction of Γ (by Lemma 6.6.1). For k ≥ 1, assume Nk has the desired form above, then one has κ1 .. −1 . AkM Bk k+1 κk N = Γ N Γ−1 = −1 . (6.6.13) k+1 k+1 k k+1 0 Mk+1CkMk+1 Mk+1Dk κk . Ik 0 .. κ1 171 −1 The proof therefore reduces to showing that Mk+1CkMk+1 is of the form κk+1 ∗ ∗ 0 . . ∗ ∗ . (6.6.14) 0 1 0 ··· 0 κk+1 Since Ck = [Nk]k and Mk+1 = Γ([Nk]k). Thus −1 −1 Mk+1CkMk+1 = Γ([Nk]k)[Nk]kΓ([Nk]k) (6.6.15) which is of the desired form, again by virtue of Lemma 6.6.1. This completes the induction. n Definition 6.16. For 0 ≤ k ≤ 2 , the k-th (standard) parabolic subgroup of SL(n), denoted Pk, is the subgroup : P1 P2 P3 Pk = P = 0 P4 P5 : det(P ) = 1,P1,P6 k × k upper triangular 0 0 P6 In the above, P2,P3,P5 are arbitrary, and P4 is invertible. We also have P0 = SL(n) + and P n = B (n). b 2 c Theorem 6.20. For each k, κk is an invariant of the coadjoint action of Pk on sln. Proof. Firstly, note that the statement of the theorem is equivalent to the same statement with κk replaced by its partial sum σk = κ1 + ··· + κk. This is due to the nesting of the subgroups: P1 ⊃ P2 ⊃ · · · ⊃ Pk. 172 Now, if X ∈ sln has the following (k, n − 2k, k) × (k, n − 2k, k) block structure: X1 X2 X3 X = X X X , (6.6.16) 4 5 6 X7 X8 X9 then, by following the first k steps of the normal form procedure, one obtains κ1 .. . Ak Bk κk Γ XΓ−1 = N = , (6.6.17) ≤k ≤k k 0 φk(X) Ck κk . Ik 0 .. κ1 where Γ≤k = ΓkΓk−1 ··· Γ2Γ1. Computing the trace of Nk yields tr(Nk) = 2σk + trφk(X) (6.6.18) −1 = 2σk + tr(X5 − X4X7 X8) (6.6.19) −1 = 2σk + tr(X5) − tr(X4X7 X8). (6.6.20) Since Nk is conjugate to X, one yields tr(X) − tr(X ) + tr(X X−1X ) σ (X) = 5 4 7 8 . (6.6.21) k 2 Observe that this quantity does not depend on anything above the diagonal of X, in particular, on X2, X3, X6 and the above diagonal entries of X5. Nor does this quantity require us to know what is in the top-left and bottom-right blocks (X1 and X9) Furthermore, suppose we have two matrices in Pk: 173 P1 P2 P3 Q1 Q2 Q3 P = 0 P P and Q = 0 Q Q . (6.6.22) 4 5 4 5 0 0 P6 0 0 Q6 Then (P1X1 + P2X4 + P3X7)Q1 ∗ ∗ PXQ = (P X + P X )Q (P X + P X )Q + (P X + P X )Q ∗ . 4 4 5 7 1 4 4 5 7 2 4 5 5 8 4 P6X7Q1 P6(X7Q2 + X8Q4) P6(X7Q3 + X8Q5 + C9Q6) (6.6.23) Thus, any arbitary combination of left and right multiplications by matrices in P1 do not cause X2, X3, or X6 to come into play in the formation of σk. We can assume, therefore, that X2, X3 and X6 are all zero to begin with, or, when considering Nk, this means we can disregard Ak, Bk and Ck, since Nk is obtained by conjugation of X by a matrix in B+ ⊂ Pk. Let P ∈ Pk be arbitrary. Since −1 −1 −1 −1 −1 −1 −1 −1 PXP = (P Γ≤k)(Γ≤kXΓ≤k)(P Γ≤k) = (P Γ≤k)Nk(P Γ≤k) , (6.6.24) −1 it suffices to show that σk(PNkP ) = σk(Nk) for all P ∈ Pk. First, we compute P −1: −1 −1 −1 −1 −1 −1 P −P P2P P (P2P P5 − P3)P 1 1 4 1 4 6 −1 P = 0 P −1 −P −1P P −1 . 4 4 5 6 −1 0 0 P6 Then ∗ ∗ ∗ −1 −1 −1 −1 −1 PNkP = P P P φ (X)P − P P P P ∗ , (6.6.25) 5 1 4 k 4 5 1 2 4 −1 −1 −1 P6P1 −P6P1 P2P4 ∗ 174 and we compute tr(PN P −1) − tr(P φ (X)P −1 − P P −1P P −1) σ (PN P −1) = k 4 k 4 5 1 2 4 k k 2 tr((P P −1)(P P −1)−1(−P P −1P P −1)) + 5 1 6 1 6 1 2 4 2 tr(X) − tr(P φ (X)P −1) + tr(P P −1P P −1) = 4 k 4 5 1 2 4 2 tr(P P −1P P −1P P −1P P −1) − 5 1 1 6 6 1 2 4 2 tr(X) − tr(φ (X)) + tr(P P −1P P −1) − tr(P P −1P P −1) = k 5 1 2 4 5 1 2 4 2 tr(X) − tr(φ (X)) = k . 