Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux Dorian Croitoru

Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux

by Dorian Croitoru

M.A., University of Pittsburgh, 2005

Submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2010

The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Author...... Department of Mathematics July 21, 2010 Certiﬁedby...... Alexander Postnikov Associate Professor of Applied Mathematics Thesis Supervisor Accepted by ...... Bjorn Poonen Chairman, Department Committee on Graduate Students Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux

Dorian Croitoru

Submitted to the Department of Mathematics on July 21, 2010 in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis consists of two parts. In the ﬁrst part, we start by investigating the classical permutohedra as Minkowski sums of the hypersimplices. Their volumes can be expressed as polynomials whose coeﬃcients – the mixed Eulerian numbers – are given by the mixed volumes of the hypersimplices. We build upon results of Postnikov and derive various recursive and combinatorial formulas for the mixed Eulerian numbers. We generalize these results to arbitrary root systems Φ, and obtain cyclic, recursive and combinatorial formulas for the volumes of the weight polytopes (Φ-analogues of permutohedra) as well as the mixed Φ-Eulerian numbers. These formulas involve Cartan matrices and weighted paths in Dynkin diagrams, and thus enable us to extend the theory of mixed Eulerian numbers to arbitrary matrices whose principal minors are invertible. The second part deals with the study of certain patterns in standard Young tableaux of shifted shapes. For the staircase shape, Postnikov found a bijection between vectors formed by the diagonal entries of these tableaux and lattice points of the (standard) associahedron. Using similar techniques, we generalize this result to arbitrary shifted shapes.

Thesis Supervisor: Alexander Postnikov Title: Associate Professor of Applied Mathematics Acknowledgements

I am greatly indebted to my advisor Alexander Postnikov for all his support, patience and understanding throughout the years, for believing in me when I lacked conﬁdence and for choosing interesting and challenging research problems.

I would like to thank Richard Stanley for kindly and generously sharing his math knowledge as well as interesting life stories. His knowledge, wisdom and experience are a continuous source of inspiration to me.

I am grateful to Dan Kleitman for all the interesting and fun conversations both math- ematical and not, as well as for serving on my thesis committee.

My early interest in mathematics was primarily shaped by eﬀorts of my high school teachers Petru Gherman and Marcel Teleuca.

Finally, thanks go to my friends and family for their indefatigable love, patience and support.

3 Contents

1 Introduction 5

2 Permutohedra, Mixed Eulerian Numbers and beyond 7 2.1 Permutohedra ...... 7 2.2 Classical Mixed Eulerian Numbers ...... 9 2.3 Root Systems, Hypersimplices and Mixed Eulerian Numbers ...... 11 2.4 Recursive Formulas For Mixed Eulerian Numbers ...... 13 2.5 Examples...... 17 2.6 A cyclic formula for the volumes of weight polytopes ...... 19 2.7 Generalization To Positive Deﬁnite Matrices ...... 22 2.8 Weighted Paths in Dynkin Diagrams ...... 27

3 Shifted Young Tableaux 29 3.1 Shifted Young diagrams and tableaux ...... 29 3.2 A generating function for diagonal vectors of shifted tableaux ...... 30

3.3 A bijection between diagonal vectors and lattice points of Pλ ...... 31

3.4 Vertices of Pλ and extremal Young tableaux ...... 33

4 Chapter 1

Introduction

My main research interests lie in geometric and algebraic combinatorics and in particular in the geometry of convex polytopes. Convex polytopes lie at the crux of combinatorics. Studying their classical invariants, such as their volumes, numbers of lattice points, f-vectors and Erharht polynomials has been a central problem, not only in its own interest, but more importantly because of the vast connections of these invariants with other areas of mathematics such as algebraic geometry and representation theory. The ﬁrst problem investigated here is stuying invariants of a certain class of polytopes called permutohedra. The classical

permutohedron P (x1, . . . , xn+1) is deﬁned as the convex hull of the (n+1)! points obtained by

permuting the coordinates of (x1, ..., xn+1). Permutohedra and more importantly, their various generalizations, have been studied sistematically by Postnikov and others. Special cases of permutohedra include many interesting polytopes such as graphical zonotopes and graph associahedra in graph theory, as moment polytopes (and in particular matroid polytopes) in algebraic geometry, and as alcoved polytopes and weight polytopes in representation theory. It is a general principle that the volumes, numbers of lattice points and other invariants of these polytopes should have alternative descriptions in terms of other objects (both combinatorial and non combinatorial) such as trees, Young tableaux, degrees of toric varieties, Weyl group elements with special properties, and so on. For example, P (n − 1,..., 1, 0) is the zonotope

corresponding to the complete graph Kn+1, and it is known that n!V ol (P (n − 1, ..., 1, 0)) is

the number of spanning trees of the complete graph Kn+1. A more interesting example is k n+1−k k the hypersimplex ∆n+1,k = P 1 , 0 (1 means k ones), which are matroid polytopes,

the moment polytopes corresponding to a toric varieties Xp for generic points p in the Grass-

manian, and also appear as weight polytopes of the fundamental representations of gln. In this work I have investigated volumes of permutohedra by computing the mixed volumes of hypersimplices, called the mixed Eulerian numbers. These numbers include many classical combinatorial numbers such as the Catalan numbers, binomial coeﬃcients and Eulerian numbers. I have found various recursive, combinatorial and cyclic formulas that enable one to

5 compute the mixed Eulerian numbers easily. All of these results generalize to arbitrary root systems, and some results even to arbitrary positive definite matrices. The second problem that I have explored in this thesis is studying patterns in Young tableaux of shifted shapes. Generalizing a result of Postnikov, I have shown that diagonal vectors of these tableaux are in bijection with lattice points of a certain polytope, which can be represented as a Minkowski sum of coordinate simplices (depending on the Young shape). These polytopes are generalized permutohedra, and are often combinatorially equivalent to the associahedron. The thesis is organized as follows. Chapter 2 is devoted to Permutohedra, Mixed Eulerian numbers and their generalizations. In Section 2.1, we give a brief overview on permutohedra and their volumes. In Section 2.2, we introduce the (classical) mixed Eulerian numbers, motivate their study and discuss known results about them. Section 2.3 generalizes the setup to any affine root system Φ. Next, in Section 2.4 we derive recursive formulas for the mixed Eulerian numbers, and in particular show that they are positive integers for any root system. We illustrate these results in Section 2.5, as well as provide new simple proofs for known results on the mixed Eulerian numbers. In Section 2.6 we prove a cyclic relation between volumes of weight polytopes associated to root subsystems of an extended affine root system. In Section 2.7 we use the dependence of the mixed Eulerian numbers solely on the Cartan matrix AΦ of the root system to generalize the theory to arbitrary positive definite matrices. We specialize some of these general results to AΦ in Section 2.8 and obtain an alternate characterization of the mixed Eulerian numbers in terms of weighted paths in Dynkin diagrams. Chapter 3 is about Shifted Young Tableaux. Section 3.1 reviews basic definitions. In Section 3.2 we use a result of Baryshnikov and Romik to derive a generating function for the diagonal vectors of shifted Young tableaux. We use this later in Section 3.3 to establish a one-to-one correspondence between diagonal vectors of shifted λ-tableaux (λ is the shape of the tableaux) and lattice points of a certain polytope Pλ. This polytope is a n Minkowski sum of simplices in R and its combinatorial structure only depends on the length of the partition λ. In particular, if the length of λ is n, Pλ turns out to be combinatorially equivalent to the associahedron Assn. In Section 3.4 we describe the vertices of Pλ in terms of certain binary trees, and give a simple construction of the corresponding “extremal” λ-shifted tableaux.

6 Chapter 2

Permutohedra, Mixed Eulerian Numbers and beyond

2.1 Permutohedra

The classical permutohedron P (x1, . . . , xn+1) is deﬁned as the convex hull of the (n+1)! points obtained by permuting the coordinates of the point (x1, ..., xn+1). According to G. Ziegler, permutohedra appeared for the ﬁrst time in the work of Schoute in 1911 ([20]), though the

term permutohedron was only coined much later. For generic x1, ..., xn+1, P (x1, ..., xn+1) is n- n+1 dimensional, lying in the hyperplane t1 +...+tn+1 = x1 +...+xn+1. For a polytope P ⊆ R included in a hyperplane t1+...+tn+1 = c, deﬁne its volume as the usual n-dimensional volume

of the projection of P onto tn+1 = 0. Permutohedra as well as their various generalizations have been studied extensively in [16, 17]. In particular their combinatorial structure has been described in terms of certain posets – building posets – which will also appear in Chapter 3. Special cases of permutohedra appear as graphical zonotopes and graph associahedra in graph theory, as moment polytopes in algebraic geometry, as alcoved polytopes arising from aﬃne Coxeter arrangements (see [13]), and as weight polytopes of fundamental representations of Lie groups. Of particular interest are invariants associated to them such as their volumes, numbers of lattice points or their Ehrhart polynomials. For example, P (n, n − 1, n − 2,..., 0)

is the graphical zonotope corresponding to the complete graph Kn+1:

X P (n, n − 1, n − 2,..., 0) = [ei, ej] 1≤i

In other words, it is the Minkowski sum of all the line segments [ei, ej] between eiand ej, n+1 where ei denotes the ith standard basis vector in R . As such, a basic result of zonotope theory tells us that n!V ol (P (n, n − 1, n − 2, ..., 0)) is the number of spanning trees of Kn+1 (see [22, Ex.4.32]), namely (n + 1)n−1, whereas its number of lattice points is the number of

7 spanning forests of Kn+1. An important example of a permutohedron is the hypersimplex

k n+1−k ∆n+1,k = P 1 , 0

(ab means a sequence of b a’s), which is the intersection of the unit hypercube with the hyperplane t1 + ··· + tn+1 = k. This is the matroid polytope corresponding to the uniform matroid of rank k consisting of all the k−subsets of [n + 1]. It appears in algebraic geometry as the moment polytope of the toric variety Xp = Tp of a point p ∈ Grk,n+1 whose Plucker ∗ n+1 coordinates are all non-zero (the action of the torus T = (C ) on Grk,n+1 is given by (t1, ..., tn+1) · (x1, ..., xn+1) = (t1x1, ..., tn+1xn+1)). As such, it is known that the normalized (n+1)−1 volume of ∆n+1,k is the degree of Xp as a subvariety of CP k (see [9, 8]). On the other hand, it is well-known that 1 V ol(∆ ) = A , n+1,k n! n,k

where An,k is the number of permutations of size n with k − 1 descents (i.e. An,k’s are the Eulerian numbers). This is a famous old result, dating back to Euler. However, the ﬁrst published proof seems to be due to Laplace ([14]). A simple half-page proof constructing a

triangulation of ∆n+1,k into An,k unit simplices was found by Stanley ([24]).

