Combinatorics of affine Springer fibers and combinatorial wall-crossing

Combinatorics of Affine Springer Fibers and

Combinatorial Wall-Crossing

by

Guangyi Yue

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

May 2020

c

○ Massachusetts Institute of Technology 2020. All rights reserved.

Author . . . .

Department of Mathematics

April 20, 2020

Certified by . . . .

Roman Bezrukavnikov

Professor of Mathematics

Thesis Supervisor

Accepted by . . . .

Davesh Maulik

Chairman, Department Committee on Graduate Theses

Combinatorics of Affine Springer Fibers and Combinatorial

Wall-Crossing

by

Guangyi Yue

Submitted to the Department of Mathematics on April 20, 2020, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

This thesis deals with several combinatorial problems in representation theory.

The first part of the thesis studies the combinatorics of affine Springer fibers of type A. In particular, we give an explicit description of irreducible components of ℱ 𝑙𝑡𝑆 and calculate the

relative positions between two components. We also study the lowest two-sided Kazhdan-Lusztig cell and establish a connection with the affine Springer fibers, which is compatible with the affine matrix ball construction algorithm. The results also prove a special case of Lusztig’s conjecture. The work in this part include joint work with Pablo Boixeda.

In the second part, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition. This result gives a special situation where column regularization, can be used to understand the complicated Mullineux map, and also proves a special case of Bezrukavnikov’s conjecture. Furthermore, we prove a condition under which the two maps are exactly the same, generalizing the work of Bessenrodt, Olsson and Xu. The combinatorial constructions is related to the Iwahori-Hecke algebra and the global crystal basis of the basic 𝑈𝑞(sl𝑏)-module and we provide several

conjectures regarding the 𝑞-decomposition numbers and generalizations of results due to Fayers. This part is a joint work with Panagiotis Dimakis and Allen Wang.

Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics

Acknowledgements

I am grateful to my advisor Roman Bezrukavnikov for suggesting research problems and providing support and advice during the past five years. Also I am benefited from very helpful discussions with Richard Stanley and Zhiwei Yun. This thesis is based on joint work and papers with Pablo Boixeda, Panagiotis Dimakis and Allen Wang, and I want to thank them for collaboration.

Contents

2.3 Bruhat Order and Demazure Product . . . 26

3 Affine Matrix Ball Construction and Kazhdan-Lusztig Cells of Affine Type A 29 3.1 Affine Matrix Ball Construction – Algorithm . . . 30

3.2 Column-Type Permutations . . . 33

3.3 Inverses under Affine Matrix Ball Construction . . . 37

3.4 Knuth Classes . . . 40

3.5 Structure of Lowest Two-sided Cell . . . 42

4 Combinatorics of Affine Springer Fibers 57 4.1 Affine Springer Fibers ℱ 𝑙𝑡𝑆 . . . 57

4.2 Finite Flag Varieties and the Robinson-Schensted Correspondence . . . 62

4.3 Combinatorial Lemmas . . . 64

4.4 𝐺(𝒪)-orbits and Irreducible Components of ℱ 𝑙𝑡𝑆 . . . 68

4.6 Relationship with Affine Matrix Ball Construction . . . 84

5 Mullineux Involution and the Column Regularization 95 5.1 Two Equivalent Definitions of Mullineux Transpose . . . 95

5.2 Regularization and Column Regularization . . . 102

5.3 Relationship between the Two Operations . . . 106

5.4 Conjectures . . . 120

6 Combinatorial Wall-Crossing and Bezrukavnikov’s Conjecture 125 6.1 Two Series of Transformations . . . 126

6.2 Uniqueness of Monotonicity . . . 138

6.3 Bezrukavnikov’s Conjecture . . . 140

A Diagrams of 𝒜 and the Associated Components 𝑌𝑤 141

List of Figures

2-1 Illustration of Example 2.1.9 for 𝜆 = (6, 5, 3, 3, 2, 1, 1) and 𝑏 = 4. . . 22

3-1 Illustration for a zig-zag. The red boxes are its inner corner-posts, blue boxes are its outer corner-posts and the star denotes its back corner-post. . . 31

3-2 Affine matrix ball construction for column-shape permutations - Step 1 and 2. 35 3-3 Affine matrix ball construction for column-shape permutations - Step 3 and 4. 36 4-1 Illustration of Example 4.6.5 of 𝜑2_(𝑡𝑐_𝑌 𝑠0) ⊂ 𝐷𝑦, where 𝑛 = 4, 𝑐 = (2, −1, 0, −1), and 𝑦 = [3, 9, −8, 6]. In the figure, 𝑐1 = (0, 0, 1, −1), 𝑐2 = (0, 0, 1, −1), 𝑐3 = (0, −1, 2, −1), 𝑐4 = (0, −1, 2, −1). . . 89

5-1 Residues of boxes in 𝜆 = (5, 4, 2). . . 97

5-2 Build (5, 4, 2)M4T _{= (4, 2, 2, 2, 1) from co-good sequences. . . . 98}

5-3 Illustration of the proof of Lemma 5.1.10. . . 99

5-4 The truncated 3-rims for (7, 5, 1, 1) and (7, 2, 1). . . 100

5-5 The 5-rectangular decomposition of (12, 9, 9, 7, 5, 2, 2, 1). . . 101

5-6 An example of a Colreg_2,3-valid partition (3, 2, 2, 1) and an Colreg_2,3-invalid one (3, 2, 2). . . 103

5-7 An example of column semi-regularization Colseg_2,5: (13, 10, 9, 7, 5, 2, 2, 1)Colseg2,5 ₌ (10, 10, 7, 6, 2, 2, 1). . . 106

5-8 Illustration of the statement of Lemma 5.3.2 on (8, 8, 7, 6, 6, 1) with 𝑎 = 2 and 𝑏 = 5. . . 108

5-9 Illustration of 𝐸𝑖𝑐 = (𝑖𝑐, 𝑗 ′_{) and a box (𝑖}′′_{, 𝑗}′_{) in 𝒮 north of it. The rectangle} 𝑟𝛽 is colored in yellow. . . 109

5-10 Illustration of the shape of 𝒮 and ℰ in the proof of Lemma 5.3.2. Boxes shaded black are in 𝒮. Boxes shaded in red are in ℰ . . . 110 5-11 Illustration of the argument in the proof of Lemma 5.3.2 on (8, 8, 7, 6, 6, 1)

with 𝑎 = 2 and 𝑏 = 5. The boxes in 𝒮 are shaded with black lines and ℰ is shaded with red lines. They intersect at (5, 5), which determines the hook 𝐻3,5 with 𝑎3,5 = 𝑏 − 𝑎 − 1 = 2 and 𝑙3,5 = 𝑎 = 2, outlined in blue. . . 110

