Limit shapes of standard Young tableaux and sorting networks via the Edelman–Greene correspondence

SAMU POTKA

Licentiate Thesis Stockholm, Sweden 2018 KTH Institutionen för Matematik TRITA-SCI-FOU 2018:43 100 44 Stockholm ISBN 978-91-7729-959-2 SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentiats- examen i matematik onsdagen den 17 oktober 2018 kl 10.15 i sal F11, Kung- liga Tekniska högskolan, Lindstedtsvägen 22, Stockholm.

c Samu Potka, 2018

Tryck: Universitetsservice US AB iii

Abstract

This thesis consists of the following two articles. • New properties of the Edelman–Greene bijection. Edelman and Greene constructed a correspondence between re- duced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the per- mutation’s (Rothe) diagram. Properties of the Edelman–Greene bijection restricted to 132-avoiding and 2143-avoiding permuta- tions are presented. We also consider the Edelman–Greene bijec- tion applied to non-reduced words. • On random shifted standard Young tableaux and 132- avoiding sorting networks. We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is de- termined, with the integers in the SYT as heights. This implies via properties of the Edelman–Greene bijection results about ran- dom 132-avoiding sorting networks, including limit shapes for tra- jectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency. iv

Sammanfattning

Denna avhandling består av följande två artiklar. • New properties of the Edelman–Greene bijection. Edelman och Greene konstruerade en korrespondens mellan re- ducerade uttryck av den omvända permutationen och standard- Young-tablåer. Vi bevisar att för varje reducerat uttryck änd- ras formen av insättningstablåns region som innehåller de mins- ta möjliga elementen exakt som den övre vänstra komponen- ten av permutationens Rothediagram. Egenskaper av Edelman– Greenebijektionen begränsad till 132- och 2143-undvikande per- mutationer är presenterade. Vi betraktar också Edelman–Greene- bijektionen tillämpad på oreducerade ord. • On random shifted standard Young tableaux and 132- avoiding sorting networks. Vi studerar skiftade standard-Young-tablåer (SYT). Gränsfor- men av ytan av likformigt fördelade skiftade SYT med trappform är bestämd där heltalen i Youngtablån ger höjder. Den impli- cerar genom egenskaper hos Edelman–Greenebijektionen resul- tat om slumpmässiga 132-undvikande sorteringsnätverk, inklude- rande gränsformer för banor och mellanliggande permutationer. Dessutom betraktar vi det förväntade antalet på varandra följan- de grannar i SYT. Det är bevisat att genomsnittligt innehåller varje rad och varje kolonn av en skiftade SYT med trappform precis ett sådant grannpar. Contents

Contents v

Acknowledgements vii

I Introduction and summary 1

1 Introduction 3 1.1 Permutations ...... 3 1.2 Reduced words ...... 5 1.3 Standard Young tableaux ...... 6 1.4 The Edelman–Greene bijection ...... 10 1.5 Random sorting networks ...... 14

2 Summary of results 19 2.1 Paper A ...... 19 2.2 Paper B ...... 20

References 23

II Papers 29

Paper A New properties of the Edelman–Greene bijection (joint with Svante Linusson) Preprint: arXiv:1804.10034

v vi CONTENTS

Extended abstract: Séminaire Lotharingien Combin., 80B.79, 2018.

Paper B On random shifted standard Young tableaux and 132-avoiding sorting networks (joint with Svante Linusson and Robin Sulzgruber) Preprint: arXiv:1804.01795. Acknowledgements

First of all, I am indebted to my advisor Svante Linusson for his inspiration, guidance, support, patience and hinsert any quality in a great advisori. I am grateful to each member of the combinatorics group for the great atmosphere and all the fun activities. A former member also needs to be mentioned: It was fun working with you Robin – good luck in Toronto! I also want to thank my fellow PhD students. Finally, I thank my family for their endless support, for being there; my friends for all the (positive) distractions; and Kati – for everything.

Kiitos!

vii

Part I

Introduction and summary

1

1 Introduction

In the heart of this thesis are two seemingly unrelated objects; reduced words of permutations, and standard Young tableaux. The bridge between these two was provided by Paul Edelman and Curtis Greene in [9]. We will review the concepts and related background in this chapter.