More recently, Postnikov has computed the volume of P (x1, . . . , xn+1) explicitly as a

homogeneous polynomial of degree n in x1, . . . , xn+1. To state the theorem, recall that

there is a natural bijection between sequences of nonnegative integers c1, . . . , cn+1 such that 2 c1 + ··· + cn+1 = n, and lattice paths in Z from (0, 0) to (n, n) with “up” or “right” steps: The path L corresponding to (c1, ..., cn) has ci vertical steps along the line x = i − 1. Let th th Ic1...cn+1 ⊆ [n] be the set of indices i such that both the (2i − 1) and 2i steps of L are

below the x = y axis (see Figure 2.1), and Dn(Ic1...cn ) be the number of permutations in Sn+1

with descent set Ic1...cn .

Theorem 2.1.1. [16, Theorem 3.2] The volume of P (x1, . . . , xn+1) is given by

c xc1 x n+1 X |Ic1...cn+1 | 1 n+1 V ol(P (x1, . . . , xn+1)) = (−1) Dn(Ic1...cn+1 ) ··· . c1! cn+1! c1+...+cn+1=n,ci≥0

Example 2.1.2. The path in Figure 2.1 corresponds to the composition (2, 1, 0, 0, 2, 0, 0, 2). 7 7 We have I21002002 = {4, 6, 7} and there are 3 ·3+ 2 ·2 = 189 permutations in S8 with descents 2 2 2 in positions 4,6,7. Hence, by Theorem 1, the coeﬃcient of x1x2x5x8 in V ol (P (x1, ..., x8)) is 189 − 8 .

8 7 6

4 5

3 2

Figure 2.1.1: Lattice path corresponding to (c1, . . . , c8) = (2, 1, 0, 0, 2, 0, 0, 2)

2.2 Classical Mixed Eulerian Numbers

Theorem 2.1.1 is a strong result, however it is not presented in a way that can be naturally generalized to other root systems. Let us instead introduce the new variables u1 = x1 − x2, u2 = x2 − x3, . . . , un = xn − xn+1. We have the following Minkowski sum decomposition:

P (x1, ..., xn+1) = P (u1 + ... + un + xn+1, ..., un + xn+1, xn+1)

= u1∆n+1,1 + u2∆n+1,2 + ··· + un∆n+1,n + xn+1(1,..., 1)

Since the Minkowski sum of Q and a point v is just Q translated by v, we may ignore the term xn+1(1, ..., 1) when taking volumes of both sides in above to obtain

V ol (P (x1, ..., xn+1)) = V ol (u1∆n+1,1 + u2∆n+1,2 + ··· + un∆n+1,n) X = V ol(∆n+1,i1 ,..., ∆n+1,in )ui1 ··· uin , (2.2.1) n (i1,...,in)∈[n]

where V ol (∆n+1,i1 ,..., ∆n+1,in ) is the mixed volume of the polytopes ∆n+1,i1 , ∆n+1,i2 ,...,

∆n+1,in . The (Brunn-Minkowski) theory of mixed volumes of polytopes is one of the corner- stones of classical convexity theory, and was pioneered by Minkowski in [15]; the last equality in (2.2.1) is essentially a restatement of Minkowski’s main theorem in the case of the hypersimplices. Mixed volumes of integer polytopes have important connections to algebraic geometry. For example, by a famous theorem of Bernstein, they count common zeroes of generic polynomials whose Newton polytopes are the given polytopes (see [2]). Computing mixed volumes of integer polytopes is very diﬃcult in general, but for standard coordinate simplices Post- nikov has managed to ﬁnd combinatorial formulas by using Bernstein’s result and ingenious linear algebra techniques ([16]). An excellent treatment of the Brunn-Minkowski theory is

9 contained in [19]; see also [4] for formulas and inequalities involving mixed volumes. Since the mixed volume of n polytopes does not depend on their listed order, we may combine similar terms in (2.2.1) to obtain

c1 cn X u1 un V ol (P (x1, . . . , xn+1)) = Ac1...cn ··· , (2.2.2) c1! cn! c1+...+cn=n,ci≥0

c1 cn ci where Ac1...cn = n!V ol ∆n+1,1,..., ∆n+1,n , and ∆n+1,i denotes ci copies of ∆n+1,i. The coeﬃcients Ac1...cn are called the (classical) mixed Eulerian numbers. They are positive integers because hypersimplices are integer polytopes of full dimension (see [8]).

Example 2.2.1. The kth hypersimplex ∆n+1,k is the Newton polytope of the k-th elementary

symmetric polynomial in x1, ..., xn+1. Bernstein’s theorem says that A120 equals the number 3 of distinct solutions in CP of the system

 a x + a x + a x + a x = 0  1 1 2 2 3 3 4 4 b1x1x2 + b2x2x3 + b3x3x4 + b4x1x3 + b5x2x4 + b6x3x4 = 0   c1x1x2 + c2x2x3 + c3x3x4 + c4x1x3 + c5x2x4 + c6x3x4 = 0

where ai, bj, ck are generic.

It is natural to ask whether these coeﬃcients have a simple combinatorial interpretation. This is one of the main problems that will be explored here. The question becomes more relevant given the following list of known results.

Theorem 2.2.2. [16, 6, 24] The mixed Eulerian numbers have the folowing properties:

(1) Ac1...cn = Acn...c1 . l (2) For 1 ≤ k ≤ n, A0k−1,n,0n−k = An,k (the usual Eulerian number). Here and below 0 denotes a sequence of l zeros. n (3) Ak,0...0,n−k = k .

(4) For 1 ≤ k ≤ n, i = 0, . . . , n the number A0k−1,n−i,i,0n−k−1 is the number of permutations w ∈ Sn+1 with k descents and w(n + 1) = i + 1.

c1 c2 cn 1 2n (5) If c1+···+ci ≥ i for i = 1, . . . , n then Ac1...cn = 1 2 . . . n . There are Cn = n+1 n such sequences (c1, . . . , cn).

(6) Let ∼ be the equivalence relation on sequences (c1, . . . , cn) given by (c1, . . . , cn) ∼

(d1, . . . , dn) if and only if (c1, . . . , cn, 0) is a cyclic shift of (d1, . . . , dn, 0). Then the sum of mixed Eulerian numbers in each equivalence class is n!, and the number of equivalence classes 1 2n is Cn = n+1 n . (7) P 1 A = (n + 1)n−1. c1...cn c1!···cn! c1c2...cn

10 Some comments about the above theorem are in place. Part (1) of the theorem follows n+1 because the volume-preserving isometry of R given by

(x1, ..., xn) 7→ (1 − x1, ..., 1 − xn)

maps ∆n+1,i to ∆n+1,n+1−i, i = 1, . . . , n. Part (2) is essentially a restatement of the classical

formula n! · V ol(∆n+1,k) = An,k. Part (4) was originally proved by Ehrenborg, Readdy and

Steingrimsson in [6]. The idea behind their proof is that uk∆n+1,k+uk+1∆n+1,k+1 (the volume Pn n−i i of which is i=0 A0k−1,n−i,i,0n−k−1 uk uk+1) turns out to be a slice of another cube, and so can more or less be handled directly. However it’s not clear how one could generalize their method

to compute other mixed Eulerian numbers (for example of form A0...0,a,b,c,0...0), because the Minkowski sum of even three (rescaled) hypersimplices is, in general, a complicated polytope. Part (6) is an interesting result which was conjectured by Stanley and proved by Postnikov in [16, Theorem 16.4]. This claim has a simple geometric explanation in terms of alcoves of the aﬃne Weyl group in type A. In some sense, it comes from symmetries of the extended Dynkin diagram of type A, which allow one to express the volume of the fundamental alcove (which is easy to compute) as a sum of volumes of n permutohedra. We will generalize this

result in Section 2.6. It is known that there are Cn equivalence classes ∼ and each of them

contains exactly one Catalan sequence (c1, . . . , cn) such that c1 + ··· + ci ≥ i, ∀i = 1 . . . n. The right side in (7) is exactly the volume of the permutohedron P (n, n − 1,..., 0), which as n−1 we have seen, is the number (n + 1) of spanning trees of Kn+1.

Problem 2.2.3. It is known that there are Cn equivalence classes ∼ and each of them

contains exactly one Catalan sequence (c1, . . . , cn) such that c1 + ··· + ci ≥ i, i = 1 . . . n.

For each Catalan sequence (c1, . . . , cn), can we ﬁnd some statistic on permutations of Sn

(or Sn+1) whose distribution gives the mixed Eulerian numbers in the ∼-equivalence class

of (c1, . . . , cn)? For example, part (4) of 2.2.2 implies that for i = 1, . . . , n − 1 the mixed Eulerian numbers of the sequences in the ∼-class of (n − i, i, 0n−2) are given by counting

permutations w ∈ Sn+1, w(n + 1) = i + 1 with a ﬁxed number of descents.

Rather than handle the (classical) mixed Eulerian numbers directly, we generalize the setup to other root systems, and then discuss particular cases.

2.3 Root Systems, Hypersimplices and Mixed Eulerian Num- bers

Let Φ be a reduced root system of rank n spanning a real vector space V . Fix a choice

of simple roots α1, . . . , αn in Φ. The roots are ordered in accordance with the standard labelling of the Dynkin diagram of Φ (see [12], p.58 for example). Let Λ be the associated

11 weight lattice, WΦ - the Weyl group of Φ, and (·, ·) - the correponding WΦ-invariant inner

product on V . Let λ1, . . . , λn ∈ V be the fundamental dominant weights of Φ; they form the 2αi dual basis of the simple coroots . For x ∈ V , we can deﬁne the weight polytope PΦ(x) (αi,αi) as the convex hull in V of the orbit WΦx. Write x = u1λ1 + ··· + unλn, ui ∈ R. Note that the coeﬃcients u are given by i 2αi ui = x, (2.3.1) (αi, αi)

We have the following Minkowski sum decomposition

PΦ(x) = u1PΦ(λ1) + ··· + unPΦ(λn).

The polytopes PΦ(λ1),...,PΦ(λn) are called the Φ-hypersimplices. Taking volumes of both sides we obtain:

c1 cn X Φ u1 un V ol (PΦ(x)) = Ac1...cn ··· , (2.3.2) c1! cn! c1,...,cn≥0,c1+···+cn=n

where

Φ c1 cn Ac1...cn = n! · V ol (PΦ(λ1) ,...,PΦ(λn) )

is the mixed volume of c1 copies of PΦ(λ1), c2 copies of PΦ(λ2),..., cn copies of PΦ(λn). We deﬁne

1 1 VΦ(u1, ..., un) = V ol (PΦ(x)) = V ol (PΦ(u1λ1 + ... + unλn)) | det (α1 ... αn) | | det (α1 ... αn) |

Thus, VΦ is a homogeneous polynomial of degree n = rankΦ. It is the volume of PΦ(x), if the volume form were normalized so that the box spanned by α1, ..., αn had unit volume. The Φ coeﬃcients Ac1...cn are called the mixed Φ-Eulerian numbers. As before, they are positive because the Φ-hypersimplices have full dimension.