2 To conclude the proof, recall that tr(X) = tr(Nk) = 2σk(X) + tr(φk(X)). Thus, tr(X) − tr(φk(X)) = 2σk(X), which yields −1 σk(PNkP ) = σk(X), as desired. Pk Lemma 6.21. For σk(X) = i=1 κk(X), then the gradient ∇σk is given by −1 −1 −1 Idk X7 X8 X7 X8X4X7 1 −1 ∇σk(X) = 0 0 X X , 2 4 7 0 0 Idk where X1 X2 X3 X = X X X . 4 5 6 X7 X8 X9 Proof. Note that σk(X) can be written equivalently as tr(X ) + tr(X ) + tr(X X−1X ) σ (X) = 1 9 4 7 8 . k 2 175 To compute the gradient of σk, first compute, for X, Y ∈ sln: σk(X + εY ) − σk(X) ε tr(X + εY ) + tr(X + εY ) + tr((X + εY )(X + εY )−1(X + εY )) = 1 1 9 9 4 4 7 7 8 8 2ε tr(X ) + tr(X ) + tr(X X−1X ) − 1 9 4 7 8 2ε tr(εY ) + tr(εY ) + tr((X + εY )(X + εY )−1(X + εY )) − tr(X X−1X ) = 1 9 4 4 7 7 8 8 4 7 8 2ε tr(εY ) + tr(εY ) + tr((X + εY )((I + εY X−1)X )−1(X + εY )) − tr(X X−1X ) = 1 9 4 4 7 7 7 8 8 4 7 8 2ε tr(εY ) + tr(εY ) + tr((X + εY )X−1(I + εY X−1)−1(X + εY )) − tr(X X−1X ) = 1 9 4 4 7 7 7 8 8 4 7 8 2ε tr(εY ) + tr(εY ) + tr((X + εY )X−1(I − εY X−1 + O(ε2))(X + εY )) − tr(X X−1X ) = 1 9 4 4 7 7 7 8 8 4 7 8 2ε ε(tr(Y ) + tr(Y ) + tr(X X−1Y − X X−1Y X−1X + Y X−1X )) + O(ε2) = 1 9 4 7 8 4 7 7 7 8 4 7 8 . 2ε Therefore, σ (X + εY ) − σ (X) tr(Y ) + tr(Y ) + tr(X X−1Y ) − tr(X X−1Y X−1X ) + tr(Y X−1X )) lim k k = 1 9 4 7 8 4 7 7 7 8 4 7 8 . ε→0 ε 2 Then, ∇σk(X) is defined to be such that tr(Y ∇σk(X)) = hY, ∇σk(X)i (6.6.26) −1 −1 −1 −1 tr(Y1)+tr(Y9)+tr(X4X7 Y8)−tr(X4X7 Y7X7 X8)+tr(Y4X7 X8)) = 2 . (6.6.27) Let M = σk(X), with the same block structure as X, then Y1 Y2 Y3 M1 M2 M3 YM = Y Y Y M M M (6.6.28) 4 5 6 4 5 6 Y7 Y8 Y9 M7 M8 M9 Y1M1 + Y2M4 + Y3M7 ∗ ∗ = ∗ Y M + Y M + Y M ∗ , 4 2 5 5 6 8 ∗ ∗ Y7M3 + Y8M6 + Y9M9 (6.6.29) 176 then tr(YM) is equal to tr(Y1M1) + tr(Y9M9) + tr(Y8M6) + tr(Y7M3) + tr(Y4M2) + tr(Y2M4 + Y3M7 + Y5M5 + Y6M8). (6.6.30) Thus, we have M4 = M7 = M8 = 0 (so that M ∈ pk) and M5 = 0. Furthermore, one must have the following: 1 M1 = M9 = 2 Idk, 1 −1 tr(Y8M6) = 2 tr(X4X7 Y8) for all Y8. Looking at matrix dimensions, one sees 1 −1 that M6 = 2 X4X7 , 1 −1 tr(Y4M2) = 2 tr(Y4X7 X8) for all Y4. Looking at matrix dimensions, one sees 1 −1 that M2 = 2 X7 X8, 1 −1 −1 tr(Y7M3) = 2 tr(X4X7 Y7X7 X8) for all Y7. Again, following from the dimen- 1 −1 −1 sions, and the commutation property for trace, we have M3 = 2 X7 X8X4X7 . Thus, −1 −1 −1 Idk X7 X8 X7 X8X4X7 1 −1 ∇σk(X) = 0 0 X X . 2 4 7 0 0 Idk 1 k For the usual Casimirs of the Lie-Poission bracket, Ik(X) = k tr(X ), we can perform 177 a similar calculation: k k−1 k−2 k−1 2 k Ik(X+εY )−Ik(X) tr(X +ε(YX +XYX +···+X Y )+O(ε )) tr(X ) limε→0 ε = limε→0 kε − kε tr(YXk−1) + tr(XYXk−2) + ··· + tr(Xk−1Y ) = lim + O(ε) ε→0 k tr(YXk−1) + tr(XYXk−2) + ··· + tr(Xk−1Y ) = k ktr(YXk−1) = k = tr(YXk−1) = tr(Y ∇Ik(X)) k−1 hence ∇Ik(X) = X . 178 Chapter 7 Future Directions 7.1 Some Natural Extensions of this Work As indicated at the end of Section 5.4, there is a natural conjecture that would provide the final link in the table below: de-tropicalisation continuum limit discrete space continuous space continuous space discrete time discrete time continuous time RSK Continuous Algorithm 2 Geometric RSK 5 (Schensted Insertion) gRSK . 