1.1 Permutations

Permeating through mathematics, permutations of n ∈ N are bijections from the set [n] = {1, 2, . . . , n} to itself. Let Sn be the set of permutations of n. We usually write permutations in one-line notation: suppose σ ∈ Sn; we then write σ = σ(1) σ(2) . . . σ(n). For example, 2 3 1 4 is a permutation of 4. A permutation σ ∈ Sn can also be viewed as a permutation matrix n×n M(σ) ∈ {0, 1} such that M(σ)i,j = 1 if i = σ(j), and M(σ)i,j = 0 oth- erwise. See the example below. The corresponding permutation is written on top of the matrix. 2 3 1 4 0 0 1 0 1 0 0 0   0 1 0 0 0 0 0 1

There are other conventions as well. A shared feature, however, is that each row and column has exactly one 1-entry and the rest are 0s. Another tool for visualizing permutations, introduced in 1800 by Hein- rich August Rothe [29, 21], is the (Rothe) diagram D(σ) of a permutation σ,

3 4 CHAPTER 1. INTRODUCTION

2 0 0 which is the set D(σ) = {(i, j) ∈ N : i < σ(j), i 6= σ(j ) for all 1 ≤ j < j}. In other words, if we shade each cell (i, j) of the permutation matrix weakly below or to the right of a 1-entry, the diagram is given by the cells left unshaded. Again, see the example below.

0 0 1 0

1 0 0 0 D(2 3 1 4) = 0 1 0 0

0 0 0 1

Diagrams of permutations have appeared, for instance, in the study of Schu- bert polynomials and degeneracy loci [14]. Permutation diagrams lead naturally to the final topic of this section, pattern avoidance. Percy MacMahon is said to have been the first to study permutation patterns [26]. The subject was picked up again by Knuth in [19]. Since then, a plethora of results have appeared. We refer to [18] for a thorough account. For instance, one classical result is that permutations of n avoiding any fixed pattern of length 3 are counted by the Catalan numbers. So, what are permutation patterns? The permutation σ ∈ Sn contains the pattern τ = τ(1) . . . τ(k) ∈ Sk if there exist indices 1 ≤ i1 < i2 ··· < ik ≤ n such that σ(i) < σ(j) if and only if τ(i) < τ(j) for all i < j, i, j ∈ {i1, . . . , ik}. In other words, σ in one-line notation contains a subsequence in the same relative order as τ. If σ does not contain τ, it is called τ-avoiding. As an example, consider again σ = 2 3 1 4, which contains the pattern 213 but is 132-avoiding. In fact, 132-avoidance is of interest in both papers in this thesis. A dominant permutation σ ∈ Sn is such that its code c(σ) = (c1, . . . , cn), ci = |{j > i : σ(j) < σ(i)}|, is weakly decreasing. It is straightforward to check that the dominant permutations are exactly the 132-avoiding permutations. William Fulton made the below observation (phrased slightly differently in terms of dominant permutations).

Proposition 1.1 ([14, Proposition 9.19]). Suppose σ ∈ Sn. Then σ is 132- avoiding if and only if D(σ) consists of the connected component containing (1, 1). We use Proposition 1.1 in Paper A. Another class of permutations we encounter is the vexillary permutations of Alain Lascoux and Marcel-Paul Schützenberger [23, 24], which also appear 1.2. REDUCED WORDS 5 in the same geometric context as permutation diagrams. These are the 2143- avoiding permutations. Note that 132-avoidance implies 2143-avoidance, so dominant permutations are a special class of vexillary permutations. One may characterize the diagrams of vexillary permutations in terms of integer partitions [14], which we will define later on.

1.2 Reduced words

Permutations of n form a group with composition of functions as the group operation. This group is known as the symmetric group Sn. The symmetric group Sn provides an example of the more general Cox- eter groups, introduced by Coxeter in 1934 [5], which are groups G that can be generated by some S = {r1, . . . , rm} ⊂ G satisfying the relations m (rirj) ij = 1G (the identity element of G), where mii = 1 and mij ≥ 2 for i 6= j. If ri1 . . . rik = g ∈ G with k minimal, i1 . . . ik is called a reduced word (or reduced expression) of g. We define the (Coxeter) length of g by `(g) = k. It is well-known that Sn can be presented as a Coxeter group having S = {s1, . . . , sn−1} and mij = 2 for |i−j| > 1, mij = 3 otherwise, where si = (i, i+1) = 1 . . . i+1 i . . . n, i ∈ [n−1], is the adjacent transposition sending i to i+1 and i+1 to i. Namely, every permutation σ in Sn can be expressed as a product σ = si1 . . . sik . The smallest possible k equals the number of inversions inv(σ) of σ, that is, pairs i < j such that σ(i) > σ(j). This follows from the fact that multiplying by any adjacent transposition either increases or decreases the number of inversions. Hence, if si1 . . . sik = σ, i1 . . . ik is a reduced word of σ precisely when k = inv(σ). Thus `(σ) = inv(σ). For example, the permutation 4 3 2 1 has the reduced word 123121. Note that inv(4 3 2 1) = `(4 3 2 1) = 6. We let R(σ) denote the set of reduced words of the permutation σ. Continuing with the same example,

R(4 3 2 1) = {123121, 121321, 212321, 231231, 213231, 123212, 312312, 132312, 312132, 132132, 321232, 231213, 213213, 232123, 323123, 321323}.