The main purpose of this Chapter is to study the mixed Φ-Eulerian numbers. Postnikov has expressed these coeﬃcients as complicated sums over weighted doubly-labelled trees [16]. Our purpose is to ﬁnd simpler formulas, or combinatorial interpretations involving paths in the Dynkin diagrams, or the geometry of the root system Φ.

Example 2.3.1. Φ = An. Recall that the standard realization of An is given by α1 = e1 − e2, α2 = e2 − e3, ..., αn = en − en+1 inside the hyperplane V = {(x1, ..., xn+1)|x1 + ... + xn+1 =

0}. If P is the box generated by α1, ..., αn then the volume of the image of P under the

projection (x1, ..., xn+1) → (x1, ..., xn) is

12  1 −1 ···   . .   .. ..  det   = 1    1 −1  1 hence V ol on V agrees with the volume form deﬁned in Section 2.1. The Weyl group is n+1 Sn+1 which acts on V ⊂ R by permuting the coordinates. The fundamental dominant weights are λ = (1i, 0n+1−i) − i (1, 1, ..., 1), hence the Φ-hypersimplices are P (λ ) = i n+1 WAn i i n+1−i i i Pn+1 1 , 0 − n+1 (1, ..., 1) = ∆n+1,i − n+1 (1, ..., 1), i.e. the classical hypersimplices Φ translated into V . Therefore, for Φ = An, Ac1...cn are just the classical mixed Eulerian numbers.

Lemma 2.3.2. If Φ is the direct sum of two root systems Φ1 and Φ2, spanned by α1, ..., αm

and αm+1, ..., αn respectively, then

VΦ (u1, ..., un) = VΦ1 (u1, ..., um) VΦ2 (um+1, ..., un)

Proof. This follows because PΦ(x) is the direct product of PΦ1 (x1) with PΦ2 (x2), where xi is the projection of x onto the subspace of V spanned by Φi.

2.4 Recursive Formulas For Mixed Eulerian Numbers

In this section we derive recursive formulas for the mixed Eulerian numbers. One of the main tools that we employ is the following result due to Postnikov.

Proposition 2.4.1. [16, Proposition 18.6] For i = 1, . . . , n, we have

n ∂ X |WΦ|(λi, λj) V (u , . . . , u ) = V (u , . . . , u , u , . . . , u ), ∂u Φ 1 n |W |(α , λ ) Φj 1 j−1 j+1 n i j=1 Φj j j

where Φj = Φ − {j} is the root subsystem of Φ with node αj removed.

2(αi,αj ) Recall that the Cartan matrix associated to Φ is given by AΦ = = (αj ,αj ) 1≤i,j≤n T 2α1 2αn 2(λi,αj ) (α1 ... αn) ... . Since = δij, it follows that (α1,α1) (αn,αn) (αj ,αj )

 2  (α ,α ) 1 1 2(λ , λ ) −1 T  ..  i j AΦ = (λ1 ... λn) (λ1 ... λn)  .  = .   (αj, αj) 2 i,j (αn,αn)

1 Now, letting V Φ = VΦ, 2.3.2 can be rewritten as follows: |WΦ|

13     ∂/∂u1(V Φ) V Φ1 (u2, ..., un)  .  −1  .   .  = A  .  ,   Φ   ∂/∂un(V Φ) V Φn (u1, ..., un−1) or     ∂/∂u1(V Φ) V Φ1 (u2, ..., un)  .   .  AΦ  .  =  .  . (2.4.1)     ∂/∂un(V Φ) V Φn (u1, ..., un−1)

The polynomials V Φ are completely (over)determined by the recurrence (2.4.1) and the 1 initial condition V A1 (u1) = 2 u1. Tables of polynomials VΦ (u1, ..., un) for irreducible root

systems of rank up to 4 can be found in Appendix A. The table is missing VC4 , but this polynomial is easily readable from VB4 as the following observation shows.

Lemma 2.4.2. For any n,

1 V (u , ..., u ) = 2V u , ..., u , u Bn 1 n Cn 1 n−1 2 n

Proof. This follows since the root systems Bn and Cn are dual to each other. Indeed, assume

α1, ..., αn is a set of simple roots for Bn in some space V , (corresponding to the nodes of

the Dynkin diagram). Then α1, ..., αn−1, 2αn can be taken as a set of simple roots for a root

system of type Cn in V . Let x ∈ V . Since the Weyl groups of Bn and Cn are the same (as

subgroups of GL(V )), we have PBn (x) = PCn (x). By deﬁnition, the volume of PBn (x) is

| det (α1 ... αn) |VBn (u1, ..., un) 2αi where ui = x, . Similarly, the volume of PC (x) is (αi,αi) n

| det (α1 ... αn−1 2αn) |VCn (u1, ..., un−1, vn) where now v = x, 2(2αn) = 1 u . Equating the two volumes implies the result. n (2αn,2αn) 2 n Theorem 2.4.3. For any Φ, the mixed Φ-Eulerian numbers are positive integers.

Φ Φ Proof. We have seen that Ac1...cn > 0. We show that Ac1...cn ∈ Z, by using the recursion (2.4.1). This recursion implies that for each i, ∂ V is a linear combination of the poly- ∂ui Φ 1 |WΦ| nomials VΦ1 (u2, ..., un), ..., VΦn (u1, ..., un−1) with coeﬃcients of the form det A · . It is Φ |WΦj | straightforward to show that these numbers are always integers by using explicit tables for det AΦ and |WΦ| (see [12, p.66,68]). For example, suppose Φ = E7. Then

Φj ∈ D6, A6, A1 ⊕ A5, A2 ⊕ A1 ⊕ A3, A4 ⊕ A2, D5 ⊕ A1, E6

14 hence 5 4 7 4 det AΦ|WΦj | ∈ 2 · 2 6!, 2 · 7!, 2 · 2!6!, 2 · 3!2!4!, 2 · 5!3!, 2 · 2 5!2!, 2 · 2 3 5

which always divides |W | = 210345 · 7. Thus, ∂ V is an integer linear combination E7 ∂ui Φ uc1 ...ucn of V , ..., V . Integrating in u , it follows that for c > 0, the coeﬃcient AΦ of 1 n Φ1 Φn i i c1...cn c1!···cn! in V is an integer linear combination of numbers of the form AΦj . Hence the result Φ d1...dn−1 follows by induction on the rank n of Φ (we have already seen this result for Φ = An).

Rather than deal with the entries (AΦ)i,j directly, it’s more convenient to consider the natural weight function wt on the edges of the Dynkin diagram of Φ, deﬁned as follows: 1 wt(i → j) is 2 times the number of edges in the diagram from i to j, where undirected edges count both ways. For example, in type G2, there are 3 directed edges from node 2 to node 1, so we set wt(1 → 2) = 1/2 and wt(2 → 1) = 3/2. In types A, D, E all edges are undirected so wt(i → j) = 1/2 if i and j are connected, and 0 otherwise. In other words,

wt(i → j) = 2 (AΦ)i,j. The function wt will also appear later in Section 2.8. Next, given a weak composition (c1, ..., cn) of n, we identify it with the weight function w : i 7→ ci on the Φ 1 Φ nodes of the Dynkin diagram. We write A for A . Let wi→j denote the labelling w |WΦ| c1...cn which is identical to w except wi→j(i) = w(i) − 1, wi→j(j) = w(j) + 1. Theorem 2.4.4. Fix i such that w(i) > 0. The mixed Eulerian numbers satisfy the following recursive formula

Φ X Φ 1 Φ−{i} Aw − wt(i → j)Aw = A . (2.4.2) i→j 2 w|Φ−{i} i,j−connected

c c −1 u 1 u i cn Proof. The result follows by comparing the coeﬃcients of 1 ··· i ··· un in the ith c1! (ci−1)! cn! equation of (2.4.1).

Looking at these recursions, it seems that one could generalize the mixed Eulerian numbers Φ to graphs. However, they overdetermine Aw, and one can show that the only simple connected graphs which admit such positive coeﬃcients come from the Dynkin diagrams.

Consider the case Φ = An. Then Φ − {i} = Ai−1 ⊕ An−i. Let d1, ..., dn ≥ 0 be integers with d1 + ... + dn = n − 1. Applying Theorem 2.4.4 for w corresponding to the composition d1, ..., di + 1, ..., dn of n, we obtain

Proposition 2.4.5. For every nonnegative integers d1, ..., dn which sum to n − 1, we have

2Ad1...di−1,di+1,di+1...dn − Ad1...,di−1+1,di,...dn − Ad1...di,di+1+1,di+2...dn (2.4.3)  n+1  Ad ...d · Ad ...d if d1 + ... + di−1 = i − 1, di = 0 = i 1 i−1 i+1 n 0 otherwise.

15 These recurrences (together with A1 = 1) determine all of the mixed Eulerian numbers uniquely. On the other hand, Proposition 2.4.1 gives us directly another set of recurrences which characterize the coeﬃcients Ac1...cn . Following the setup of Example 2.3.1, we have ij (λi, λj) = i − n+1 for i < j. Therefore

n ∂ X ij V (u , . . . , u ) = min {i, j} − V (u , . . . , u )V (u , ..., u ) ∂u An 1 n n + 1 Aj−1 1 j−1 An−j j+1 n i j=1

c uc1 u i−1 To ﬁnd A , choose any i such that c > 0. Extracting the coeﬃcient of 1 · ... · i−1 · c1...cn i c1! ci−1! c −1 ci+1 u i u cn i · i+1 · ... · un on both sides of the last equation, we obtain (formally) (ci−1)! ci+1! cn!

1 X ij 1 1 A = (j − ) · A · A (n + 1)! c1...cn n + 1 j! (n − j + 1)! c1...cj−1 cj+1...ci−1(ci−1)...cn 1≤j

However, remember that Ac1...cn is only deﬁned when c1 + ... + cn = n, so we have to set it equal to 0 otherwise. Therefore in the above sums, the terms for j < i only appear when c1 + ... + cj−1 = j − 1 and cj = 0, and similarly, the terms for j ≥ i only appear when cj = 0 and c1 + ... + cj = j (with the exception for the case i = j, in which case the restrictions are ci = 1, c1 + ... + ci = i). We call such indices j good. Therefore,

X j(n + 1 − i)n + 1 A = A · A c1...cn n + 1 j c1...cj−1 cj+1...ci−1(ci−1)...cn j

X n A = (n − i + 1) A · A (2.4.4) c1...cn j − 1 c1...cj−1 cj+1...ci−1(ci−1)...cn j

In particular, by choosing i to be maximal such that ci > 0 in the last formula, we obtain the following result.