5-12 Illustration of the statement of Lemma 5.3.4. The ladders ℓ1,𝑦 are labeled. A

hook of shape (5.4) is outlined in blue. Key boxes in ℓ1,𝑦 ∖ 𝜆 are drawn in

dotted lines. . . 111 5-13 Illustration of the proof of Lemma 5.3.4. . . 112 5-14 Illustration of the definition of 𝜔(𝜆) and 𝜓(𝜆). . . 113 5-15 Illustration of the statement of Proposition 5.3.8 on 𝜆 = (11, 8, 7, 5, 3, 2, 2, 2),

with 𝑎 = 2 and 𝑏 = 5. The yellow rectangles are 𝑟1, 𝑟2, and 𝑟3 of the

5-rectangular decomposition. . . 113 5-16 Illustration of the inductive step for the case when 𝑎 = 2 and 𝑏 = 5. The

left side shows a rectangle with width 𝑟𝑥 = 𝑎, and the right side shows a

rectangle with width 𝑟𝑥 _{< 𝑎. In this case, the boxes (𝑖 + 𝑎, 𝑗 − (𝑏 − 𝑎)) and}

(𝑖 + 𝑎 − 1, 𝑗 − (𝑏 − 𝑎)) are both not in 𝜆. . . 114 5-17 Illustration of Lemma 5.3.11 on 𝜆 = (13, 10, 9, 7, 5, 2, 2, 1) with 𝑎 = 2 and

𝑏 = 5. The shaded boxes are removed in the corresponding operator. . . 116

5-18 Illustration of 𝐺1 for (8, 5, 3, 1) when 𝑎 = 2 and 𝑏 = 5. The boxes in

{rim of 𝜆} ∖ {5-rim of 𝜆} are colored in yellow and each point to a corre-sponding box of 𝐺1. Notice that the boxes in the first row of ℓ1,7 and ℓ1,8 slide

exactly to the boxes of 𝐺1 when applying Colseg2,5. . . 117

5-19 The hand box h𝑖,𝑗(︀𝜆Colseg𝑎,𝑏)︀ = (𝑖, 𝑗′) comes from a box in the first row of 𝜆,

but the foot box does not. The boxes of the 𝑏-gap are shaded in the figure, which comes from the first row of 𝜆. 𝐻𝑖,𝑗(︀𝜆Colseg𝑎,𝑏)︀ is outlined by black lines

and 𝐻𝑖−𝑎+1,𝑗(𝜆) is colored in red. . . 119

6-2 Illustration of the proof of Theorem 6.1.5. . . 131 6-3 Illustration of Step 1 in the proof of Theorem 6.1.5. . . 131 6-4 Illlustration of Remark 6.1.7. . . 132 6-5 Illustration of Step 4 in the proof of Theorem 6.1.5: 𝐿𝑐0 and 𝐿𝑐1 have the

same residue. . . 134 6-6 Illustration of Step 4 in the proof of Theorem 6.1.5: 𝐿𝑐0 and 𝐿𝑐1 have different

residues. . . 135 6-7 Illustration of the proof of Corollary 6.1.12: quotient before wall-crossing. . . 137 6-8 Illustration of the proof of Corollary 6.1.12: quotient after wall-crossing. . . . 138

A-1 Illustration of 𝒜 = LKC𝑤0 when 𝑛 = 3 and the corresponding irreducible

components. . . 141 A-2 Illustration of all irreducible components of ℱ 𝑙𝑡𝑆 when 𝑛 = 3 up to translation.141

A-3 Illustration of 𝒜 = LKC𝑤0 when 𝑛 = 4 and the corresponding irreducible

components. . . 142 A-4 Illustration of 𝒜 = LKC𝑤0 when 𝑛 = 5. . . 143

A-5 Illustration of some irreducible components of ℱ 𝑙𝑡𝑆 when 𝑛 = 4 in 𝐺(𝒪)-orbits

Chapter 1

Introduction

This thesis contains two parts.

The first part deals with a combinatorial problem of affine Springer fibers. The Robinson-Schensted correspondence is a bijection between pairs of Young tableau and the symmetric group:

𝑤 ∈ 𝑆𝑛↦→ (insertion tableau 𝑃, recording tableau 𝑄).

This well-known correspondence are realized by many equivalent combinatorial algorithms, for example the row-insertion algorithm and the matrix ball construction [15, 52].

Robinson-Schensted correspondence appears in the study of (finite) Springer fibers. Namely, given a nilpotent 𝑁 of type 𝜆, the irreducible components of the Springer fiber of 𝑁 are la-beled by standard Young tableau of shape 𝜆. Moreover, the relative position between two components labeled by tableaux 𝑃 and 𝑄 respectively are exactly the permutation given by the Robinson-Schensted algorithm for 𝑃 and 𝑄. The Kazhdan-Lusztig cells of the sym-metric group can also be read easily from the Robinson-Schensted correspondence, where the two-sided cells are given by the shape of the Young tableau, right cells are given by the insertion tableaux 𝑃 , and left cells are given by the recording tableaux 𝑄. If we fix the first (resp. second) component in the relative position map, the image is exactly a right (resp. left) cell. These nice interpretations are studied by Spaltenstein [46], Steinberg [48], and further extended by van Leeuwen [50]. It is very natural to find an analogue in the affine setting.

First of all, the Robinson-Schensted correspondence is generalized by Shi [42] as a sur-jection of the affine symmetric group onto pairs of Young tabloids of the same shape. Still the shape (as a partition), the insertion tabloid, the recording tabloid gives two-sided cells, right cells, left cells respectively. Later Honeywill [22] added the third piece of data, weights, to make it a bijection:

𝑤 ∈ ̃︁𝑆𝑛↦→ (insertion tabloid 𝑃, recording tabloid 𝑄, dominant weight 𝜌).

Chmutov, Pylyavskyy, Yudovina [9] generalized the matrix ball construction given by Viennot and Fulton to give a simpler and more intuitive realization of Shi’s algorithm. In addition, the weights 𝜌 get a natural interpretation. This generalized algorithm, named the affine matrix ball construction, has a variety of nice applications. In particular, it is used to understood the structure of bi-directed edges in the Kazhdan-Lusztig cells in affine type A in [8]. Most importantly, fibers of the inverse map of affine matrix ball construction possess a Weyl group symmetry, which motivates a solution to the relative position map being no longer injective on pairs of irreducible components of the affine Springer fiber.

On the geometry side, Lusztig [35] conjectured in a more general setting (not necessarily type A) that the image of the map (see precise definition in [35]):

Irr(ℱ 𝑙𝑁) × Irr(ℱ 𝑙𝑁) → 𝑊𝑎𝑒 (1.1)

is exactly the 𝑆-cell of type 𝛾 where 𝑁 is a regular semi-simple topologically nilpotent of type 𝛾. The notion of 𝑆-cells coincide with two-sided cells in type A, which could be parameterized by partitions. In Chapter 4, we study the irreducible components of the affine Springer fiber ℱ 𝑙𝑡𝑆 in type A of the particular nil-element 𝑡𝑆 (of type (1𝑛)), following [7], and parametrize

the components modulo the lattice action by tabloids of column-shape. The method taken is very different from the finite situation since for finite flags, the nilpotent naturally acts on subspaces or quotient flags hence we get a saturated chain of partitions. Since affine flags are saturated periodic chains of lattices in C((𝑡))𝑛, so we could restrictions would not work any more. One possible solution may be studying the action of 𝑡𝑆 on quotient lattices, but this is much more complicated than the finite situation. The other approach we take

towards the problem is to study the fixed points, since we could obtain an upper bound of the relative position of two components using the maximum fixed point of each. We then label the components by computing the image of the relative position under affine matrix ball construction. The map 𝑟 is a surjection onto the lowest two-sided cell of ̃︁𝑆𝑛 containing

all column-type permutations. Finally we establish a bijection between pairs of irreducible components modulo common translations with triples (𝑃, 𝑄, 𝜌) of column-type where 𝜌 is not necessarily dominant:

Irr(ℱ 𝑙𝑡𝑆)/Λ Irr(ℱ 𝑙𝑡𝑆) ×ΛIrr(ℱ 𝑙𝑡𝑆) 𝑆̃︁_𝑛

𝑇 (1𝑛_{) Ω} 1𝑛 𝑓 𝑝𝑟𝑖 𝑟 𝐹 𝑝𝑟𝑖 Ψ (1.2)

The commutative diagram above not only proved Lusztig’s conjecture in type A and the regular semi-simple nilpotent being column-type, but also obtained a more detailed corre-spondence with left (resp. right) cells and solved the non-injectivity problem of the map in Equation (1.1).