1 4 7 (Ghost) (Hirota’s) Dynamics 3 6 Toda Lattice Box-Ball System Discrete Toda Maslov tropicalisation stroboscope Figure 7.1. Roadmap of the dissertation, with numbers corresponding to connec- tions between adjacent cells. Specifically, this would allow for traversing the table from the right column to the middle, providing a very natrual link from continuous-time geometric RSK to the Deift-Nanda-Tomei [DNT83] / Symes [Sy80] construction with log X. In addition, we would like to apply ultradiscretisation to an analogue of this the stroboscope for full Kostant-Toda to determine a potentially interesting new class of 179 box-ball system. 7.2 Generalised Dressing Transformations Our applications of Proposition 5.2 were restricted to the case where η(t) is linear. But extension to the case where η(t) is a general continuous function is natural to implement. Indeed this is the setting of Kashiwara’s continuous crystals. Kashiwara [Ka93] originally introduced such objects as a means to generalise the study of Kac- Moody algebras. The term “crystal” here enters because a semiclassical limit of such continuous extensions yields Littelman’s alcove walks [GL05] within Weyl chambers for general crystallographic groups. Under tropicalisation, the path Tw0 η becomes Pα1 ◦ · · · ◦ Pαm η(t) (7.2.1) associated to a decomposition of w0 in terms of simple reflections, and where ∨ Pαη(t) = η(t) − inf α (η(s))α, (7.2.2) t≥s≥0 ∨ mimicking the usual crystallographic reflection sα(x) = x − α (x)α associated to the Weyl group W , hence the name “continuous crystal”. Having gone to continuous paths it is natural to pass to random ensembles of contin- uous crystals. O’Connell did this in the Brownian case and was able to establish the following elegant result: Theorem 7.1. ([O13]) If η(t) = B(t) + λt, with B(t) Brownian, then x(t) is a dif- ∆i(b(t)) fusion with generator Lλ. Here x(t) is the vector with entries xi(t) = log , ∆i−1(b(t)) where ∆i(b(t)) is the determinant of the top-right k × k minor of b(t). The diffusion Lλ is in the nature of a quantum extension of the KPZ equation. 180 In this random setting the semiclassical limit of Tw0 remarkably turns out to be a multi-dimensional generalisation of the classical Pitman operator [Bi09]. Pitman showed that for a real Brownian motion, Bt, started at 0, the process Xt := Bt − 2 inf0≤s≤t Bs is distributed as the norm of a 3-dimensional Brownian motion. The multi-dimensional extension has applications to random walks on non-commutative spaces (such as Hermitian matrices, which is the so-called Dyson Brownian motion) and from there to the very active areas of KPZ and random matrix universality. Given our novel take on gRSK and particularly its relation to box-ball systems, it will be of interest to study how the extension to continuous crystals and random ensembles manifests itself in our setup. 7.3 Geometric RSK and Box-Ball Systems for General Semisimple Lie Algebras It is known ([Ko78] and [EFS93]), that the classical Toda lattice for sln has generalisa- tions to other complex semisimple Lie algebras, with these generalisations extending to dressing transformations and the factorisation theorem. It is unclear what the meaning of geometric RSK, or perhaps box-ball systems, would be in these generalisations. With our explorations in this dissertation (particularly in Chapter5), we now have a route for pursuing their analogues via discrete B¨acklund transformations and ultradiscretisation. 