16 Theorem 2.4.6. Let c1, ..., cn be a composition of n, and suppose i = max{j|cj > 0}. Then the classical mixed Eulerian number Ac1...cn is given recursively, by

X n A = (n − i + 1) A · A + iA , c1...cn j − 1 c1...cj−1 cj+1...ci−1(ci−1)...cn c1...(ci−1)...cn−1 j

where the last term only appears if cn ≤ 1. This recurrence together with A1 = 1 deter- mines all the mixed Eulerian numbers uniquely.

2.5 Examples.

In this section we apply Theorem 2.4.6 to give new proofs of parts (2)-(5) of Theorem 2.2.2.

Example 2.5.1. (volumes of hypersimplices) Let A(n, k) = A0...n...0, where n is in position n+1 k. Thus A(n, k) is the normalized volume of the kth hypersimplex in R , i.e. the volume n of the slice of the unit cube in R lying inside k − 1 ≤ x1 + ... + xn ≤ k. We have A(1, 1) = 1 and for n ≥ 2 2.4.5 becomes A(n, k) = (n − k + 1) · A(n − 1, k − 1) + kA(n − 1, k), where the last term appears unless k = n. This is exactly the recurrence characterizing the Eulerian numbers, hence A(n, k) is the number of permutations in Sn with k − 1 descents. Thus we recover Euler’s famous result.

Example 2.5.2. (mixed volumes of the opposite hypersimplices) Let A(n, k) = Ak0...0(n−k) for k = 0, 1, ..., n. We have A(n, 0) = A(0, n) = 1. Theorem 2.4.6 implies A(n, k) = n n k Ak0...0A0...(n−k−1) = k for 0 < k < n − 1, and A(n, n − 1) = nA(n−1)0...0 = n. Therefore, n in all cases we have A(n, k) = k , as claimed in part (3) of Theorem 2.2.2.

Example 2.5.3. (mixed volumes of two adjacent hypersimplices) Let

A(n, i, k) = A0...i,n−i,...0

with n − i in position k. We have A(n, n, k) = A(n, k − 1),A(n, 0, k) = A(n, k) and for n ≥ 2, 0 < i < n the recurrence 2.4.5 becomes A(n, i, k) = (n−k+1)A(n−1, i, k−1)+kA(n−1, i, k).

Claim. A(n, i, k) equals the number of permutations π ∈ Sn+1 with k − 1 descents such that π(n + 1) = n − i + 1.

Proof. For i = 0 or i = n the result follows easily from the previous section. It remains

to show that the number B(n, i, k) of permutations π ∈ Sn+1 with k − 1 descents and last coordinate n−i+1 satisﬁes the same recurrence as A(n, i, k). Consider such a permutation π

17 written as a sequence of numbers. If we remove 1 from π, and then decrease all the remaining

digits by 1, we obtain a new permutation τ ∈ Sn such that τ(n) = n − i and τ has k − 1 or k − 2 descents. If τ has k − 1 descents then 1 must have been inserted in a descent position of π − {1}, or in the beginning. If τ has k − 2 descents, 1 must have been inserted in one of the n − 1 − (k − 2) = n − k + 1 ascent positions of π − {1}.

This establishes part (4) of Theorem 2.2.2.

Example 2.5.4. Suppose c1, ..., cn satisﬁes c1 + ... + ci ≥ i, ∀i = 1, ..., n. Let i = max{j|cj >

0}. Since there are no good indices j < i and cn = n − (c1 + ... + cn+1) ≤ 1, we have Ac1...cn = ci ci−1 c1−1 iAc1...ci−1(ci−1)...cn−1 . An easy inductive argument implies Ac1...cn = i (i − 1) ...1 A1, hence part (5) of Theorem 2.2.2 follows.

Theorem 2.4.6 gives another characterization of the mixed Eulerian numbers in type A. It’s not obvious at all that 2.4.3 and 2.4.5 are equivalent recursions. The set of good indices of a composition has a simple geometrical interpretation. There is a natural bijection between n-tuples c1, ..., cn of non-negative integers which sum to n, and plane lattice paths S between

(1, 1) and (n + 1, n + 1) with “up” and “right” steps. Fix i such that ci > 0. Let’s modify S by moving 1 unit down the part of S which lies to the right of the x = i. Call the new path S0. It is easy to see that the set of good indices j is precisely the set of x coordinates of points where S0 crosses the diagonal x = y.

Example 2.5.5. The lattice path in Figure 2.5.1 corresponds to the composition (c1, . . . , c8) = (1, 0, 3, 0, 0, 1, 3, 0). For i = 7, the set of good indices is {2, 5}. In this case formula 2.4.5 gives

8 8 A = 2 A A + A A + 7A . 10300130 1 1 300120 4 1030 120 1030012

4 3 Using 2.4.6 repeatedly, we ﬁnd A1030 = 2 1 A1A20 + 3A102 = 2 · 4 · 1 · 1 + 3 1 = 17 7 7 3 7 and A1030012 = 1 A1A30011 + 4 A1030A11 = 7 · 1 · 4 · 5 + 4 · 17 · 2! = 1330, and ﬁnally 3 2 2 A10300130 = 16 · 1 · 4 · 5 + 140 · 17 · 1 · 2 + 7 · 1330 = 20430.

The mixed Eulerian numbers include the factorials, binomial coeﬃcients, numbers of permutations with various restrictions, numbers of the form 1c1 ...ncn . While ﬁnding a simple closed formula for Ac1...cn is unlikely (there is no such formula already for A0...0,k,n−k,0..0), it seems reasonble to try the following

Problem 2.5.6. Find a way to label the n vertical segments of the path S with numbers 1, . . . , n with certain order restrictions depending on how S behaves (e.g. how S crosses the diagonal x = y), such that the number of labelings is Ac1...,cn .

18 8

1 2 3 4 5 6 7 8 9

Figure 2.5.1: The lattice path corresponding to the composition (c1, . . . , c8) = (1, 0, 3, 0, 0, 1, 3, 0).

2.6 A cyclic formula for the volumes of weight polytopes

In this section we investigate the geometry of alcoves in the aﬃne Coxeter arrangement of a root system Φ to obtain a generalization of Theorem 2.2.2(6). Our approach is similar to Postnikov’s in [16, Proposition 16.6], and uses symmetries of extended Dynkin diagrams. Recalling the setup of Section 2.3, we introduce additional notation. It is well-known that

there is a well-deﬁned highest root αn+1 = m1α1 + ··· + mnαn, where m1, ..., mn are positive

integers and m1 + ... + mn is the height of αn+1. We also let mn+1 = 1. See [3, Chapter 6] for more on these coeﬃcients.

Theorem 2.6.1. Let Φi denote the root system in V spanned by {αj|j 6= i}. For any u1, ..., un+1 we have

n n+1 −2 n+1 ! X mi | det (α1 ... αn) | X mi (αi, αi) VΦ (u1, ..., uˆi, ..., un+1) = ui (2.6.1) |W | i n!m1 ··· mn 2 i=1 Φi i=1

Proof. Recall that the aﬃne Coxeter arrangement of Φ in V consists of all hyperplanes of the form

(αi, x) = k, k ∈ Z, i = 1, 2, ..., n + 1.

These hyperplane arrangements as well as polytopes arising from them have been studied

(mostly in type An) in [18, 13]. These hyperplanes subdivide V into regions called alcoves.

The reflections in these hyperplanes generate the affine Weyl group W Φ, which is the semidi- n rect product of WΦ(reflections fixing the origin) and Z (translations) . The fundamental

19 Φ alcove A0 is given by

Φ A0 = {y ∈ V |0 ≤ (αi, y) , ∀i ≤ n, (−αn+1, y) ≤ 1}

2 It is a simplex with vertices v1, ..., vn+1 given by vn+1 = 0 and vi = λi for i ≤ n (the mi(αi,αi) latter points lying on (−αn+1, y) = 1). Hence its volume is

1 Φ 2λ1 2λn V olA0 = det (α ,α ) ... (α ,α ) (2.6.2) n!m1 ··· mn 1 1 n n T 1 −1 1 = det (α1 ... αn) = . n!m1 ··· mn n!m1 ··· mn det (α1 ... αn)

Φ On the other hand, consider any point x = u1λ1 + ... + unλn in the interior of A0 . The W Φ-orbit of x has a unique point in each alcove. If we look at the elements of W Φx closest to vi (i.e. the ones in the alcoves adjacent to vi), they are the vertices of the weight polytope Pi

(centered at vi) corresponding to the root system Φi in V generated by the roots {αj|j 6= i} (also centered at v ). The walls passing through v subdivide P into |W | congruent pieces, i i i Φi Φ one of which is Pi ∩ A0 . Figure 2.6.1 illustrates the geometry in type B2: The shaded region is the fundamental alcove, and there are 3 adjacent weight polytopes at x). Therefore,

n+1 X V olPi V olAΦ = (2.6.3) 0 |W | i=1 Φi

Now, how do we compute V ol(Pi) as a polynomial in u1, ..., un? Recall that x = u1λ1 + ... + 2(x,αi) unλn implies ui = , i.e. we only need x and the simple roots in order to know where to (αi,αi) evaluate V (u , ..., u ). We let u = 2 ((x, α ) + 1) for convenience, so that Φ 1 n n+1 (αn+1,αn+1) n+1

n+1 X mi (αi, αi) u = 1 (2.6.4) 2 i i=1

Since Pi is centered at vi, the coeﬃcients of the linear expansion of x in terms of the funda-

mental dominant weights in Φi are

2 (x − vi, αj) wj = = uj, ∀j 6= i (αj, αj)

(This follows easily since (vi, αj) = 0 for j 6= n + 1 and (vi, αn+1) = 1;i 6= j). The volume of the parallelotope formed by the αj’s (j 6= i) is

| det (α1 ...αi... αn+1) | = mi| det (α1 ... αn) |,

Therefore,

20 V ol(P ) = m | det (α ... α ) |V (u , ..., uˆ , ..., u ) . i i 1 n Φi 1 i n+1 Combining the latter with (2.6.2) and (2.6.3) we arrive at

n+1 1 X mi 2 = det (α1 ... αn) VΦ (u1, ..., uˆi, ..., un+1) n!m1 ··· mn |W | i i=1 Φi

The last formula holds for all u1, ..., ui+1 satisfying (2.6.4). Since the right side is a homoge-

neous polynomial of degree n in u1, ..., un+1, we obtain

n+1 n+1 !n 2 X mi 1 X mi (αi, αi) det (α1 ... αn) VΦ (u1, ..., uˆi, ..., un+1) = ui |W | i n!m1 ··· mn 2 i=1 Φi i=1

for any u1, ..., un+1. The theorem is proved.