The natural conjecture is that for regular semi-simple topologically nilpotent element 𝑁 ∈ sl𝑛(C((𝑡))) of type 𝜆 ⊢ 𝑛 other than (1𝑛), we still have a bijection from the components

pair to the corresponding triples which is also compatible with the inverse of affine matrix ball construction.

The second part of this thesis studies the wall-crossing transformation in a combinatorial way. Wall-crossing functors appear in the context of infinite-dimensional representations of complex semisimple Lie algebras, and Beilinson and Ginzburg studied its relation with translation functors in [3]. More recently, wall-crossing functors have appeared in the study of quantized symplectic resolutions of singularities as perverse equivalences between different categories of modules, for more details one can look at [1, 6, 31]. These perverse equivalences induce bijections between irreducible objects in the corresponding derived categories, which are referred to as the combinatorial wall-crossing. In the classical case of Lie algebra rep-resentations they are related to the cactus group actions [19, 32]. It is called combinatorial wall-crossing because in case of rational Cherednik algebras of type 𝐴, the derived categories

are parametrized by rational numbers, and the bijection among two categories parametrized by consecutive rational numbers (with denominator bounded above) is like crossing a wall. Our work is motivated by combinatorial wall-crossing for representations of rational Chered-nik algebras in large positive characteristic, where the combinatorial wall-crossing is given by the extended Mullineux involution due to Losev [31]. This collection of permutations on the set of partitions is our main object of study.

Based on Kleshchev’s work in [14, 29], the irreducible 𝑝-modular representations of the symmetric group 𝑆𝑛 are labeled by the 𝑝-regular partitions of 𝑛; we denote the irreducible

representation corresponding to the 𝑝-regular partition 𝜆 by 𝜌𝜆. Then the Mullineux

invo-lution M𝑝 is the involution on the set of 𝑝-regular partitions satisfying

𝜌_𝜆M𝑝 = 𝜌_𝜆⊗ sgn

where sgn is the sign representation.

There are a few combinatorial ways to define M𝑝 in [14, 29], where 𝑝 is not necessarily

prime, and this is the foundation of our investigation.

In Chapter 5, we study the relationship of the Mullineux involution and the general-ized column regularization. One of the purpose is to simplify the complicated algorithm of Mullineux involution despite it has a simple representation-theoretic definition. This was first studied by Walker, Bessenrodt, Olsson, Xu, and Fayers, where they studied the com-binatorial properties of Mullineux involution by relating it to the (column) regularization map Reg_𝑏 (resp. Colreg_𝑏). These two operations are naturally related by the Iwahori-Hecke

algebra ℋ = ℋF,𝑞(𝑆𝑛) of the symmetric group where F is a field and 𝑞 ∈ F ∖ {0}. Let 𝑏

be the minimal integer satisfying 1 + 𝑞 + · · · + 𝑞𝑏−1 = 0 and the decomposition numbers

𝑑𝜆,𝜇 = [︀𝑆𝜆 : 𝐷𝜇]︀ be the multiplicity of the simple module 𝐷𝜇 (𝜇 is 𝑏-regular) inside the

Specht module 𝑆𝜆 of ℋ. Previous works show that the identities for these decomposition

numbers involve certain combinatorics of partitions. James proved that 𝑑𝜆,𝜇 = 0 unless

𝜇 E 𝜆Reg1,𝑏 _{and 𝑑}

𝜆,𝜆Reg𝑏 = 1 [23]. And later, Lascoux, Leclerc and Thibon proved the identity

𝑑𝜆,𝜇= 𝑑𝜆T_,𝜇M𝑏 [30]. Therefore, it is natural to study the relationship between the Mullineux

In combinatorics, Walker proved that when the partition 𝜆 is horizontal or row-stable, it satisfies 𝜆M𝑏T _{= 𝜆}Colreg𝑏 [53, 54]. Later, Bessenrodt, Olsson, and Xu showed in [4] that

Walker’s conditions can be broadened to the short-legged (or shallow) partitions, namely for every hook in the partition divisible by 𝑏, the length of the corresponding arm is at least (𝑏−1) times that of the leg. Even better, Bessenrodt, Olsson, and Xu proved that those partitions are the only ones satisfying 𝜆M𝑏T_{= 𝜆}Colreg𝑏. Fayers generalized Bessenrodt, Olsson, and Xu’s

result by considering partitions that are not necessarily 𝑏-regular. The Specht module 𝑆𝜆 is reducible when 𝜆Reg𝑏M𝑏 ̸= 𝜆T Reg𝑏and Fayers proved in [13] that the identity 𝜆Reg𝑏M𝑏 _{= 𝜆}T Reg𝑏

holds if and only if hooks divisible by 𝑏 must be either shallow or steep. However, in all these previous works, the operators Reg_𝑏 and Colreg_𝑏only involved a single parameter 𝑏. The second parameter was added in the joint paper [11] with P. Dimakis, where the motivation

is explained in the Chapter 6. Parameters of (resp. column) regularization Reg_𝑎,𝑏 (resp.

Colreg_𝑎,𝑏) were extended to any positive rational number 𝑎_𝑏 in the unit interval. The goal of this chapter is to choose suitable parameters 𝑎 and finding the condition under which Mullineux transpose is identical to the generalized column regularization, generalizing the result of Bessenrodt, Olsson, and Xu in [4]. The main result we obtained is that for given positive integers 𝑎 < 𝑏, if 𝜆 is a partition satisfying 𝜆Colreg𝑎,𝑏 ∈ 𝒫 and all hooks 𝐻

𝑖,𝑗 in 𝜆 with 𝑏 | 𝐻𝑖,𝑗 satisfy: (︂ 𝑏 𝑎 − 1 )︂ 𝑙𝑖,𝑗 < 𝑎𝑖,𝑗+ 1, (1.3)

then we have 𝜆M𝑏T _{= 𝜆}Colreg𝑎,𝑏_{. This theorem is proved combinatorially by completely}

characterizing the shape of the partitions satisfying the inequalities.