An alternative way of thinking about reduced words comes from defining a partial order on Coxeter groups. First of all, it is important to remark that as opposed to composing functions from the right, we consider the adjacent transpositions si acting on positions i and i + 1, and perform their 6 CHAPTER 1. INTRODUCTION composition from the left. (Equivalently one could compose from the right and consider them acting on values.) As an example, consider S4 and the reduced word 1213. Composing s1s2s1s3 from the left yields the permutation 3 2 4 1. In terms of permutation matrices, we would have, for example, 2 1 3 4 2 3 1 4 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 s1 =  s1s2 =  0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1

3 2 1 4 3 2 4 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 s1s2s1 =  s1s2s1s3 =  1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 where we can see that si corresponds to swapping the columns i and i + 1. The weak right Bruhat order on (G, S) is defined by letting x  y if y = xri1 . . . rik for some rij ∈ S such that `(yri1 . . . rij ) = `(x) + j for any 0 ≤ j ≤ k. The covering relation is then given by the k = 1 case of the above definition. Note that the length function ` is a rank function (meaning that `(x) < `(y) if x ≺ y and `(y) = `(x) + 1 if y covers x), making the weak order a graded poset. Observe also that the maximal chains 1G ≺ x1 ≺ · · · ≺ x`(x) = x are in bijection with reduced words of x. The weak order for G = S4 is illustrated in Figure 1.1. For stronger properties of the weak order, and more on Coxeter groups, see, for example, the book of Anders Björner and Francesco Brenti [3]. Of special importance to us is the reverse permutation w0 = n n − 1 ... 2 1 which is the top element in the weak order on Sn since it is the n
unique permutation having the largest possible number of inversions, 2 . In the case of σ = w0 ∈ Sn, we often write R(σ) = R(n). We already saw R(4) above. Another example, R(3) is just {121, 212}.

1.3 Standard Young tableaux

We will start by defining the objects describing shapes of standard Young tableaux. These are tuples of natural numbers called (integer) partitions 1.3. STANDARD YOUNG TABLEAUX 7

Figure 1.1: The Hasse diagram of the weak order on S4.

λ = (λ1, λ2, . . . , λk) with λi ∈ N for all i ∈ [k] and λi ≥ λi+1 for each i ∈ [k − 1]. The partition λ is said to be a partition of n ∈ N, denoted by Pk λ ` n, if i=1 λi = n. The length of λ is the number of parts λi, that is, `(λ) = k. As an example, consider the partition λ = (4, 4, 2, 1) ` 11. Then `(λ) = 4. Partitions appeared already in a 1674 letter of Leibniz, who was interested in computing the number of partitions of a fixed n, but Euler was the first to discover a major result in his pentagonal number theorem [1]. Partitions can be visualized by their Young diagrams λdg = {(i, j): i ∈ [`(λ)], j ∈ [λi]}. Young diagrams are named after Alfred Young who introduced Young tableaux in 1900 [40]. The elements of λdg are called cells. In the English convention, a Young diagram is drawn as a collection of boxes corresponding to its cells with, similarly to matrix notation, i increasing downwards and j to the right. The Young diagram of (4, 4, 2, 1) is shown 8 CHAPTER 1. INTRODUCTION below.

Young tableaux are fillings of Young diagrams. More precisely, if λ ` n, a standard Young tableau of shape λ is a bijection T : λdg → [n] such that T (i+1, j) > T (i, j) and T (i, j +1) > T (i, j) whenever the respective cells lie in λdg. In other words, in the English convention the filling with 1, 2, . . . , n is increasing along rows and down columns. Below is a standard Young tableau of shape (4, 4, 2, 1).

1 3 5 6

2 7 10 11

4 8

9 We also encounter shifted standard Young tableaux. Given a strict par- tition λ = (λ1, λ2, . . . , λk), meaning that λi > λi+1 for all i ∈ [k − 1], we define its shifted Young diagram as sh λ = {(i, j + i − 1) : i ∈ [`(λ)], j ∈ [λi]}. Thus the shifted Young diagram is obtained from the normal Young diagram by shifting row i to the right by i−1 steps for each i ∈ [`(λ)]. Since (4, 4, 2, 1) is not a strict partition, we cannot use the same example again. Consider λ = (4, 2, 1) instead. Then

λsh = .