Equation (2.6.1) can be somewhat simpliﬁed, by using the following classical formula for the size of the Weyl group of an irreducible root system (see [3, Proposition 7 on p.190 ]):

|WΦ| = n!m1m2...mn det AΦ

We have

h i T 2α1 2αn det AΦ = det (α1 ... αn) ... (α1,α1) (αn,αn) n Y 2 = det (α ... α )2 1 n (α , α ) i=1 i i

With these last formulas, equation (2.6.1) becomes

n n+1 n+1 !n Y X mi 1 X (αi, αi) VΦ (u1, ..., uˆi, ..., un+1) = mi (αi, αi) ui , |W | i |WΦ| i=1 i=1 Φi i=1 for any irreducible root system Φ. In types A, D, E all roots have the same length, hence the latter simpliﬁes to

n+1 n+1 !n X 1 X m V (u , ..., uˆ , ..., u ) = m u i Φi 1 i n+1 |W | i i i=1 Φ i=1

21 x

Figure 2.6.1: Alcoves in the aﬃne Coxeter arrangement of type B2.

2.7 Generalization To Positive Deﬁnite Matrices

Let Φ be an aﬃne root system with Cartan matrix AΦ. Then equation 2.4.1 can be rewritten as     ∂/∂u1VΦ VΦ1 (u2, ..., un)  .   .  AΦ  .  =  .  .     ∂/∂unVΦ VΦn (u1, .., un−1)

Since VΦ(u1, ..., un) is a homogeneous polynomial of degree n = rank(Φ), we obtain   n VΦ−{1}(u2, ..., un) X ∂ −1  .  nVΦ = uj VΦ = [u1 ... un] A  .  . ∂u Φ . j=1 j   VΦ−{n}(u1, ..., un−1)

Since the Cartan matrix of Φ−{j} is just the jth principal minor of AΦ, the last equation motivates the following construction:

Definition 2.7.1. Let A be a positive definite n by n matrix. We define the homogeneous

polynomial PA(u1, ..., un) by the following recursion:   PA11 (u2, ..., un) −1  .  u PA(u1, ..., un) = [u1 ... un] A  .  ,P[a](u) = . (2.7.1)   a PAnn (u1, ..., un−1)

By the above discussion, we have

Theorem 2.7.2. Let Φ be any root system, and AΦ its Cartan matrix. Then

PAΦ (u1, ..., un) = n!V Φ (u1, ..., un) .

We also provide a new direct proof of Theorem 2.7.2.

22 Proof of Theorem 2.7.2. Consider the point x = u1λ1 + ... + unλn in the weight lattice of Φ. We may assume without any loss that x is dominant. The Weyl chambers divide

the polytope PWΦ (x) into |WΦ| congruent subpolytopes; denote by P the piece containing x. For i = 1, 2, ..., n let Pi be the projection of x onto the hyperlane orthogonal to

αi, and let Qi be the projection of x onto the line spanned by λi. One can see that

the points x, P1, ..., Pi−1,Qi,Pi+1, ..., Pn lie on the hyperplane Hi which passes through x

and is orthogonal to λi. Thus, these points together with the origin 0 form a pyramid

with base xP1...Pi−1QiPi+1...Pn and altitude 0Qi, and P decomposes into n such pyra-

mids. Now, the projection to the aﬃne hyperplane Hi (considered as having the origin

at Qi), naturally induces a root system structure isomorphic to Φ − {i} on Hi: the sim- 0 ple roots are αj, and the fundamental dominant weights are just the projections λj of λj onto Hi, j 6= i. Since projections are linear transformations, x = u1λ1 + ... + unλn implies 0 0 0 0 0 x = x = u1λ1+...+ui−1λi−1+ui+1λi+1+...+unλn, and one can see that xP1...Pi−1QiPi+1...Pn 0 0 is one of the |WΦ−{i}| congruent pieces of PWΦ−{i} (x ), the weight polytope of x with the respect to the root system Φi = Φ − {i} in Hi. Therefore,

V ol (xP1...Pi−1QiPi+1...Pn) =

1 = V ol(P (u λ0 + ... + u λ0 + u λ0 + ... + u λ0 )) WΦi 1 1 i−1 i−1 i+1 i+1 n n |WΦi | V ol(α1, ...αˆi..., αn) = VΦi (u1, ...uˆi..., un) . V ol(α1, ..., αn)

The last quotient is 1 over the length of the projection of αi onto λi (because λi is the

orthogonal complement of the hyperplane spanned by αj, j 6= i). Thus

V ol(α1, ...αˆi..., αn) (αi, λi) 2 = 1/ λi = kλik . V ol(α1, ..., αn) (λi, λi) (αi, αi)

Therefore, the volume of the ith pyramid 0xP1...Pi−1QiPi+1...Pn is

1 1 (x, λi) 2 k0Qik V ol(xP1...Pi−1QiPi+1...Pn) = λi · kλik VΦi (u1, ...uˆi..., un) n n (λi, λi) (αi, αi) 2(x, λi) = VΦi (u1, ...uˆi..., un). (αi, αi)n

Summing the last expression over i = 1, ..., n we obtain the volume of P :

n X 2(x, λi) V (u , ..., u ) = V (u , ...uˆ ..., u ), Φ 1 n (α , α )n Φi 1 i n i=1 i i

23 which upon multiplying both sides by n! can be written as

n X 2(x, λi) PA (u1, ..., un) = PA (u1, ...uˆi..., un) Φ (α , α ) Φi i=1 i i n n X X 2(λj, λi) = uj PA (u1, ...uˆi..., un) (α , α ) Φi i=1 j=1 i i   P (u , ..., u ) AΦ1 2 n −1  .  = [u1 ... un] A  .  , as claimed. Φ   P (u , ..., u ) AΦn 1 n−1

We now derive some basic properties of the polynomials PA(u1, ..., un).

  A1  .  Proposition 2.7.3. Suppose A is a block diagonal matrix, A =  ..  with block   Ak sizes n1, . . . , nk. Then

 n1 + ··· + nk PA(u1, . . . , un) = PA1 (u1, . . . , uk) ··· PAk (un1+···+nk−1+1, . . . , un1+···+nk ). n1, . . . , nk

Proof. The general case follows from the case k = 2 by induction. Henceforth, we assume ! B A = where A, B, C are of size n, m, n − m respectively. We induct on n, the case C n = 1 being trivial. By using the inductive hypothesis and the recurrence formula for the

24 polynomials PA, we have   PA (u2, . . . , un) −1 ! 11 B  .  PA = [u1 ... un]  .  C−1   PAnn (u1, . . . , un−1)  n−1  PB11 PC m−1  .  !  .  B−1   = [u ... u ]  P P n−1  1 n −1  Bmm C m−1  C    .   .  n−1 PBPCn−m,n−m m   P (u , . . . , u ) B11 2 m −1  .  n − 1 = [u1 ...um] B  .  PC (um+1, . . . , un) · +   m − 1 PBmm (u1, . . . , um−1)   P (u , . . . , u ) C11 m+2 n −1  .  n − 1 + [um+1 ... un] C  .  PB(u1, . . . , um) ·   m PCn−m,n−m (um+1, . . . , un−1) n − 1 n − 1 = P (u , . . . , u )P (u , . . . , u ) · + B 1 m C m+1 n m − 1 m n = P (u , . . . , u )P (u , . . . , u ). m B 1 m C m+1 n

  d1  .  n! Corollary 2.7.4. If A =  .. , then PA = u1 ··· un.   d1···dn dn

n! Proposition 2.7.5. The coeﬃcient of u1 ··· un in PA is |A| . In particular, for any root system Φ, the mixed volume of the n Φ-hypersimplices is AΦ = |WΦ| . 1...1 det AΦ

Proof. If we define QA = |A| · PA, then 2.7.1 implies that the coefficient of u1 ··· un in QA Pn satisfies the recurrence [u1 ··· un] QA = j=1[u1 ··· uˆj ··· un]QAjj and [u1]Q[a] = 1. Therefore [u1 ··· un]QA = n!. The second part of the Lemma follows since the coefficient of u1 ··· un in Φ V ol (PWΦ ) is A1...1.

Thus, the determinant of A is encoded by the coefficient of u1 ··· un in PA. It would be interesting to find out what other invariants associated to A are represented by coefficients

of PA(u1, ..., un).

25 For an n by n matrix A, and two sequences of numbers a1, . . . , ak and b1, . . . , bl we

denote by |Aa1...ak,b1...bk | the minor of A corresponding to deleting rows a1, . . . , ak and columns

b1, . . . , bk. We write |Aa1...ak | instead of |Aa1...ak,a1...,ak |.

Theorem 2.7.6. We have

n+1 X +σ+i(π)+i(π,σ) |Aσ1,π1 | · |Aπ1σ2,π1π2 | · · · |Aπ1...πn−1σn,π1...πn | P = (−1)( 2 ) u ··· u A |A| · |A | · · · |A | σ1 σn π,σ π1 π1...πn−1 (2.7.2) n where the sum is over all permutations π ∈ Sn and sequences σ = (σ1, . . . , σn) ∈ [n] such that σk 6= π1, . . . , πk−1. Here i(π, σ) denotes the number of (π, σ)-inversions, i.e. the number

of pairs of indices (i, j), i < j such that σj < πi.

Proof. The theorem follows by repeatedly applying the recurrence relation. Indeed, 2.7.1 can be rewritten as

n X |Aj,i| P = P (u ... uˆ . . . u )(−1)i+j u . (2.7.3) A Ai 1 i n |A| j i,j=1

Pn k+l+ |Ail,ik| Next, we have PA = PA · (−1) u , where is 0 if k, l > i or i k,l=1;k.l6=i ik |Ai| l k, l < i and 1 otherwise (this accounts for the fact that we haven’t changed the labelling Pn of the rows and columns of Ai). Let’s perform one more step: PAik = r,s=1,r,s6=i,k PAikr · r+s+δ |Aiks,ikr| (−1) us. Again, δ compesates for the fact that the rows and columns of Aik are |Aik| still labelled as in A, hence δ is the number of pairs among (i, r), (k, r), (i, s), (k, s) which are in order ((a, b) is in order if a < b). Now it’s easy to see that if we apply 2.7.3 recursively n − 1 times, we obtain exactly the right side of 2.7.2 except that the exponent of −1 is

π1 + ··· + πn + σ1 + ··· + σn + ξ + µ, where ξ is the number of pairs (i < j) such that πi < πj,

and µ is the number of pairs (i < j) such that πi < σj. In other words, the exponent of −1 n n n+1 is 1 + 2 + ··· + n + σ1 + ··· + σn + 2 − i(π) + 2 − i(π, σ) ≡ 2 + σ1 + ··· + σn + i(π) + i(π, σ)(mod 2). The proof is complete.

It’s interesting to note that similar expressesions to the right side of equation 2.7.2 appear in the famous work in non-commutative algebra on quasi-determinants by Gelfand and Retakh ([10]).

n Corollary 2.7.7. The coeﬃcient of ui in PA(u1, . . . , un) is

n+1 X +ni+i(π)+(π) |Ai,π1 | · |Aπ1i,π1π2 | · · · |Aπ1...πn−1i,π1...πn | (−1)( 2 ) , |A| · |Aπ1 | · · · |Aπ1...πn−1 | π∈Sn,πn=i

where (π) is the number of ordered pairs (r < s) such that πr > i.