In Chapter 6, we study the behavior of one-row partition (𝑛) under composition of a series of wall-crossing transformations intensively. A strong monotonicity property motivates the generalized version of column regularization on partitions, which was introduced in Chapter 5. We proved that the combinatorial wall-crossing and a certain composition of generalized column regularization procedures have the same effect on the one-row partition.

The most important consequence of this result, stated as Theorem 6.2.4, is that this one-row partition case is the only case where monotonicity holds at each step of the composition of the transformations. We were kindly informed by Losev that he has an alternative proof of

Theorem 6.2.4 using Heisenberg actions and perverse equivalences while our method is purely combinatorial. Also Theorem 6.2.4 answers a question by Bezrukavnikov which is motivated by potential applications to the study of nabla operators and Haiman’s 𝑛! conjecture in [18]. In addition, this result is a special case of Bezrukavnikov’s conjecture and the work in Chapter 4 also shed light on solving other cases of the conjecture.

The rest of the thesis is organized as follows. The second chapter is a review of prelimi-naries on both topics. Chapter 3 deals with the algorithm of affine matrix ball construction and the properties of the lowest two-sided cell containing column-shape permutations. In particular, we study the Knuth class containing 𝑤0 and relate it to the fundamental box. In

Chapter 4, we follow [7] and compute relative positions between irreducible components of

ℱ 𝑙𝑡𝑆 by fixed points, and establish a bijection between components modulo common

trans-lations with triples of two Young tabloids and a weight. Chapter 5 is about the Mullineux involution and generalized column regularization. We prove that under the shallow hook conditions, the two maps are equivalent. We also provide several conjectures regarding the 𝑞-decomposition numbers and generalizations of results due to Fayers. In the last chapter, we study the wall-crossing transformation and the one-row case of Bezrukavnikov’s conjecture and prove the monotonicity property combinatorially. In Appendix A, we explicitly draw the Hasse diagram of the left Knuth class 𝒜 containing 𝑤0 and the corresponding irreducible

components 𝑌𝑤 with the associated 𝛼𝑖-fibrations when 𝑛 = 3, 4, 5. Also we include examples

of the irreducible components in the 𝐺(𝒪)-orbits that are not irreducible. Finally we present computations of Bezrukavnikov’s conjecture when 𝑛 = 5 in Appendix B.

Chapter 2

Preliminaries

2.1

Notational Preliminaries

2.1.1

Partitions

A partition 𝜆 of 𝑛 ∈ N is a finite tuple of weakly decreasing positive integers 𝜆 = (𝜆1, ..., 𝜆𝑘)

where 𝜆1 ≥ ... ≥ 𝜆𝑘 > 0 and |𝜆| :=

∑︀𝑘

𝑖=1𝜆𝑖 = 𝑛. The exponential version of a partition is

𝜆 = (𝜆𝑠1

1 , ..., 𝜆 𝑠𝑘

𝑘 ), where the superscript 𝑠𝑖 indicates the number of repetitions of the part 𝜆𝑖

and 𝜆1 > ... > 𝜆𝑘. Denote all partitions by 𝒫 and partitions of 𝑛 by 𝒫𝑛. Denote ℓ(𝜆) = 𝑘 to

be the number of nonzero parts of 𝜆. Given two positive integers 𝑖 ≤ 𝑗, denote 𝜆[𝑖,𝑗] to be

the subpartition (𝜆𝑖, . . . , 𝜆𝑗). And for simplicity, we define the concatenation of two finite

positive integer sequences 𝜆 and 𝜇 as the tuple 𝜆 ⊕ 𝜇 =(︀𝜆1, . . . , 𝜆𝑙(𝜆), 𝜇1, . . . , 𝜇𝑙(𝜇))︀. We use

matrix coordinates throughout this thesis by fixing the x-axis pointing to the south and the y-axis pointing to the east.

The Young diagram of a given partition 𝜆 is the set of unit boxes whose southeast vertices are given by:

{(𝑖, 𝑗) ∈ N × N | 1 ≤ 𝑖, 1 ≤ 𝑗 ≤ 𝜆𝑖} ,

which is a left-justified collection of boxes with the first row having 𝜆1 boxes, second row

having 𝜆2 boxes and so on. Therefore southeast vertex of the box on the 𝑖-th row, 𝑗-th

𝜆 is given by:

{(𝑖, 𝑗) ∈ N × N | 1 ≤ 𝑗, 1 ≤ 𝑖 ≤ 𝜆𝑗} .

Given a box (𝑖, 𝑗) ∈ 𝜆, the arm 𝑎𝑖𝑗 = 𝑎𝑖𝑗(𝜆) is the set of boxes (𝑖, 𝑗′) ∈ 𝜆 with 𝑗 <

𝑗′. We use 𝑎𝑖𝑗 to denote either the above set or the number of elements of the above set

interchangeably. Similarly, the leg 𝑙𝑖𝑗 = 𝑙𝑖𝑗(𝜆) is the set of boxes (𝑖′, 𝑗) ∈ 𝜆 with 𝑖 < 𝑖′. We use

𝑙𝑖𝑗 to denote either the above set or the number of elements of the above set interchangeably

as well. Finally the hook 𝐻𝑖𝑗 = 𝐻𝑖𝑗(𝜆) is the union of sets (𝑖, 𝑗) ∪ 𝑎𝑖𝑗 ∪ 𝑙𝑖𝑗. The number

of elements of the hook is also denoted by 𝐻𝑖𝑗 = 1 + 𝑎𝑖𝑗 + 𝑙𝑖𝑗. The northeast-most (resp.

southwest-most) box in 𝐻𝑖,𝑗 is called the hand (resp. foot) box associated to (𝑖, 𝑗), denoted

by h𝑖,𝑗 = h𝑖,𝑗(𝜆) = (𝑖, 𝑗 + 𝑎𝑖,𝑗) (resp. f𝑖,𝑗 = f𝑖,𝑗(𝜆) = (𝑖 + 𝑙𝑖,𝑗, 𝑗)).

Let 𝜆 ∈ 𝒫𝑛. A box 𝐴 ∈ 𝜆 is called a removable box of 𝜆 if 𝜆 ∖ 𝐴 ∈ 𝒫𝑛−1. A box 𝐵 /∈ 𝜆

is called an addable box of 𝜆, if 𝜆 ∪ 𝐵 ∈ 𝒫𝑛+1. The rim of 𝜆 consists of the boxes (𝑖, 𝑗) ∈ 𝜆

such that (𝑖 + 1, 𝑗 + 1) /∈ 𝜆. The boundary of 𝜆 is defined to be the rim of ˜𝜆 where

˜

𝜆 = 𝜆 ⋃︁

(𝑖,𝑗) addable to 𝜆

(𝑖, 𝑗).

A skew shape 𝜆/𝜇, where 𝜇 ⊂ 𝜆, is the collection of boxes in 𝜆 but not in 𝜇. If 𝜆/𝜇 does not contain any 2 × 2 squares, then it is called a ribbon. Note that every (𝑖, 𝑗) ∈ 𝜆 corresponds to a ribbon of size 𝐻𝑖𝑗 containing in the rim of 𝜆.