7.4 Geometric Quantisation Notions of “quantisation” have played a role in our description of RSK and its ge- ometrisation, through the use of Maslov’s dequantisation [LMRS11] and ultradiscreti- sation. For others ([O13], [Gi97]), there is a natural role played in terms of canonical 181 quantisation of the classical Toda lattice. We believe our setup can be pursued to provide applications to physics and to the representation theory of Lie groups, including B+. We plan to explore this in our future work. Appendices 182 183 Appendix A Fukuda: The Advanced Box-Ball System and the Carrier Algorithm In this section, we review some of the results in the literature on the relationship between RSK and an advanced version of the box-ball system. This exploration will be based on Fukuda’s paper, “Box-ball systems and Robinson-Schensted-Knuth correspondence” [Fu04]. A.1 The Box-Ball System with Labels We begin by providing the basic box-ball system with labelled balls. Such a box- ball system is the same as the original box-ball system, except the balls are labelled (coloured) by numbers 1, 2, . . . , n. The evolution is essentially the same as the orig- inal box-ball evolution, however all of the balls of colour 1 are moved first, then of colour 2, and so on until finally the balls of colour n are moved. Below is an example of the time evolution of such a labelled box-ball system for n = 5, with each step shown for each of the ball colours: t: ··· 3 1 1 2 5 1 2 1 3 4 ··· ··· 3 2 5 1 1 2 1 3 4 1 ··· ··· 3 5 2 1 1 2 1 3 4 1 ··· ··· 3 5 2 1 1 2 1 4 1 3 ··· ··· 3 5 2 1 1 2 1 1 3 4 ··· t + 1: ··· 3 2 1 1 5 2 1 1 3 4 ··· Figure A.1. A single time-step of the advanced box-ball system, split into the successive movements of each colour/label. 184 The first row is the initial box-ball state, the second is the result of moving the balls of colour 1, the third is the result of moving the balls of colour 2, and so on until the last is the result of moving the ball(s) of colour 5 from the previous row. A.2 The Advanced Box-Ball System One additional modification to the original is to allow each box to also have a label, called its carrying capacity, which is a finite integer representing how many balls the box can accommodate. For the purpose of the box-ball evolution, if a box is not at capacity, a ball may be placed in it. Other than that, the algorithm remains the same. An advanced box-ball system can be viewed as a sequence of stacks of boxes, 1 3 2 1 4 5 ··· 1 4 5 5 5 4 5 5 ··· Figure A.2. An advanced box-ball system with carrying capacities. The ordering within a box does not matter, so one is free to arrange balls in a stack in ascending order (as in the figure above). Alternatively, the advanced box-ball system can be represented by a sequence of numbers and walls, where 0’s serve as placeholders for vacant spaces in a stack, again with numbers ascending in order. The above is represented as ··· 1 2 4 1 5 0 0 0 0 5 1 3 4 5 4 5 5 0 0 5 0 0 0 0 0 0 0 0 0 ··· The following is a single time evolution of the above advanced box-ball system, sep- arated by the successive colours being moved: 185 ··· 1 2 4 1 5 0 0 0 0 5 1 3 4 5 4 5 5 0 0 5 0 0 0 0 0 0 0 0 0 ··· ··· 0 2 4 0 5 0 0 1 1 5 0 3 4 5 4 5 5 0 1 5 0 0 0 0 0 0 0 0 0 ··· ··· 0 0 4 2 5 0 0 1 1 5 0 3 