26 2.8 Weighted Paths in Dynkin Diagrams

While Theorem 2.7.2 gives an explicit formula for the polynomials PA(u1, ..., un), it is diﬃcult

to compute the coeﬃcients of PA (or even check whether they are positive). However, in the −1 case of a Cartan matrix A = AΦ, one can perform a trick which expresses the entries of AΦ in terms of weighted paths in the Dynkin diagram of Φ as follows. We deﬁne the weight of

each edge of Φ to be 1/2. For example, in type G2, there are 3 directed edges from node 2 to node 1, so we set wt(1 → 2) = 1/2 and wt(2 → 1) = 3/2. In types A, D, E all edges are unlabelled so wt(i → j) = 1/2 if i and j are connected, and 0 otherwise. The weight of a (directed) path is deﬁned as the product of weights of its edges.

Theorem 2.8.1. For any root system Φ, the Φ-mixed Eulerian numbers are given by

|W |c1!...cn! X X AΦ = wt(P )...wt(P ) (2.8.1) c1...cn 2nn! 1 n π=π1...πn∈Sn P1,...,Pn

where the sum is over all (directed) paths Pi : πi → σi in the Dynkin diagram such that

Pi avoids π1, ..., πi−1, and such that exactly cj of these paths end at j.

2(αi,αj ) Proof. Consider the Cartan matrix A = AΦ = = 2(I − BΦ), where the ij entry (αj ,αj ) i,j −1 1 2 of BΦ is just wt(i → j). Then A = 2 (I + BΦ + BΦ + ...). It follows that the ij entry −1 1 2 1 P of AΦ is 2 δij + (BΦ)ij + (BΦ)ij + ... = 2 P wt(P ), where the sum is over all paths P k in Φ starting at i and ending at j (the term (BΦ)ij is the sum of weights of such paths of length k). More generally, we may consider any principal minor Ai1...ik = 2(I − Bi1...ik ) −1 1 2 and by a similar argument we obtain Ai ...i = δij + (Bi1...ik )ij + (Bi ...i )ij + ... = 1 k ij 2 1 k 1 P 2 Q wt(Q), where the sum is over all paths Q in Φ from i to j avoiding i1, ..., ik. Now, recall that 1 AΦ is the coeﬃcient of uc1 ...ucn in V ol (P (u λ + ··· + u λ )) = c1!...cn! c1...cn 1 n WΦ 1 1 n n |WΦ| n! PAΦ (u1, ..., un) (cf. Theorem 3). Proceeding as in the proof of 2.7.6, we have

n n X X −1 PA(u1, ..., un) = PA (u1, ...uˆπ ..., un) A uσ π1 1 π1σ1 1 π1=1 σ1=1 X −1 −1 = PA (u1, ...uˆπ uˆπ ...un) A A uσ uσ π1π2 1 2 π1 π2σ2 π1σ1 2 1 π1,σ1;π2,σ26=π1 = ... X −1 −1 −1 = A Aπ ... Aπ ...π uσ1 ...uσn , (2.8.2) π1σ1 1 π2σ2 1 n−1 π σ n n n π∈Sn,σ∈[n]

n where the sum is over all π ∈ Sn, σ ∈ [n] such that σi 6= π1, ..., πi−1. By the above discussion,

−1 −1 −1 X 1 A A ... A = wt(P1)...wt(Pn−1), π1σ1 π1 π2σ2 π1...πn−1 n πnσn 2 P1,...,Pn−1

27 where the sum is over all collections of directed paths Pi : πi → σi such that Pi avoids

π1, ..., πi−1. We may as well include the weight 1 of the empty path Pn : πn → σn = πn (the only path avoiding π1, ..., πn−1), to each product in the sum. The theorem now easily follows c1 cn by extracting the coeﬃcients of u1 ...un on both sides of 2.8.2.

Φ Remark. By reversing all the paths in 2.8.1, one gets a similar formula for Ac1...cn where Pi : σi → πi avoids π1, ...πi−1, and cj of these paths start at j.

Example 2.8.2. Let’s illustrate Theorem 2.8.1 by computing the usual mixed-Eulerian num- 1 ber A030. Here, the root system is A3, all 4 edge weights of the Dynkin diagram are 2 . We are interested in triples of paths P1 : 2 → π1,P2 : 2 → π2,P3 : 2 → π3 = 2 with P2

avoiding π1 (and P3 is just the empty path which we may ignore). There are 2 possibili-

ties (π1 = 1, π2 = 3, and vice versa) which, by symmetry, yield the same contribution to

A030. Consider pairs of paths P1 : 2 → 1,P2 : 2 → 3, with P2 avoiding 1. There is one path P of each odd length k, hence P wt(P ) = 1 + 1 + ... = 2 . If a is the number of 2 P2 2 2 8 3 k k−1 paths P1 of length k, then ak satisﬁes ak = 2ak−2, a1 = 1. Thus, ak = 2 2 for odd k, and P wt(P ) = P ak = P 1 = 1. Therefore, Theorem 2.8.1 implies P1 1 k=2n+1≥1 2k n≥0 2n+1

4!0!3!0! 2 A = 2 · · · 1 = 4, 030 233! 3

which, in accordance with Euler’s classical result, is also the number of permutations in

S3 with 2 descents.

28 Chapter 3

Shifted Young Tableaux

This chapter is based on [5]. In this chapter, we study vectors formed by entries on the diagonal of standard Young tableaux of shifted shapes. We will establish a connection between such vectors and the lattice points of certain generalized permutohedra which are Minkowski sums of coordinate simplices.

3.1 Shifted Young diagrams and tableaux

Deﬁnition 3.1.1. Let λ = (λ1, . . . , λn) be a partition into (at most) n parts. The shifted Young diagram of shape λ (or just λ-shifted diagram) is the set

 2 Dλ = (i, j) ∈ R |1 ≤ j ≤ n, j ≤ i ≤ n + λj .

We will think of Dλ as a collection of boxes with n+1−i+λi boxes in row i, for i = 1, 2, . . . , n and such that the leftmost box of the ith row is also in the ith column. A shifted standard

Young tableau shape λ (or just λ-shifted tableau) is a bijective map T : Dλ → {1,..., |Dλ|} which is increasing in the rows and columns, i.e. T (i, j) < T (i, j + 1),T (i, j) < T (i + 1, j) n+1 (|Dλ| = 2 + λ1 + ··· + λn is the number of boxes in Dλ). The diagonal vector of such a tableau T is diag(T ) = (T (1, 1),T (2, 2),...,T (n, n)).

Figure 3.1.1 shows an example of a shifted standard Young tableau for n = 4, λ = (4, 2, 1, 0). Its diagonal vector is (1, 4, 7, 17). We are interested in describing the possible diagonal vectors of λ-shifted Young tableaux. The problem was solved in the case λ = (0, 0,..., 0) (the empty partition) by A. Postnikov, in [16, Section 15]. Speciﬁcally, it was shown that diagonal vectors of the shifted triangular shape

D∅ are in bijection with lattice points of the (n − 1)-dimensional associahedron Assn−1(to be deﬁned in section 2). Moreover, a simple explicit construction was given for the ”extreme”

diagonal vectors, i.e. the ones corresponding to the vertices of Assn−1. We use similar

29 1 2 3 5 8 9 12 13

4 6 10 11 16

7 14 15

Figure 3.1.1: A λ-shifted Young tableau of shape λ = (4, 2, 1, 0). techniques to generalize Postnikov’s results to arbitrary shifted shapes.

3.2 A generating function for diagonal vectors of shifted tableaux

For a non-negative integer vector (a1, ..., an), let Nλ(a1, . . . , an) be the number of standard

λ-shifted tableaux T such that T (i + 1, i + 1) − T (i, i) − 1 = ai for i = 1, . . . , n and where we n+1 set T (n + 1, n + 1) = 2 + λ1 + ··· + λn + 1 .

Theorem 3.2.1. We have the following identity:

a1 an X t1 tn Nλ(a1, . . . , an) ··· = a1! an! a1,...,an≥0

1 Y = Qn · (ti + ··· + tj−1) · sλ(t1 + ··· + tn, t2 + ··· + tn, . . . , tn) (λi + n − i)! i=1 1≤i

where sλ denotes the Schur symmetric polynomial associated to λ.

Proof. Consider a vector x = (x1, x2, ..., xn) with x1 > x2 > ... > xn. Deﬁne the polytope

Pλ(x) = {(pij)(i,j)∈Dλ | 0 ≤ pij ≥ pi(j+1), pij ≥ p(i+1)j, pii = xi}.

By deﬁnition, Pλ(x) is exactly the section of the order polytope of shape Dλwhere the values

along the main diagonal are x1,..., xn. If λ = ∅, this polytope is known as the Gelfand-Tsetlin

polytope, which has important connections to ﬁnite-dimensional representations of gln(C) (see [11]). Our proof strategy is to compare two diﬀerent formulas for the volume of Pλ(x), one of which is more direct and the other is a summation over standard λ-shifted Young tableaux. On the one hand, by [1, Proposition 12] we have

1 Y vol(Pλ(x)) = Qn · (xi − xj) · sλ(x). (3.2.1) (λi + n − i)! i=1 1≤i

30 Baryshnikov and Romik proved this result directly by an inductive argument (on the number of boxes plus the number of parts of λ) using iterated integrations. On the other hand, there is a natural map φ from Pλ(x) to the set of standard λ-shifted Young tableaux defined as 0 0 0 0 follows: Let p =(pij)(i, j)∈Dλ ∈ Pλ(x) be a point such that pij = pi j ⇔ (i, j) = (i , j ). Arrange the pij’s in decreasing order and define the tableau T = φ(p) by writing k in box th (i, j) if pij is the k element in the above list. By the definition of Pλ(x), it is clear that T is a standard λ-shifted Young tableau. Given standard λ-shifted tableau T with diagonal −1 vector diag(T ) = {d1, . . . , dn}, it is easy to see that φ (T ) is isomorphic to the set

|T | {(yi) ∈ R | y1 > y2 > ··· > y|T | > 0, ydi = xi} which is the direct product of (inﬂated) simplices

{x1 = y1 > y2 ··· > yd2−1 > x2} × · · · × {xn = ydn > ydn+1 ··· > y|T | > 0}

Therefore, (x − x )a1 (x − x )an−1 xan vol(φ−1(T )) = 1 2 ····· n−1 n · n . a1! an−1! an! Summing over all λ-shifted tableaux T , we obtain

X −1 vol(Pλ(x)) = vol(φ (T )) T a1 an−1 an X (x1 − x2) (xn−1 − xn) xn = Nλ(a1, . . . , an) ····· · . a1! an−1! an! a1,...,an≥0

Comparing the last formula to (3.2.1), and making the substitutions

t1 = x1 − x2, . . . , tn−1 = xn−1 − xn, tn = xn we obtain the identity in the theorem.