Definition 2.1.1. Given two positive integers 𝑎 < 𝑏 and a partition 𝜆. A hook 𝐻𝑖,𝑗 in 𝜆 is

(𝑎, 𝑏)-shallow if it satisfies:

(︂ 𝑏

𝑎 − 1

)︂

𝑙𝑖,𝑗 < 𝑎𝑖,𝑗+ 1. (2.1)

Dually, a hook 𝐻𝑖,𝑗 is (𝑎, 𝑏)-steep if it satisfies:

(︂ 𝑏

𝑎 − 1

)︂

𝑎𝑖,𝑗 < 𝑙𝑖,𝑗+ 1. (2.2)

𝑡 ∈ P; and similarly, 𝐻𝑖,𝑗 is (𝑎, 𝑏)-steep iff 𝑎𝑖,𝑗 ≤ 𝑡𝑎 − 1 and 𝑙𝑖,𝑗 ≥ 𝑡(𝑏 − 𝑎) for some 𝑡 ∈ N>0.

In particular, the above inequalities hold when 𝑡 = 𝐻𝑖,𝑗/𝑏.

Lemma 2.1.3. A hook divisible by 𝑏 can be both (𝑎, 𝑏)-shallow and (𝑎, 𝑏)-steep only if 𝑏 < 2𝑎.

Proof. By the above remark, by taking 𝑡 = 𝐻𝑖,𝑗/𝑏, we know 𝑡(𝑏 − 𝑎) ≤ 𝑎𝑖,𝑗 ≤ 𝑡𝑎 − 1 and

𝑡(𝑏 − 𝑎) ≤ 𝑙𝑖,𝑗 ≤ 𝑡𝑎 − 1 for some 𝑡 ∈ N>0. Hence 𝑡(𝑏 − 𝑎) ≤ 𝑡𝑎 − 1, and we conclude 𝑏 < 2𝑎.

For two partitions 𝜆 and 𝜇 of the same size, the dominance order is defined as 𝜆_{E 𝜇 if}

∑︀𝑘

𝑖=1𝜆𝑖 ≤

∑︀𝑘

𝑖=1𝜇𝑖 is satisfied for all 𝑘. Note that 𝜆E 𝜇 iff 𝜇

T_{E 𝜆}T_.

Fix a number 𝑏 ∈ N≥2. We will call a Young diagram 𝜆 𝑏-regular if there exist no 𝑖 ∈ N

such that 𝜆𝑖 = 𝜆𝑖+1 = ... = 𝜆𝑖+𝑏−1 > 0. Also for a box 𝐴 = (𝑖, 𝑗) the residue of 𝐴 with

respect to 𝑏, denoted by res 𝐴, is the residue class (𝑗 − 𝑖) mod 𝑏.

Definition 2.1.4. Given a partition 𝜆 and a positive integer 𝑏, 𝜆 can be uniquely written as a union of multisets

𝜆 = 𝜈 ∪ 𝜇

where each part of 𝜈 has multiplicity less than 𝑏 and each part of 𝜇 has multiplicity being a multiple of 𝑏. Denote Reg_𝑏(𝜆) = 𝜈 as the regular part of 𝜆 and the irregular part Irr𝑏(𝜆)

is defined by 𝜇 = 𝑏 ⋆ Irr𝑏(𝜆), where the operator 𝑏 ⋆ is to repeat each part of the partition 𝑏

times. This decomposition is called the 𝑏-regular decomposition of 𝜆. Next, we define the core and quotient of a partition following [18].

Definition 2.1.5. A partition 𝜆 is a 𝑏-core if it does not contain any ribbon of length 𝑏. The 𝑏-core Core𝑏(𝜆) of any partition 𝜆 is the partition that remains after one removes as many

𝑏-ribbons in succession as possible. The result is independent of choices of removals.

Definition 2.1.6. The 𝑏-content of the partition 𝜆 is a tuple (𝑐0, . . . , 𝑐𝑏−1) where 𝑐𝑖 is the

number of boxes in 𝜆 with residue 𝑖.

Proposition 2.1.7 ( [37]). The 𝑏-content determines the 𝑏-core of a partition.

Definition 2.1.8. For any box 𝐴 = (𝑖, 𝑗) ∈ 𝜆, let 𝐵 and 𝐶 be the boxes at the end of the arm 𝑎𝑖𝑗 and the leg 𝑙𝑖𝑗 respectively. Then 𝐻𝑖𝑗 is divisible by 𝑏 precisely when res 𝐵 = 𝑘 and

res 𝐶 = 𝑘 + 1 for some 𝑘 ∈ {0, 1, ..., 𝑏 − 1}. Now for some fixed 𝑘 the boxes 𝐴 with res 𝐵 = 𝑘 and res 𝐶 = 𝑘 + 1 form an "exploded" copy of a partition which we denote 𝜆𝑘. The quotient

of a partition 𝜆 is defined to be the 𝑏-tuple of partitions, Quot_𝑏(𝜆) = (𝜆0, 𝜆1, ..., 𝜆𝑏−1).

Example 2.1.9. Let 𝜆 = (6, 5, 3, 3, 2, 1, 1) and 𝑏 = 4, and the residue of the rim of 𝜆 are labeled in Figure 2-1. After removing the 4 pieces of 4-ribbons, we obtain Core4(𝜆) = (4, 1).

2 3 0 1 2 3 0 1 2 3 0 1

Figure 2-1: Illustration of Example 2.1.9 for 𝜆 = (6, 5, 3, 3, 2, 1, 1) and 𝑏 = 4.

Quot₄(𝜆) = ((1), (2, 1), ∅, ∅), as shown in the above picture.

Young Tableaux and Tabloids

Now fix a positive integer 𝑛 and denote [𝑛] = {1, . . . , 𝑛} and [𝑛] = {1, . . . , 𝑛} where 𝑖 = 𝑖+𝑛Z is the residue class of 𝑖. A (Young) tableaux of shape 𝜆 ∈ 𝒫𝑛is a filling of the Young diagram

of 𝜆 with [𝑛], i.e. a bijection 𝜆 → [𝑛]. A tableaux is called standard if it is increasing along each row and each column. We denote the collection of tableaux of shape 𝜆 by YT(𝜆) and standard tableaux of shape 𝜆 by SYT(𝜆). Two tableaux of the same shape are considered row-equivalent if one can be obtained from the other by permuting elements within rows. A (Young) tabloid of shape 𝜆 is such an equivalence class. And we denote its collection by

T(𝜆). Note T(1𝑛_{) is exactly the symmetric group 𝑆}

𝑛.

2.1.2

Affine Permutations

The symmetric group 𝑆𝑛 is the Weyl group of type 𝐴𝑛−1. And we have various ways to

understand it:

2. The collection of window notations [𝑤1, . . . , 𝑤𝑛] containing each element of [𝑛] exactly

once;

3. 𝑛 × 𝑛 permutation matrices;

4. Group generated by 𝑠1, . . . , 𝑠𝑛−1 and Coxeter relations 𝑠2𝑖 = 1 and (𝑠𝑖𝑠𝑗)𝑚𝑖𝑗 = 1 for

1 ≤ 𝑖 ̸= 𝑗 ≤ 𝑛 − 1 where 𝑚𝑖𝑗 = 3 if |𝑗 − 𝑖| = 1 and 𝑚𝑖𝑗 = 2 otherwise.