As with the non-shifted case, shifted Young tableaux of shape λ ` n are increasing fillings of the shifted Young diagram λsh with 1, 2, . . . , n. As an example, take 1 2 4 5

3 6 .

7 1.3. STANDARD YOUNG TABLEAUX 9

Outside of combinatorics, standard Young tableaux appear, for example, in the theory of symmetric functions – some examples include Schur func- tions and Littlewood–Richardson coefficients, representation theory, and ge- ometry – for instance Schubert calculus [15]. On the representation theory side, partitions λ ` n index the complex irreducible representations of Sn and moreover, the dimension of the representation indexed by λ is given by f λ = |SYT(λ)|, where SYT(λ) is the set of standard tableaux of shape λ [13, 41]. For an introduction, see, for instance, [32]. For strict parti- tions and shifted standard tableaux, the connection to projective irreducible representations of Sn is quite similar [35, 31]. The paragraph above mentioned f λ, the number of standard Young tableaux of shape λ. This can actually be computed using the celebrated hook-length formula [39, 12]. The conjugate partition λ0 of λ is the partition whose Young diagram is {(j, i):(i, j) ∈ λdg}, that is, is obtained from the diagram of λ by transposing. If (i, j) is a cell in the Young diagram of a 0 partition λ, the hook-length of (i, j) is defined as hλ(i, j) = λi −j +λj −i+1. In other words, it is the number of cells weakly below (i, j) in the column j plus the number of cells to the right of (i, j) on the row i. The hook-lengths of the cells of (4, 4, 2, 1)dg are shown below.

7 5 3 2 6 4 2 1 3 1 1

We are now ready to state the formula.

Theorem 1.2 (The hook-length formula. [12]). If λ is a partition, let |λ| = |λdg|. Then, λ |λ|! f = Q . u∈λdg hλ(u) In the next section, we will see the significance of Theorem 1.2 for enu- merating reduced words of permutations. For shifted tableaux, there is a similar hook-length formula, but the definition of hooks is slightly different. Given a cell (i, j) in the shifted sh Young diagram of a strict partition, we define its shifted hook-length hλ (i, j) as the number of cells in row i that lie to the right of column j, respectively 10 CHAPTER 1. INTRODUCTION in column j and weakly below row i, plus the number of cells in the (j + 1)- th row of the shifted Young diagram. In other words, if there is a cell (j, j) in column j, we add the number of cells on row j + 1 to the ordinary hook-length. The hook-lengths of the cells of (4, 2, 1)sh are given below. 6 5 4 1 3 2 1 One may also think of shifted hook-lengths in terms of ordinary hook-lengths in the diagram obtained by gluing the shifted diagram to its transpose as follows. We use this trick in Paper B. 6 5 4 1 3 2 1

The shifted hook-length formula is analogous to the ordinary hook-length formula. Theorem 1.3 (The shifted hook-length formula. [39]). If λ is a strict par- sh sh tition, let |λ| = |λ | and let fλ be the number of shifted standard Young tableaux of shape λ. Then,

sh |λ|! fλ = Q sh . u∈λsh hλ (u) Theorem 1.3 is used in Paper B. There are several proofs of Theorem 1.2 and Theorem 1.3. We would like to mention some, which all provide means of constructing uniformly random (shifted) standard Young tableaux. Both formulas have been proved probabilistically, Theorem 1.2 in [16] and Theorem 1.3 in [30], and bijectively as well. One bijective proof of Theorem 1.2 was given by [27]. Fischer followed with a proof of Theorem 1.3 based on [27] in [10].

1.4 The Edelman–Greene bijection

As mentioned in the beginning of this chapter, there is a bridge connect- ing standard Young tableaux and reduced words of permutations. Richard 1.4. THE EDELMAN–GREENE BIJECTION 11