3.3 A bijection between diagonal vectors and lattice points

of Pλ

In this section we recall the setup from [16, Section 6]. Let n ∈ N and let e1, . . . , en denote n the standard basis of R . For a subset I ∈ {1, 2, . . . , n}, let ∆I = Conv{ei|i ∈ I}, which is a coordinate simplex of dimension |I| − 1. A class of generalized permutohedra is given by n polytopes in R of the form

y X Pn ({yI }) = yI ∆I ∅6=I⊆{1,...,n}

31 y In other words, Pn ({yI }) is the Minkowski sum of the simplices ∆I rescaled by yI ≥ 0. It y is straightforward to see that if yI = yJ , whenever |I| = |J|, then Pn ({yI }) is the classical

permutohedron Pn(z[n], z[n−1], ..., z{1}), where

X X z[n] = yI , z[n−1] = yI , . . . , z{1} = y{1}. I⊆[n] I⊆[n−1] An extensive study of generalized permutohedra, including their combinatorial structures, volumes, numbers of lattice points was carried by Postnikov and others in [16, 17]. One particular example of a generalized permutohedron, the associahedron, in its Loday realization, is deﬁned as X Assn = ∆[i,j] 1≤i≤j≤n It is also known as the Stasheﬀ polytope [21], and it has generalizations to any Lie type via cluster algebras (see [7], for example).

Proposition 3.3.1. For any subsets I1,...,Ik ⊆ [n], and any non-negative integers a1, . . . , an, a1 an the coeﬃcient of t1 ··· tn in

k   Y X  ti (3.3.1) j=1 i∈Ij Pk is non-zero if and only if (a1, . . . , an) is an integer lattice point of the polytope j=1 ∆Ij .

a1 an Proof. It’s easy to see that the coeﬃcient of t1 ··· tn in (3.3.1) is non-zero if and only

if (a1, . . . , an) can be written as a sum of vertices of the simplices ∆I1 ,..., ∆Ik . By [16, Pk Proposition 14.12], this happens if and only if (a1, . . . , an) is a lattice point of j=1 ∆Ij .

a1 an Proposition 3.3.2. The coeﬃcient of t1 ··· tn in sλ(t1 +···+tn, t2 +···+tn, . . . , tn) is non- zero if and only if (a1, . . . , an) is a lattice point of the polytope λ1∆[1,n]+λ2∆[2,n]+···+λn∆{n}.

Proof. Recall that

X w1 wn sλ(t1 + ··· + tn, t2 + ··· + tn, . . . , tn) = (t1 + ··· + tn) ··· tn , (3.3.2) T

where the sum ranges over all semi-standard Young tableaux T of shape λ and weight w =

(w1, . . . , wn), i.e. wi is the number of i’s appearing in T (see [23]). Let T be a SSYT of shape

λ and weight w. Then w1 + ··· + wi ≤ λ1 + ··· + λi, ∀i = 1 . . . n, because if we consider the boxes containing the numbers 1, 2, . . . , i in T , there can be no more than i of them in the same column. Hence the number of such boxes is at most the size of the ﬁrst i rows in the

Young diagram of λ, which is λ1 + ··· + λi.

32 a1 an w1 wn It follows that any monomial t1 ··· tn appearing in (t1 + ··· + tn) ··· tn also appears λ1 λn λ1 λn in (t1 + ··· + tn) ··· tn . On the other hand, (t1 + ··· + tn) ··· tn does appear in the right side of (3.3.2) as the term corresponding to the tableau T with 1’s in the ﬁrst row, 2’s in the

a1 an second row, etc. Therefore, the coeﬃcient of t1 ··· tn in sλ(t1 + ··· + tn, t2 + ··· + tn, . . . , tn) λ1 λn is non-zero if and only if it is non-zero in (t1 +···+tn) ··· tn , which by Proposition 3.3.1, is

non-zero if and only if (a1, . . . , an) is a lattice point of λ1∆[1,n] + λ2∆[2,n] + ··· + λn∆{n}.

We now have all the tools to establish the ﬁrst main result of this chapter.

Theorem 3.3.3. The number of (distinct) diagonal vectors of λ-shifted Young tableaux is equal to the number of lattice points of the polytope

X Pλ := ∆[i,j] + λ1∆[1,n] + λ2∆[2,n] + ··· + λn∆{n}. 1≤i≤j≤n−1

Proof. By Theorem 3.2.1, and Propositions 3.3.1, 3.3.2 it follows that Nλ(a1, . . . , an) 6= 0 if and only if (a1, . . . , an) is an integer lattice point of the polytope

X ∆[i,j] + λ1∆[1,n] + λ2∆[2,n] + ··· + λn∆{n}. 1≤i≤j≤n−1

In particular, if λ has n parts (i.e. λn > 0), we see that Pλ is combinatorially equivalent

to Assn.

3.4 Vertices of Pλ and extremal Young tableaux

In what follows we describe the vertices of the polytope Pλ by using techniques developed y P in [16]. Given a generalized permutohedron Pn ({yI }) = ∅6=I⊆{1,...,n} yI ∆I , assume that its building set B = {I ⊆ [n]|yI > 0} satisﬁes the following conditions: 1. If I,J ∈ B and I ∩ J 6= ∅, then I ∪ J ∈ B.

2. B contains all singletons {i}, for i ∈ [n].

A B-forest is a rooted forest F on the vertex set [n] such that

1. For any i, desc(i, F ) ∈ B. Here and below, desc(i, F ) denotes the set of descendants of i in F (including i).

2. There are no k ≥ 2 distinct incoparable nodes i1, . . . , ik in F such that

k [ desc(ij,F ) ∈ B j=1

33 .

3. {desc(i, F )|i- root of F } = {I ∈ B|I−maximal}.

We will need the following result of Postnikov:

y Proposition 3.4.1. [16, Proposition 7.9] Vertices of Pn ({yI }) are in bijection with B-forests. y More precisely, the vertex vF = (t1, . . . , tn) of Pn ({yI }) associated with a B-forest F is given P by ti = J∈B:i∈J⊆desc(i,F ) yJ , for i ∈ [n].

Remark 3.4.2. It’s not hard to see that Proposition 3.4.1 remains essentially true even if we allow the building set B not to contain the singletons {i}. This is because a term of the form y{i}∆{i} = u{i}ei in a Minkowski sum just translates the other Minkowski summand.

The combinatorial structure of Pλ clearly only depends on its building set, i.e. the number of non-zero parts of the partition λ. Assume λ has k positive parts, so that the building set of Pλ is

Bn,k = {[i, j]|1 ≤ i ≤ j ≤ n − 1} ∪ {[i, n]|1 ≤ i ≤ k}.

We ﬁrst deal with the case k = n. Let T be a plane binary tree on n nodes. For a node v of T , denote by Lv,Rv the left and right branches at v. There is a unique way to label the nodes of T such that for any node v, its label is greater than all labels in Lv and smaller than all labels in Rv. This labelling is called the binary search labelling of T .

Proposition 3.4.3. [16, Proposition 8.1] The Bn,n-forests are exactly plane binary trees on n nodes with the binary search labeling.

If k = 0, then the building set of Pλ is the same as Bn−1,n−1 hence Bn,0-forests are plane binary trees on n − 1 nodes. The rest of the theory for k = 0 is the same as for the case k = n, but with n replaced by n − 1.

Assume now k ≥ 1. Let T be a Bn-forest. It’s easy to see that desc(x, T ) has form [a, n] if and only if the path from the root to x always goes to the right. In this case, desc(x, T ) = [x − |Lx|, n]. We want to check when desc(x, T ) ∈ Bk. This will happen if and only if x − |Lx| ≤ k or x − |Lx| = n (cf. Remark 3.4.2). But x − |Lx| increases as x moves down to the right starting from the root of the tree, and x − |Lx| = n can only happen when x = n and |Lx| = 0. It follows that {desc(x, T )|x ∈ [n]} ⊆ Bk, ∀x if and only if n ( n − 1 − |Ln−1| ≤ k and |Ln| = 0 {desc(x, T )|x ∈ [n]} ⊆ Bk ⇔ n − |Ln| ≤ k and |Ln| > 0

This argument together with Proposition 3.4.3 implies

34 Proposition 3.4.4. Let k ≥ 1. The Bk-forests are exactly plane binary trees on n nodes

with the binary search labeling and such that either |Ln| ≥ max{n − k, 1}, or |Ln| = 0 and

|Ln−1| ≥ n − 1 − k.

Corollary 3.4.5. For 1 ≤ k ≤ n, the number of vertices of Pλ is

(C0Cn−1 + C1Cn−2 + ··· + Ck−1Cn−k) + (C0Cn−2 + ··· + Ck−1Cn−1−k) (3.4.1)

1 2n th where Cn = n+1 n denotes the n Catalan number, and C−1 is taken to be 0.

Proof. By Propositions 3.4.1 and 3.4.4, the number of vertices of Pλ is equal to the number of plane binary trees T on n nodes such that right-most node v in T has a non-empty (left) subtree Lv of size at least n − k, or v has no descendants and its parent u has at least n − k descendants. In the ﬁrst case, if |L| = i, then there are Ci ways to choose L and Cn−1−i ways to choose the tree T \L ∪ {v}. In the second case, if the size of the left subtree of u is |Lu| = j then there are Cj ways to choose Lu and Cn−2−j ways to choose T \Lu ∪ {u, v}. Summing over i = max{1, n − k}, ..., n − 1 and j = n − 1 − k, ..., n − 2 yields the desired formula.

Remark. There is no diﬀerence in the combinatorial structure of Pλ whether λ has n or n − 1 0 parts. Indeed, if λ = (λ1, ..., λn) and λ = (λ1, ..., λn−1) then Pλ is just the translation of Pλ0

by λnen. In either case, the number of vertices of Pλ is Cn = C0Cn−1 + ... + Cn−1C0. On the

other extreme, if λ has k = 0 parts, then Pλ = Assn−1 has Cn−1 vertices.

To describe the vertices of Pλ, recall that plane binary trees T on n nodes are in bijective

correspondence with the Cn subdivisions of the shifted Young diagram D∅ into n rectangles. This can be deﬁned inductively as follows: Let i be the root of T (in the binary search

labeling). Then draw an (|Li| + 1) × (|Ri| + 1) rectangle. Then attach the subdivisions

corresponding to the binary trees Li,Ri to the left and, respectively, bottom of the rectangle. th For a subdivision Ξ of D∅ into n rectangles, the i rectangle is the rectangle containing th th the i diagonal box of D∅. If T is the binary tree corresponding to Ξ, then the i rectangle of Ξ has size (|Li| + 1) × (|Ri| + 1). In particular, |Ln| + 1 is the length of the (bottom-right) vertical strip of the subdivision Ξ.