The affine symmetric group ̃︁𝑆𝑛 is the affine Weyl group of type ˜𝐴𝑛−1, we have the

analogous descriptions as follows:

1. All bijections 𝑤 : Z → Z satisfying:

𝑤(𝑖 + 𝑛) = 𝑤(𝑖) + 𝑛 𝑛 ∑︁ 𝑖=1 𝑤(𝑖) = (𝑛 + 1)𝑛 2 ;

2. The collection of window notations [𝑤1, . . . , 𝑤𝑛] such that {𝑤1, . . . , 𝑤𝑛} = [𝑛] and

∑︀𝑛

𝑖=1𝑤𝑖 = (𝑛+1)𝑛

2 ;

3. Z × Z permutation matrices with conditions defined in 1. or 2.;

4. When 𝑛 > 2, this is also equivalent to the group generated by 𝑠1, . . . , 𝑠𝑛−1, 𝑠0 = 𝑠𝑛 and

Coxeter relations 𝑠2

𝑖 = 1 and (𝑠𝑖𝑠𝑗)𝑚𝑖𝑗 = 1 for 0 ≤ 𝑖 ̸= 𝑗 ≤ 𝑛 − 1 where 𝑚𝑖𝑗 = 3 if

|𝑗 − 𝑖| = 1 and 𝑚𝑖𝑗 = 2 otherwise. (For convenience, we will take the indices 𝑖 to be in

[𝑛] without ambiguity.)

When 𝑛 = 2, ̃︀𝑆2 = ⟨𝑠1, 𝑠0 = 𝑠2⟩ with relations 𝑠20 = 𝑠21 = 1.

There are two well-known formulas for computing length of an affine permutation, the first one is given by Shi [42]:

Proposition 2.1.10. For 𝑤 ∈ ̃︁𝑆𝑛, we have:

ℓ(𝑤) = ∑︁ 𝑛≥𝑖>𝑗≥1 ⃒ ⃒ ⃒ ⃒ ⌊︂ 𝑤(𝑖) − 𝑤(𝑗) 𝑛 ⌋︂⃒ ⃒ ⃒ ⃒ ; (2.3) ℓ(𝑤) = # {(𝑖, 𝑗) ∈ [𝑛] × P | 𝑖 < 𝑗, 𝑤(𝑖) > 𝑤(𝑗)} . (2.4)

The extended affine symmetric group is the collection of all periodic bijections 𝑤 : Z → Z (i.e. we drop the second condition in the first description) and hence 𝑆𝑛 = Ω n ̃︁𝑆𝑛 where

Ω is the infinite cyclic group generated by 𝑠 = [2, 3, . . . , 𝑛 + 1] : Z → Z, 𝑖 ↦→ 𝑖 + 1. Let 𝜑(𝑤) = 𝑠𝑤𝑠−1 be the conjugation by 𝑠. This is an automorphism of ̃︁𝑆𝑛, i.e. 𝑠𝑠𝑖𝑠

−1 _{= 𝑠} 𝑖+1

for 𝑖 ∈ [𝑛], which corresponds to the rotation of the Dynkin diagram of type ]𝐴𝑛−1.

2.2

Kazhdan-Lusztig Cells

In the paper [26], Kazhdan and Lusztig laid a foundation for studying representation of Hecke algebras, and in particular they introduced the notion of cells, whose structure has significant relation with combinatorics.

Let (𝑊, 𝑆) be a Coxeter system where 𝑆 is the set of simple reflections and denote the length function by ℓ and the Bruhat order by ≤. The Hecke algebra ℋ of (𝑊, 𝑆) over 𝒜[𝑞, 𝑞−1] is an associative algebra with 𝒜-basis {𝑇𝑤 | 𝑤 ∈ 𝑊 } and relations:

(𝑇𝑠− 𝑞2)(𝑇𝑠+ 1) = 0 for 𝑠 ∈ 𝑆

𝑇𝑤𝑇𝑤′ = 𝑇_𝑤𝑤′ if ℓ(𝑤𝑤′) = ℓ(𝑤) + ℓ(𝑤′)

Now we define the bar involution of 𝒜 by 𝑞 ↦→ 𝑞 = 𝑞−1 and a bar involution on ℋ by:

∑︁ 𝑤∈𝑊 𝑐𝑤𝑇𝑤 ↦→ ∑︁ 𝑤∈𝑊 𝑐𝑤𝑇𝑤 = ∑︁ 𝑤∈𝑊 𝑎𝑤𝑇_𝑤−1−1.

The well known Kazhdan-Lusztig basis {𝐶𝑤 | 𝑤 ∈ 𝑊 } of ℋ are the unique elements

satisfy-ing: 𝐶𝑤 = 𝐶𝑤 𝐶𝑤 = 𝑞−ℓ(𝑤) ∑︁ 𝑤′_≤𝑤 𝑃𝑤′_,𝑤(𝑞2)𝑇_𝑤′

where 𝑃𝑤′_,𝑤 are called the Kazhdan-Lusztig polynomials, which have degree no more than

1

2(ℓ(𝑤) − ℓ(𝑤

′_{) − 1) when ℓ(𝑤) > ℓ(𝑤}′_{), and 𝑃}

When 𝑤′ _{ 𝑤, we denote 𝑤}′ ≺ 𝑤 if deg(𝑃𝑤′_,𝑤) = 1

2(ℓ(𝑤) − ℓ(𝑤

′_{) − 1) and 𝑤}′ _{− 𝑤 if}

𝑤′ ≺ 𝑤 and 𝑤 ≺ 𝑤′_.

For each 𝑤 ∈ 𝑊 , set the left an right descent set of 𝑤 as:

𝐿(𝑤) = {𝑠 ∈ 𝑆 | 𝑠𝑤 ≤ 𝑤}

𝑅(𝑤) = {𝑠 ∈ 𝑆 | 𝑤𝑠 ≤ 𝑤}.

Three preorders are defined on 𝑊 as follows. For 𝑤, 𝑤′ ∈ 𝑊 , 𝑤 ≤𝐿 𝑤′ (resp. 𝑤 ≤𝑅 𝑤′;

𝑤 ≤𝐿𝑅 𝑤′) if there is a sequence 𝑤 = 𝑤0, 𝑤1, . . . , 𝑤𝑘 = 𝑤′ in 𝑊 such that 𝑤𝑖−1 − 𝑤𝑖 and

𝐿(𝑤𝑖−1) * 𝐿(𝑤𝑖) (resp. 𝑅(𝑤𝑖−1) * 𝑅(𝑤𝑖), 𝐿(𝑤𝑖−1) * 𝐿(𝑤𝑖) or 𝑅(𝑤𝑖−1) * 𝑅(𝑤𝑖)). Then

there are the equivalence relations: 𝑤 ∼𝐿 𝑤′ (resp. 𝑤 ∼𝑅 𝑤′; 𝑤 ∼𝐿𝑅 𝑤′) if 𝑤 ≤𝐿 𝑤′ ≤𝐿 𝑤

(resp. 𝑤 ≤𝑅 𝑤′ ≤𝑅 𝑤; 𝑤 ≤𝐿𝑅 𝑤′ ≤𝐿𝑅 𝑤) and the corresponding equivalence classes are

called left cells (resp. right cells; two-sided cells). The preorder ≤𝐿(resp. ≤𝑅; ≤𝐿𝑅) induces

a partial order on the set of left (resp. right; two-sided) cells of 𝑊 .