Stanley proved in [37] that, for instance, reduced words of the longest per- mutation n n − 1 ... 2 1 are equinumerous with standard Young tableaux of the staircase shape scn = (n − 1, n − 2,..., 2, 1). He had conjectured this in 1982, which also inspired Edelman and Greene to construct a bijective correspondence, which specializes to a bijection between these two sets of objects. We review this correspondence and some of its properties in this section. Before we start, let us note that there are plenty of similarities to the celebrated Robinson–Schensted–Knuth (RSK) algorithm [7, 33, 20]. We might point these out along the way, but will not delve into the de- tails. We should also remark that a lot of the text below overlaps with the corresponding section of Paper A. As is the case with the RSK, the correspondence is based on an insertion algorithm. The difference is that instead of the letters of a (generalized) permutation, it inserts the letters of a (reduced) word into an insertion tableau. On the level of inserted elements the Edelman–Greene insertion and the RSK insertion are the same except for a difference in handling equal elements bumping. Definition 1.4 (The Edelman–Greene insertion). Suppose that the Young tableau P has strictly increasing rows P1,...,P` and x0 ∈ N is to be inserted in P . The insertion procedure is as follows for each 0 ≤ i ≤ `:

• If xi > z for all z ∈ Pi+1, place xi at the end of Pi+1 and stop. 0 0 0 • If xi = z for some z ∈ Pi+1, insert xi+1 = z + 1 in Pi+2. 0 • Otherwise, xi < z for some z ∈ Pi+1, and we let z be the least such 0 z, replace it by xi and insert xi+1 = z in Pi+2. In both this and the 0 case above we say that xi bumps z .

Repeat the insertion until for some i the xi is inserted at the end of Pi+1 and the algorithm stops. This could be a previously empty row P`+1. Our definition of the insertion differs from that of [9], where it is called the Coxeter–Knuth insertion. However, using for example the proof of [9, Lemma 6.23], one can show that the two definitions coincide for reduced words. In our formulation the tableaux are increasing in rows and columns also for non-reduced words. Now, based on the insertion algorithm above, we define how words are mapped to pairs of tableaux. Readers familiar with the RSK will recognize this is done in exactly the same way as with the RSK. 12 CHAPTER 1. INTRODUCTION

Definition 1.5 (The Edelman–Greene correspondence). Suppose that w = ∗ (0) w1 . . . wi . . . wm ∈ N . Initialize P = ∅.

(i−1) (i) • For each i ∈ [m], insert wi in P and denote the result by P .

Let P (m) = P (w) and let Q(w) be the Young tableau obtained by setting (k) (k−1) Q(w)i,j = k for the unique cell (i, j) ∈ P \ P . Set EG(w) = Q(w).

As an example, consider the reduced word w = 321232. Then the P (k), 1 ≤ k ≤ 6, form the following sequence

3 2 1 1 2 1 2 3 1 2 3

−→ 3 −→ 2 −→ 2 −→ 2 −→ 2 3

3 3 3 3 so that

1 2 3 1 4 5

P (321232) = 2 3 and EG(321232) = Q(321232) = 2 6 .

3 3

The tableau P (w) is called the insertion tableau and the tableau Q(w) the recording tableau. Note that P (w) and Q(w) are always of the same shape for a fixed w. To state one of the main results of Edelman and Greene, let the reading word r(P ) of an insertion tableau P be the word obtained by collecting the entries of P row by row from left to right starting from the bottom row.

Theorem 1.6 ([9, Theorem 6.25]). The correspondence

w 7→ (P (w),Q(w))

is a bijection between ∪σ∈Sn R(σ) and the set of pairs of tableaux (P,Q) such that P is row and column strict, r(P ) is reduced, P and Q have the same shape, and Q is standard.

It turns out that the entries of P (w) are always given by P (w)i,j = i+j−1 when w ∈ R(n), and in fact, as we will see later in Paper A, more generally if and only if w ∈ R(σ) with σ 132-avoiding. 1.4. THE EDELMAN–GREENE BIJECTION 13

Theorem 1.7 ([9, Theorem 6.26]). Suppose w ∈ R(n). Then the entries of P (w) are defined by P (w)i,j = i + j − 1 and Q(w) ∈ SYT(scn). The map EG(w): w 7→ Q(w) is a bijection from R(n) to SYT(scn).

Continuing in the setting of Theorem 1.7, if w ∈ R(n), the inverse to the Edelman–Greene bijection takes a very special form. To define it, we have to introduce Schützenberger’s jeu de taquin. For a good introduction, we refer to [38] or [32], although the terminology is slightly different. Let T be a partially filled Young diagram with increasing rows and columns, and each entry k ∈ [max(i,j)∈T T (i, j)] occurring exactly once. The evacuation path of T is a sequence of cells π1, . . . , πs such that

• π1 = (imax, jmax), the location of the largest entry of T ,

0 0 • if πk = (i, j), then πk+1 = (i , j ) ∈ T such that

T (i0, j0) = max{T (i, j − 1),T (i − 1, j)} > −∞

with the convention T (i, j) = −∞ for (i, j) 6∈ T and for unlabeled (i, j) ∈ T .