Example 3.4.6. Figure 3.4.1 depicts a subdivision of the staircase shape D∅ and the corresponding binary tree with the binary search labeling when n = 4.

We are ﬁnally in a position to prove the second main result of this chapter.

Theorem 3.4.7. Vertices of Pλ are in bijection with subdivisions of the shifted diagram

D∅ into n rectangles such that the bottom-right vertical strip of the subdivision has at least n − k + 1 boxes. Speciﬁcally, let Ξ be such a subdivision. One can obtain a subdivision Ξ∗ of k Dλ−h1ki by merging the rectangles in Ξ with the rows of the Young diagram of λ − h1 i that

35 1 4

3 2 6 4

6 1 3 5

Figure 3.4.1: A subdivision of D∅ and the corresponding labelled binary tree

they border. Then the corresponding vertex of Pλ is vΞ = (t1, . . . , tn), where ti is the number of boxes in the ith region of Ξ∗.

Proof. The ﬁrst part of the theorem follows from Proposition 3.4.4 and the discussion preceed- ing the theorem. To prove the second part, we use Proposition 3.4.1. Recall that the building set of P is B = {[i, j]|1 ≤ i ≤ j ≤ n} ∪ {[i, n]|1 ≤ i ≤ k}, and P = P y ∆ λ k λ [i,j]∈Bk ij [i,j] where yij = 1 if j 6= 1 and yin = λi . Let T be a Bk-forest, i.e. a binary tree on n nodes with the binary search labeling such that |Ln| ≥ n − k (cf. Proposition 3.4.4.) Note that desc(i, T ) = [i − |Li|, i + |Ri|]. Now Proposition 3.4.1 implies that the correponding vertex vT = (t1, . . . , tn) of Pλ is given by

X X ti = yJ = ykl

J∈Bk, i∈J⊆desc(i,F ) [k,l]∈Bk, i−|Li|≤k≤i≤l≤i+|Ri| i X = (|Li| + 1) · |Ri| + yk(i+|Ri|). k=i−|Li|

th If the i rectangle of Ξ borders the right edge of D∅ (i.e. n ∈ desc(i, T )), then ti = (|L | + 1) · |R | + Pi λ . Otherwise, t = (|L | + 1) · (|R | + 1) . In any case, t is the i i k=i−|Li| k i i i i number boxes in the ith region of Ξ∗.

Example 3.4.8. Let n = 4, λ = (4, 2, 1, 0), k = 3. Figure 3.4.2 shows how a subdivision ∗ Ξ of D∅ yields the subdivision Ξ of Dλ−h1ki = D(3,1,0). The corresponding vertex of Pλ is ∗ given by counting boxes in the regions of Ξ : vΞ∗ = (1, 10, 1, 2). It follows that there is a (4,2,1,0)-shifted Young tableau T whose diagonal vector is diag(T ) = (1, 1 + 1 + 1, 1 + 1 + 1 + 10 + 1, 1 + 1 + 1 + 10 + 1 + 2) = (1, 3, 14, 16). On the other hand, one can directly construct λ-shifted Young tableaux with diagonal ∗ vector vΞ∗ = (c1, c2, . . . , cn) by using the subdivision Ξ . Indeed, we know what the diagonal ∗ vector of the tableau (a1, . . . , an) should be. Consider again the subdivision Ξ of Dλ−h1ki.

36 1 1

2 2

3 3

4 4

∗ Figure 3.4.2: Constructing the subdivision Ξ of D(3,1,0) from a subdivision Ξ of D∅.

1 2 numbers

4,5,...,13 3

14 15 17

1 2 4 5 6 9 12 13

3 7 8 10 11

14 15 17

∗ Figure 3.4.3: Constructing shifted Young tableaux from a subdivision Ξ of Dλ−<1k> = D(3,1,0).

We can extend the diagram Dλ−h1ki to Dλ by ﬁrst adding a box to the left of each row of th Dλ−h1ki, and then, by deleting the last n − k boxes in the n column of Dλ−h1ki. Now, we start by putting a1, . . . , an in the diagonal boxes of Dλ. The remaining part of Dλ is divided ∗ into n regions by Ξ . Finally, for each i = 1, . . . , n, put the ci numbers ai + 1, . . . , ai+1 − 1 in the ith region of Ξ∗ in a standard way, i.e. such that entries increase along rows and down columns (as before, we set an+1 = |Dλ| + 1). In this way we obtain a λ-shifted tableau T such that diag(T ) = (a1, . . . , an). Figure 3.4.3 illustrates the above procedure for the subdivision in Example 3.4.8.

Problem 3.4.9. The normalized volume of a polytope is in some sense “dual” to its number of lattice points. Given our results on the lattice points of Pλ, it is natural to ask for a combinatorial interpretation of the normalized volume of Pλ. Are there any combinatorial objects (vectors of Young tableaux, trees, etc) which would give a triangulation of Pλ?

37 Appendix A

Φ Diagram |WΦ| VΦ (u1, ..., un) 1

A1 2 u1 1 2 2 2 u1 u2 A2 6 2 + 2u1u2 + 2 1 2 3 3 3 3 2 2 2 u1 u2 u3 u1u2 u1u2 u1u3 3! + 4 3! + 3! + 2 2 + 4 2 + 3 2 A3 24 2 2 2 u1u3 u2u3 u2u3 +3 2 + 4 2 + 2 2 + 6u1u2u3 1 2 2 2 u1 u2 B2 8 4 2 + 4u1u2 + 2 2 1 2 2 2 u1 u2 C2 8 2 2 + 4u1u2 + 4 2 2 1 2 2 u1 u2 G2 12 6 2 + 12u1u2 + 18 2 1 2 3 3 3 3 2 2 2 u1 u2 u3 u1u2 u1u2 u1u3 8 3! + 40 3! + 6 3! + 16 2 + 32 2 + 12 2 B3 48 2 2 2 u1u3 u2u3 u2u3 +12 2 + 24 2 + 12 2 + 24u1u2u3 1 2 3 3 3 3 2 2 2 u1 u2 u3 u1u2 u1u2 u1u3 4 3! + 20 3! + 24 3! + 8 2 + 16 2 + 12 2 C3 48 2 2 2 u1u3 u2u3 u2u3 +24 2 + 24 2 + 24 2 + 24u1u2u3

38 Φ Diagram |WΦ| VΦ (u1, ..., un) 4 4 4 4 u1 u2 u3 u4 4! + 11 4! + 11 4! + 4! + 24u1u2u3u4 3 3 3 3 3 3 u1u2 u3u2 u4u2 u1u2 u3u2 u4u2 +2 3! + 14 3! + 3 3! + 8 3! + 14 3! + 17 3! u3u u3u u3u u3u u3u u3u +3 1 3 + 17 3 1 + 4 1 4 + 4 1 + 8 3 4 + 2 4 3 1 2 3 4 3! 3! 3! 3! 3! 3! 2 2 2 2 2 2 2 2 2 2 2 2 u1u2 u3u2 u4u2 u1u3 u1u4 u3u4 A4 120 4 2!2! + 16 2!2! + 9 2!2! + 9 2!2! + 6 2!2! + 4 2!2! 2 2 2 2 u1u2u3 u2u3u4 u1u3u4 u1u2u4 6 2 + 2 + 16 2 + 2 2 2 2 2 u1u2u4 u1u3u4 u1u2u3 u2u3u4 +8 2 + 2 + 18 2 + 2 2 2 2 2 u1u3u4 u1u2u3 u2u3u4 u1u2u4 +12 2 + 2 + 2 + 2 4 4 4 4 u1 u2 u3 u4 16 4! + 192 4! + 368 4! + 24 4! + 192u1u2u3u4 3 3 3 3 3 u3u2 u1u2 u3u2 u4u2 u3u1 +352 3! + 128 3! + 256 3! + 160 3! + 336 3! u3u u3u u3u u3u u3u u3u +48 1 3 + 4 2 + 4 3 + 4 1 + 32 1 2 + 1 4 1 2 3 4 3! 3! 3! 3! 3! 3! 3 2 2 2 2 2 2 2 2 u3u4 u1u2 u3u2 u4u2 u3u4 B4 384 +192 3! + 64 2!2! + 320 2!2! + 96 2!2! + 2!2! 2 2 2 2 2 2 u1u3 u1u4 u1u2u4 u1u2u3 +144 2!2! + 48 2!2! + +128 2 + 288 2 2 2 2 2 2 u1u2u4 u1u2u3 u2u3u4 u1u3u4 u2u3u4 64 2 + 192 2 + 2 + 2 + 2 2 2 2 2 2 u1u2u3 u1u3u4 u2u3u4 u1u2u4 u1u3u4 +96 2 + 2 + 2 + 2 + 2 4 4 4 4 u1 u2 u3 u4 8 4! + 96 4! + 8 4! + 8 4! + 48u1u2u3u4 3 3 3 3 3 3 u1u2 u3u2 u4u2 u1u2 u3u2 u4u2 +16 3! + 3! + 3! + 64 3! + 3! + 3! 4 3 3 3 3 3 3 u1u3 u3u1 u1u4 u4u1 u3u4 u4u3 1 3 +12 3! + 3! + 3! + 3! + 3! + 3! 2 2 2 2 2 2 2 2 2 2 2 2 2 u1u2 u3u2 u4u2 u1u3 u1u4 u3u4 D4 192 +32 2!2! + 2!2! + 2!2! + 12 2!2! + 2!2! + 2!2! 2 2 2 2 2 u1u2u3 u1u2u4 u1u3u4 u1u2u3 u2u3u4 +24 2 + 2 + 2 + 2 + 2 2 2 2 2 u1u3u4 u1u2u4 u2u3u4 u1u3u4 +24 2 + 2 + 2 + 2 2 2 2 u1u2u3 u1u2u4 u2u3u4 +48 2 + 2 + 2 4 4 4 4 16u1 + 232u2 + 58u3 + 4u4 + 1152u1u2u3u4+ 3 3 2 2 3 3 128u1u2 + 512u1u2 + 384u1u2 + 672u2u3 + 336u2u3 2 2 3 3 2 2 3 +720u2u3 + 128u3u4 + 32u3u4 + 96u3u4 + 96u1u3 1 2 3 4 +208u u3 + 216 u2u3 + u2u2 + 416u3u + 48u u3 F 1152 1 3 1 3 2 4 2 4 2 4 4 3 3 2 2 2 2 +64u1u4 + 32u1u4 + 72u1u4 + 576 u1u2u3 + u2u3u4 2 2 2 2 +768u1u2u4 + 288 u1u3u4 + u1u2u4 + u2u3u4 2 2 2 +384 u1u2u4 + u1u3u4 + 192u1u3u4 2 2 2 +1152u1u2u3 + 864 u2u3u4 + u1u2u3

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