In fact the above definitions extends to extended Coxeter groups. See [57] for details. In case of type 𝐴𝑛−1 and ̃︀𝐴𝑛−1, the cells are well-studied in [34, 42, 57]. We recall the

explicit description of cells in these type ̃︀𝐴𝑛−1.

We now follow [8, 9] and use the third description of elements in ̃︁𝑆𝑛. In detail, for

𝑤 ∈ ̃︁𝑆𝑛, we draw a Z × Z matrix with row labels increasing downwards and column labels

increasing rightwards. The positions in the matrix are called boxes analogue to that of Young diagrams. If 𝑤(𝑖) = 𝑗, we draw a ball in the (𝑖, 𝑗) box and will be named also by (𝑖, 𝑗) without ambiguity. And we denote ℬ𝑤 = {(𝑖, 𝑤(𝑖)) | 𝑖 ∈ Z} which is the collection of

the balls of 𝑤. The periodicity of 𝑤 implies that (𝑖, 𝑗) ∈ ℬ𝑤 iff (𝑖 + 𝑛, 𝑗 + 𝑛) ∈ ℬ𝑤. We

say (𝑖 + 𝑘𝑛, 𝑗 + 𝑘𝑛) for 𝑘 ∈ Z are the (𝑛, 𝑛)-translates of (𝑖, 𝑗). For two balls (or boxes) (𝑖, 𝑗), (𝑘, 𝑙) ∈ ℬ𝑤, we define the southeast (partial) ordering ≤𝑆𝐸 by (𝑖, 𝑗) ≤𝑆𝐸 (𝑘, 𝑙) iff 𝑖 ≥ 𝑘

and 𝑗 ≥ 𝑙, i.e. (𝑖, 𝑗) is southeast of (𝑘, 𝑙). Other relations using compass directions can be defined similarly, and are also partial orders on Z × Z.

Definition 2.2.1. Given an (extended) affine permutation 𝑤, a subset 𝐶 ⊂ ℬ𝑤 is called

a stream if it is invariant under (𝑛, 𝑛)-translations and forms a chain under the southeast

called the density of 𝐶. A stream 𝐶 is a channel if its density is maximal among all streams of ℬ𝑤.

From Lusztig we could associate 𝜆(𝑤) = (𝑑1, 𝑑2 − 𝑑1, 𝑑3− 𝑑2, . . .) to any extended affine

permutation 𝑤 where 𝑑𝑖 is the sum of densities of 𝑖 disjoint streams in ℬ𝑤. [16, Theorem 1.5]

guarantees that 𝜆(𝑤) is a partition and is called the partition associated to 𝑤.

Remark 2.2.2. Note that in [34,42,57], the partition associated to an extended affine permu-tation is conjugate to that in [8, 9]. We adopt the latter nopermu-tation since they are consistent with the combinatorial Robinson-Schensted-Knuth algorithm.

Theorem 2.2.3 ( [34, 42, 45]). 1. For two (extended) affine permutations 𝑤, 𝑤′, 𝜆(𝑤) =

𝜆(𝑤′) iff they are in the same two-side cell.

2. The number of left (resp. right) cells in the two-sided cell corresponding to 𝜆 is the multinomial coefficient:

𝑛𝜆 =

𝑛! 𝜆1! · · · 𝜆ℓ(𝜆)!

.

3. There exists a bijection between left (resp. right) cells in a two-sided cell corresponding to 𝜆 and tabloids of shape 𝜆.

The construction of the bijection in the third part of the above theorem is firstly described by Shi [45] and is incorporated into the affine matrix ball construction [8, 9] which we will explain in detail in the next chapter.

2.3

Bruhat Order and Demazure Product

Let (𝑊, 𝑆) be a Coxeter system, the weak left order ≤𝐿, weak right order ≤𝑅, and the strong

order (or Bruhat order ) ≤ are generated by the following covering relations respectively:

𝑤 l𝐿𝑠𝑤 ⇐⇒ ℓ(𝑠𝑤) = ℓ(𝑤) + 1 for 𝑠 ∈ 𝑆;

𝑤 l𝑅𝑤𝑠 ⇐⇒ ℓ(𝑤𝑠) = ℓ(𝑤) + 1 for 𝑠 ∈ 𝑆;

It is well-known the definition of Bruhat order has the equivalent version using the subword property:

Let 𝑤 = 𝑠1. . . 𝑠𝑘 be a reduced expression. Then 𝑤′ ≤ 𝑤 iff there exists a reduced

expression 𝑤′ = 𝑠𝑖1. . . 𝑠𝑖𝑙 such that 1 ≤ 𝑖1 < . . . < 𝑖𝑙 ≤ 𝑘.

Let 𝐻 be the 0-Hecke algebra of (𝑊, 𝑆) by taking 𝑞 = 0 in ℋ and let 𝑇_𝑠′ = −𝑇𝑠 and

𝑇_𝑤′ = 𝑇_𝑠′₁. . . 𝑇_𝑠′

𝑘 for 𝑤 = 𝑠1. . . 𝑠𝑘 being a reduced expression in 𝑊 . The Demazure product *

is a binary operation on 𝑊 such that 𝑇_𝑣′· 𝑇_𝑢′ = 𝑇_𝑣*𝑢′ . More explicitly, this is equivalent to: Definition 2.3.1. We define the Demazure product of 𝑢, 𝑣 ∈ 𝑊 to be:

𝑢 * 𝑣 = max {𝑢′𝑣′ | 𝑢′ ≤ 𝑢, 𝑣′ ≤ 𝑣} .

where the set {𝑢′𝑣′ | 𝑢′ _{≤ 𝑢, 𝑣}′ _{≤ 𝑣} contains a unique maximum. Moreover there is the}

following nice property:

Lemma 2.3.2 ( [21]). Given 𝑢, 𝑣 ∈ 𝑊 , 𝑢 * 𝑣 = 𝑢′𝑣 = 𝑢𝑣′ for some 𝑢′ ≤ 𝑢 and 𝑣′ _{≤ 𝑣 and}

ℓ(𝑢 * 𝑣) = ℓ(𝑢′) + ℓ(𝑣) = ℓ(𝑢) + ℓ(𝑣′).

Hence we could define the Demazure action 𝜏_𝑠𝐿, 𝜏_𝑠𝑅: 𝑊 → 𝑊

𝜏_𝑠𝐿(𝑤) = 𝑠 * 𝑤 = ⎧ ⎨ ⎩ 𝑤, if 𝑤 > 𝑠𝑤 ; 𝑠𝑤, otherwise. (2.5) 𝜏_𝑠𝑅(𝑤) = 𝑤 * 𝑠 = ⎧ ⎨ ⎩ 𝑤, if 𝑤 > 𝑤𝑠 ; 𝑤𝑠, otherwise. (2.6) The maps{︀𝜏𝐿

𝑠 | 𝑠 ∈ 𝑆}︀ and {︀𝜏𝑠𝑅 | 𝑠 ∈ 𝑆}︀ satisfy braid relations identical that in 𝑊 , hence for

𝑢 = 𝑠1. . . 𝑠𝑘 being a reduced expression, we could define without ambiguity 𝜏𝑢𝐿 = 𝜏𝑠𝐿1. . . 𝜏

𝐿 𝑠𝑘 and 𝜏𝑅 𝑢 = 𝜏𝑠𝑅𝑘. . . 𝜏 𝑅 𝑠1 and then 𝜏 𝐿 𝑢(𝑣) = 𝑢 * 𝑣 = 𝜏𝑣𝑅(𝑢).