Next, define the tableau T ∂ by

• removing the label of T (π1),

• and sliding the labels along the evacuation path: T (π1) ← T (π2) ← · · · ← T (πs).

A single application of ∂ is called an elementary promotion. Whenever a label ` ∈ [T (π1)] slides from some cell (i, j) to (i, j+1) (respectively (i+1, j)) in applying ∂ until all labels have been removed is referred to as a right slide (respectively downslide). For w ∈ R(n), the inverse to the Edelman–Greene bijection can then be defined as follows.

Theorem 1.8 ([9, Theorem 7.18]). Suppose that Q ∈ SYT(scn). Apply (k) ∂ until all labels have been cleared and say that π1 = (ik, jk) is the first cell of the evacuation path π(k) for the k:th iteration. Then EG−1(Q) = j n . . . jk . . . j1. (2) 14 CHAPTER 1. INTRODUCTION

Consider again the example following Definition 1.5. Applying ∂ yields the sequence

1 4 5 1 5 1 1 ∂ ∂ ∂ Q = 2 6 −→ 2 4 −→ 2 4 −→ 2

3 3 3 3

1 1 ∂ ∂ ∂ −→ 2 −→ −→ .

(1) (2) (3) The largest entries are in the cells π1 = (2, 2), π1 = (1, 3), π1 = (2, 2), (4) (5) (6) −1 π1 = (3, 1), π1 = (2, 2) and π1 = (1, 3). Hence, EG (Q) = 321232 as expected. Another important operator will be the so-called evacuation S, which is in some sense dual to promotion. If T is a standard Young tableau, T S is defined by setting T S(i, j) = k if and only if (i, j) is not labeled in T ∂k but is labeled in T ∂k−1 . Thus T S records when cells become empty in iterating the elementary promotion ∂ for T . Returning to the previous example, we would have 1 2 6 S Q = 3 5 .

4 In his original work [36], Schützenberger proved a remarkable property of the operator S: it is an involution. David Little has constructed another correspondence mapping reduced words w of permutations to pairs (σw,Tw) where σ(w) is a Grassmannian permutation (having exactly one descent, that is, i such that σ(i) > σ(i+1)) and Tw is a standard Young tableau [25]. Zachary Hamaker and Benjamin Young proved that for any reduced word w, Tw = EG(w) [17]. Kraśkiewicz discovered an analogue of the Edelman–Greene correspondence for the type B Coxeter groups (signed permutations) [22].

1.5 Random sorting networks

Finally, having defined the objects we study, it is time to discuss our moti- vation. In large part, it stemmed from research on random sorting networks, 1.5. RANDOM SORTING NETWORKS 15

reduced words of w0 ∈ Sn equipped with the uniform probability measure n PU . This topic first appeared in the paper of Angel, Holroyd, Romik and Virág [2] in 2007. Based on computational evidence, they made several re- markable conjectures about random sorting networks. After receiving much attention, a proof of their conjectures was finally announced in 2018 by Dauvergne [6]. As the paper is not published yet, the text below refers to conjectures. Paper A is mostly related to the following conjecture. Let

1 1 At(dx, dy) = q dxdy 2π sin2(πt) − 2xy cos(πt) − x2 − y2

1 be the Archimedes measure with parameter t ∈ (0, 1). At t = 2 , A1/2 has density 1/(2πp1 − x2 − y2) supported on the interior of the unit disc x2 + y2 < 1 and is the projection of the surface area measure on the 2- 3 2 sphere in R into R . For general t, At is supported on the interior of an ellipse and can be obtained from A1/2 by the linear transformation (x, y) 7→ (x, −x cos(πt) + y sin(πt)). Moreover, given a permutation σ ∈ Sn, let µσ be the measure n 1 X µ = δ 2σ(i) . σ ( 2i −1,1− ) n n n i=1 Note that there is a small difference with the definitions in [2]. The reason for this is their convention to let M(σ)(n−i+1),j = 1 if σ(j) = i, that is, the permutation matrices are flipped upside down compared to our notation.

Conjecture 1.9 ([2, Conjecture 2]). Let w be an n-element random sorting network and let σt be the permutation defined by w1 . . . w n . Then, for bt(2)c all t ∈ (0, 1), Z Z fµ(σt) → fAt as n → ∞ R2 R2 2 for any compactly supported, continuous function f : R → R+ (convergence 2 in distribution in the vague topology for random Borel measures on R ).