4 5 4 5 5 0 1 5 0 0 0 0 0 0 0 0 0 ··· ··· 0 0 4 2 5 0 0 1 1 5 0 0 4 5 4 5 5 1 3 5 0 0 0 0 0 0 0 0 0 ··· ··· 0 0 0 2 5 0 1 1 4 5 0 0 0 5 0 5 5 1 3 5 4 4 0 0 0 0 0 0 0 ··· ··· 0 0 0 0 2 1 1 4 5 0 0 0 0 5 5 0 0 0 1 3 4 4 5 5 0 0 5 0 0 ··· Figure A.3. The steps of a single time evolution of an advanced box-ball system with carrying capacities. A.3 The Carrier Algorithm The advanced box-ball system evolution was shown to be essentially decribable in terms of the carrier algorithm, due to Takahashi-Matsukidaira [TM97]. The carrier algorithm is a rule of loading an unloading of digits in a sequence, governed by an insertion sequence. If C is a sequence of digits and a is a digit, the carrier algorithm prescribes a rule for inserting a into C, to produce a new sequence C0 and an output digit a0. This procedure is nicely depicted in the following diagram: a C C0 a0 Figure A.4. The Carrier Algorithm Where the rule is given as follows: min{c : c ∈ C, c a} if {c : c ∈ C, c a}= 6 ∅ a0 = i i i i i i min{ci : ci ∈ C} otherwise C0 = the sequence of digits obtained from C by replacing a by a0 186 and where the relation is defned on the digits 0, 1, . . . , n by 0 n n − 1 · · · 2 1. This is evocative of Schensted insertion. Indeed, when a 6= 0, it is clear that this is the rule of insertion into a word: if a0 6= 0, then this corresponds to a bump, and if a0 = 0, this corresponds to growing the row. Fukuda proves in [Fu04] that the advanced box-ball algorithm and the carrier algo- rithm give the same result: this is modulo reordering within boxes of capacity greater than one. The main idea is that one starts with C being an infinite sequence of zeroes, successively inserting the sequence of digits from the advance box-ball system using the carrier algorithm. A sequence of output digits is produced, and this is the sequence corresponding to the time-evolved advanced box-ball system. We simply demonstrate this for the advanced box-ball system in Figure A.3, directing the reader to [Fu04] for the proof. 187 1 2 4 1 0000000... 1000000... 1200000... 1240000... 1140000... 0 0 0 2 5 0 0 0 1145000... 1450000... 4500000... 5000000... 0 1 1 4 0 5 1 3 0000000... 5000000... 1000000... 1300000... 5 0 5 0 4 5 4 5 1340000... 1345000... 1344000... 1344500... 0 0 5 0 5 0 0 5 1344550... 3445500... 4455000... 4455500... 0 1 3 0 0 0 0 0 4555000... 5550000... 5500000... 5000000... 4 4 5 5 0 0 0 0000000... 0000000... 0000000... ··· 5 0 0 Figure A.5. Successive applications of the Carrier Algorithm with input sequence from an advanced box-ball system Thus, under the succesive application of the carrier algorithm, the sequence (1, 2, 4, 1, 5, 0, 0, 0, 0, 5, 1, 3, 4, 5, 4, 5, 5, 0, 0, 5, 0, 0, 0, 0,...) is transformed into the sequence (0, 0, 0, 2, 0, 1, 1, 4, 5, 0, 5, 0, 0, 0, 5, 0, 0, 1, 3, 0, 4, 4, 5, 5, 5, 0, 0,...). This is not the sequence in Figure A.3. However, reintroducing the walls, one obtains ··· 0 0 0 2 0 1 1 4 5 0 5 0 0 0 5 0 0 1 3 0 4 4 5 5 5 0 0 0 0 ··· 188 which is equivalent to what was found by advanced box-ball evolution, after ordering numbers in ascending order within boxes of capacity greater than one. We are now in a position, armed with the observation that, when the input digit is nonzero, the carrier algorithm is Schensted insertion, to see how Schensted insertion manifests