Informally, this implies that asymptotically the permutation matrices corresponding to the intermediate configurations coming from random sort- ing networks are supported on a family of ellipses, or, in other words, have their 1s inside an elliptic region of the matrix. In particular, at half-time the 16 CHAPTER 1. INTRODUCTION

Figure 1.2: The intermediate permutation matrices M(σt) of a 1000-element 1 1 3 random sorting network w ∈ RU (1000) at times t = 4 , 2 and 4 . permutation matrix is supported on a disc. See Figure 1.2 for an illustration.

In Paper B, we consider questions similar to their other conjectures as well, restricted to a subset of sorting networks. Let us state two more beautiful conjectures before moving on to a summary of our results. Let n
N = 2 . For α ∈ [0, 1], define the scaled trajectory of i in w by 2σ−1 (i) f (α) = αN − 1 w,i n for αN ∈ Z, and by linear interpolation otherwise. Reduced words can be visualized by their wiring diagrams. See an example in Figure 1.3. Scaled trajectories can be thought of as heights of particles i in the scaled wiring diagram of w.

n Conjecture 1.10 ([2, Conjecture 1]). Let PU be the uniform probability measure on R(n) and fw,i be the scaled trajectory of i in w ∈ R(n). Then n n n n for each n there exist random variables (Ai )i=1, (θi )i=1 such that for every ε > 0,

n  n n  PU w ∈ R(n) : max max |fw,i(α) − Ai sin(πα + θi )| > ε → 0 as n → ∞. i∈[n] α∈[0,1] See Figure 1.4. We end with their strongest conjecture, which implies both above. The n permutahedron is an embedding of Sn into R defined by σ 7→ (σ−1(1), . . . , σ−1(n)). 1.5. RANDOM SORTING NETWORKS 17

4

3

2

1

s2 s3 s2 s1 s2 s3

Figure 1.3: The wiring diagram of 232123.

Figure 1.4: The scaled trajectories of the elements 200, 400, 600 and 800 in a 1000-element random sorting network. 18 CHAPTER 1. INTRODUCTION

Every permutation σ ∈ Sn lies on the sphere

n n n(n + 1) n n(n + 1)(2n + 1)o n−2 = z ∈ n : X z = and X z2 = . S R i 2 i 6 i=1 i=1

−1 −1 For each i ∈ [n − 1], the points σ and (σ ◦ si) are connected by an edge. In other words, we connect the permutations which differ by one in exactly two coordinates. Random sorting networks correspond to random shortest −1 −1 paths from id to w0 on the permutahedron. The strongest conjecture, which implies both of the previous ones [2, Theorem 5], states that these paths are close to great circles.

n Conjecture 1.11 ([2, Conjecture 3]). Let PU be the uniform probability n−2 measure on R(n). For each n, there exists a random great circle Cn ⊂ S such that for every ε > 0,

 d (w, C )  n w ∈ R(n): ∞ n > ε → 0 as n → ∞, PU n

−1 where d∞(w, c) = maxi∈[N] infz∈c ||σi − z||∞. 2 Summary of results

This chapter contains a summary of the contributions of the papers forming this thesis.

2.1 Paper A

Paper A started as an attempt to better understand the Edelman–Greene bijection with the goal of finding a new angle of approach on the conjectures by Angel, Holroyd, Romik and Virág. Based on computer experiments with SageMath [8, 4], we noticed that the minimal entries (Pi,j = i + j − 1) of the insertion tableaux of random sorting networks seemed to form a re- gion bounded by the shapes conjectured in Conjecture 1.9. Inspired by this observation, we showed our main result that this indeed holds even deter- ministically and in a more general setting. Namely, for any permutation σ and w ∈ R(σ), the region of minimal entries in P (w) has the same shape as the upper-left connected component of D(σ). The key fact used in our proof is that r(P (w)) ∈ R(σ) and P (r(P (w))) = P (w) [9], so it is enough to assume that w is a reading word. By Proposition 1.1, our result implies that the insertion tableau only consists of minimal entries if and only if σ is 132-avoiding. We extend this result slightly by characterizing the possible insertion tableau in terms of D(σ) if σ is a vexillary permutation. In some sense, this refines the result of Edelman and Greene, and Stanley [37], that if σ is vexillary, P (w) is the same for all w ∈ R(σ). Our result on the insertion tableaux of reduced words of 132-avoiding permutations led us to define 132-avoiding sorting networks, that is, w ∈