itself in the advanced box-ball system. A.4 Schensted Insertion in the Advanced Box-Ball System Suppose one wishes to perform Schensted insertion on a pair (a, x) = ((a1, . . . , an), (x1, . . . , xn)). It is clear that if one could begin the carrier algorithm with C = 1 1 ··· 1 2 2 ··· 2 ··· n n ··· n 0 0 ··· (A.4.1) | {z } | {z } | {z } x1 x2 xn 0 and insert a1 lots 1’s, a2 lots 2’s, and so on until an lots n’s, then the final C will correspond to the output word y of Schensted insertion, and the output sequence from the carrier algorithm will encode the bumped word, b. If one takes as an initial advanced box-ball system the following ··· 1 ··· 1 2 ··· 2 ··· n ··· n 1 ··· 1 2 ··· 2 ··· n ··· n ··· x1 x2 xn a1 a2 an Pn One can see that after i=1 xi steps of the carrier algorithm initialised with the zero sequence and input sequence corresponding to the figure above, one will attain a carrier as in Equation A.4.1. What remains to be inputted will be the sequence cor- responding to a. Thus, at the end of the carrier algorithm, the output sequence will be 189 (1, ··· , 1, 2, ··· , 2, ··· n, ··· , n, 0, ··· , 0, 1, ··· , 1 2, ··· , 2, ··· n, ··· , n,) | {z } | {z } | {z } | {z } | {z } | {z } | {z } b1=0 b2 bn z y1 y2 yn where z, the number of zeroes separating the two blocks of nonzero digits, is the number of times the Schensted word is extended (i.e. each time insertion does not Pn result in a bump). Or, equivalently, z = i=1(yi − xi). Remark A.1. This does not appear to be explicitly stated in Fukuda, but it is fairly natural to see this with some familiarity with Schensted insertion. Example A.1. We apply the above to the input pair (a, x) from the following Schen- sted insertion: a = (3, 0, 2, 1, 0) x = (2, 0, 0, 2, 3) y = (5, 0, 2, 1, 0) b = (0, 0, 0, 2, 3) We build the following initial box-ball state from a and x: t: ··· 1 1 4 4 5 5 5 1 1 1 3 3 4 ··· t + 1: ··· 4 4 5 5 5 1 1 1 1 1 3 3 4 ··· Figure A.6. Schensted insertion encoded in a unit carrying capacity advance box- ball evolution. and the output sequence is (..., 0, 0, 4, 4, 5, 5, 5, 0, 1, 1, 1, 1, 1, 3, 3, 4, 0,...) which is indeed the concatenation of (4, 4, 5, 5, 5) and (1, 1, 1, 1, 1, 3, 3, 4), with an in- termediate sequence of z = 5 + 2 + 1 − 2 − 2 − 3 = 1 zeroes. 190 Index (ε + b)λ or Fλ, 39 Tλ, 146 B±, 34 U(x), 83 i Dλ, 60 r (RSK triples), 117 N±, 34 n±, 34 Π±, 34 φBBS→RSK, 109 b±, 34 φRSK→BBS, 109 `(σ), 56 π±, 34 ε, 34 τk, 85 ελ, 39 cλ, 39 εn, 34 dλ, 60 g, 34 dl, 90 %ˆ, 112 dr, 90 m κλ, 40 li , 145 m z, 113 ri , 145 0 B , 109 sr, 90 B0 , 109 n Advanced Box-Ball System, 184 ± Bm,n, 87 L(x), 83 Basic Box-Ball Evolution, 45 Mλ, 39 Birkhoff Decomposition, 135 P, 132 Box-Ball Coordinate Dynamics, 51 R, 109 Box-Ball System, 45 R , 109 n Carrier Algorithm, 185 191 Companion Embedding, 40 Conserved Shape of the BBS, 47 Crystal Embedding, 150 Ferrers Diagram, 54 Gelfand-Tsetlin Parametrisation, 142 Ghost-Box-Ball Algorithm, 111 Ghost-Box-Ball System (GBBS), 110 Ghost-Box-Ball Walls, 118 Global Exorcism, 113 Hessenberg Matrix, 35 Hook Length, 55 Length of the Longest Increasing Sub- sequence, 56 Lusztig Parameters, 131 Patience Sorting, 56 Semistandard Tableaux, 56 SST, 56 Standard Tableau, 55 Young Diagram, 54 Young Tableau, 55 192 References [AD99] M. 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