R(n) such that sw1 . . . swk is 132-avoiding for all k ∈ [N]. These are the

19 20 CHAPTER 2. SUMMARY OF RESULTS

maximum length chains in the weak Bruhat order on Sn restricted to 132- avoiding permutations, and correspond to maximum length chains in the Tamari lattice, which had already been studied by Fishel and Nelson [11]. We mostly provide reproofs on their properties, such as that they are in bijection with shifted standard Young tableaux of the shape scn. As an interesting side-note, 132-avoiding sorting networks form a commutation class [34], that is, each 132-avoiding sorting network is reachable from an- other by a sequence of commutations: sisj 7→ sjsi if |i − j| > 1. Schilling, Thiéry, White, and Williams also observed that n-element 132-avoiding sort- ing networks are in bijection with reduced words of the signed permutation −(n − 1) −(n − 2) ... −1 by si 7→ si−1 (in the type B Coxeter group s0 swaps the sign of the element in the first position). Lastly, we consider the Edelman–Greene bijection applied to non-reduced words. The motivation for this was the same: to try to better understand the bijection. We define a poset on the set of words w yielding the same Q(w), prove some of its properties, and make some conjectures.

2.2 Paper B

Paper B is in some sense a natural continuation of Paper A, where we defined 132-avoiding sorting networks. Boris Pittel and Dan Romik found the limiting surface of uniformly random standard tableaux of the square shape (nn) in [28]. We use and mimic their work to find the limit surface of uniformly random shifted standard tableaux of the staircase shape scn, which turns out to be half of the square’s. See Figure 2.1. Inspired by the conjectures of [2], and Figure 2.2 and Figure 2.3, we use the limit surface to find the limits of intermediate permutation matrices and trajectories of random 132-avoiding sorting networks. The equation of the limit surface is implicit, which unfortunately implies that the limiting tra- jectories are difficult to compute explicitly. The limit shapes of intermediate permutation matrices are determined explicitly but are rather complicated. We should remark here that [2] finds the limit shape of ordinary staircase standard tableaux but in our restricted setting the Edelman–Greene bijec- tion has a much simpler description, which allows us to prove limit results on the sorting network side. Finally, we consider adjacencies. Let w be a reduced word of w0 ∈ Sn. An index k ∈ [N − 1] is called an adjacency of w if |wk+1 − wk| = 1. The 2.2. PAPER B 21

Figure 2.1: The limit shape of uniformly random shifted SYT of staircase shape.

0 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000 0 0 0

500 500 500

1000 1000 1000

1500 1500 1500

2000 2000 2000

Figure 2.2: The intermediate permutation matrices of a random 132- 1 1 3 avoiding sorting network with 1000 elements at times α = 4 , 2 and 4 . motivation for this definition is that in the permutahedron, an adjacency π corresponds to a pair of edges with angle 3 and each non-adjacency to an orthogonal pair of edges. We show that the expected number of adjacencies in a random 132-avoiding sorting network of length N is 2(n − 2). This is done by considering adjacencies in standard Young tableaux. Let λdg be a Young diagram. A pair (T, u) of a cell u ∈ λdg and a standard tableaux T of shape λ is called a horizontal adjacency if T (u + (0, 1)) = T (u) + 1. The 22 CHAPTER 2. SUMMARY OF RESULTS

Figure 2.3: The scaled trajectories of the elements 1, 250, 500, 750 and 1000 in a random 132-avoiding sorting network with 1000 elements. pair (T, u) is called a vertical adjacency if T (u + (1, 0)) = T (u) + 1. The definitions are the same for shifted tableaux. We show that the total number of horizontal adjacencies in column c of (possibly shifted) SYT of shape λ is equal to the number of (possibly shifted) SYT of shape λ with largest entry in column c + 1 or greater. This implies, among other corollaries, that the expected number of horizontal (resp. vertical) adjacencies in column c < n − 1 (resp. row r < n − 1) of a uniformly random shifted staircase SYT is equal to 1. Horizontal and vertical adjacencies in shifted staircase SYT translate to adjacencies in the corresponding 132-avoiding sorting network.

The contributions of the author

The author did approximately 50% and 33% of the work on Papers A and B, respectively. References

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Part II

Papers

29

Paper A

New properties of the Edelman–Greene bijection (joint with Svante Linusson) Preprint: arXiv:1804.10034. Extended abstract at FPSAC 2018: Séminaire Lotharingien Combin., 80B.79, 2018.

Paper B

On random shifted standard Young tableaux and 132-avoiding sorting networks (joint with Svante Linusson and Robin Sulzgruber) Preprint: arXiv:1804.01795.