Chapter 3

Affine Matrix Ball Construction and

Kazhdan-Lusztig Cells of Affine Type A

In this chapter, we first review the affine matrix ball construction given in [9] by Chmutov, Pylyavskyy and Yudovina, which is a very nice generalization of Viennot’s geometric con-struction [52] of the Robinson-Schensted correspondence, named the matrix ball concon-struction by Fulton [15]. Then we study properties of the lowest two-sided cell in detail in preparation for the next chapter.

The well-known Robinson-Schensted correspondence is a bijection between the symmetric group and pairs of standard Young tableaux of the same shape:

𝑆𝑛 Φ  Ψ ⨆︁ 𝜆⊢𝑛 SYT(𝜆) × SYT(𝜆) (3.1) Ψ(𝑤) = (𝑃 (𝑤), 𝑄(𝑤)) (3.2)

The shape of 𝑃 (𝑤) or 𝑄(𝑤) is labelled as 𝜆(𝑤). It gives the Kazhdan-Lusztig cell structure easily by:

1. 𝑤 ∼𝐿𝑅 𝑤′ iff 𝜆(𝑤) = 𝜆(𝑤′);

2. 𝑤 ∼𝐿𝑤′ iff 𝑄(𝑤) = 𝑄(𝑤′);

In affine type A, from Theorem 2.2.3, we have the analogue result for the description of two-sided cells using the Greene-Kleitman invariants 𝜆(𝑤) for 𝑤 in the affine Weyl group ̃︁𝑆𝑛.

Shi constructed explicitly a tabloid 𝑃 (𝑤) of shape 𝜆(𝑤) for 𝑤 ∈ ̃︁𝑆𝑛 to determine the right

cell where 𝑤 lies in and therefore establishes the bijection between right cells and tabloids. An analogue bijection is obtained by Honeywill in [22] by involving the third data, weights. However the construction as well as Shi’s correspondence is a very complicated process involving a sequence of Knuth moves (star operations) to change the affine permutation into a standard form and also requires pre-computing the Greene-Kleitman invariants. This is finally solved by the affine matrix ball construction, nicely generalizes the matrix ball construction [9], which could be computed easily.

Φ : ̃︁𝑆𝑛 → Ω = ⨆︁ 𝜆⊢𝑛 {︃ (𝑃, 𝑄, 𝜌) ⃒ ⃒ ⃒ ⃒ ⃒ 𝑃, 𝑄 ∈ 𝑇 (𝜆), 𝜌 ∈ Zℓ(𝜆),∑︁ 𝑖 𝜌𝑖 = 0 }︃ (3.3) Φ(𝑤) = (𝑃 (𝑤), 𝑄(𝑤), 𝜌(𝑤)). (3.4)

Φ is an injection onto Ωdom ⊂ Ω consisting of (𝑃, 𝑄, 𝜌)’s where 𝜌 is a dominant weight

corresponding to the pair (𝑃, 𝑄). Most importantly, the affine matrix ball construction has a inverse:

Ψ : Ω → ̃︁𝑆𝑛

and Ψ(𝑃, 𝑄, 𝜌) = Ψ(𝑃′, 𝑄′, 𝜌′) iff 𝑃 = 𝑃′, 𝑄 = 𝑄′ and 𝜌, 𝜌′ belongs to the same orbit of the parabolic subgroup of ̃︁𝑆𝑛, which we will elaborate later. This inverse is crucial to our study

of relative positions between two components of the affine Springer fiber later.

3.1

Affine Matrix Ball Construction – Algorithm

First we recall the construction of Φ following [9] and apply it to the affine permutation of column-shape, i.e. 𝜆(𝑤) = (1𝑛_).

Definition 3.1.1. A zig-zag is a sequence of boxes 𝑍 = (𝐶1, . . . , 𝐶𝑘) such that either 𝑘 = 1

or 𝑘 > 1 and the following holds:

2. 𝐶𝑖+1is adjacent to and are either directly east or directly north of 𝐶𝑖, for 𝑖 = 1, . . . , 𝑘−1.

Given a zig-zag 𝑍 = (𝐶1, . . . , 𝐶𝑘), we say:

1. the back corner-post is the box in the same column of 𝐶1 and same row of 𝐶𝑘;

2. the inner corner-posts are boxes of 𝑍 such that no other boxes directly northwest to it is in 𝑍;

3. if 𝑘 ≥ 2, the outer corner-posts are boxes of 𝑍 such that no other boxes directly southeast to it is in 𝑍; if 𝑘 = 1, there are no outer corner-posts. See Figure 3-1 for an illustration.

⋆

Figure 3-1: Illustration for a zig-zag. The red boxes are its inner corner-posts, blue boxes are its outer corner-posts and the star denotes its back corner-post.

In this section we adopt the notation of partial (affine) permutations. We say (𝑈, 𝑤) is a partial permutation pair if 𝑤 : 𝑈 → Z is an 𝑛-periodic injection defined only on an 𝑛-periodic subset 𝑈 ⊂ Z, i.e.

1. 𝑖 ∈ 𝑈 ⇐⇒ 𝑖 + 𝑛 ∈ 𝑈 ; 2. 𝑤(𝑖 + 𝑛) = 𝑤(𝑖) + 𝑛.

We will also use ℬ𝑤 to denote the collection of balls in a partial permutation 𝑤. And the

to number the balls in (partial) affine permutations by numberings 𝑑 : ℬ𝑤 → Z, we say 𝑑 is

semi-periodic with period 𝑚 if 𝑑(𝑥 + (𝑛, 𝑛)) = 𝑑(𝑥) + 𝑚 and the special kind of numbering we require is defined as follows.

Definition 3.1.2. A function 𝑑 : ℬ𝑤 → Z is a proper numbering of the partial permutation

𝑤 if it is:

1. monotone: for any 𝑥, 𝑦 ∈ ℬ𝑤 where 𝑥 <𝑁 𝑊 𝑦, then 𝑑(𝑥) < 𝑑(𝑦);

2. continuous: for any 𝑦 ∈ ℬ𝑤, there exists 𝑥 ∈ ℬ𝑤 and 𝑥 <𝑁 𝑊 𝑦, 𝑑(𝑥) < 𝑑(𝑦).

Proposition 3.1.3 ( [9]). All proper numberings of a partial permutation 𝑤 are semi-periodic with period being the density of the channel of 𝑤.

Now suppose 𝑤 has a channel 𝐶. We could give 𝐶 a proper numbering ˜𝑑, which is unique

up to a constant integer. The the numbering on 𝐶 induces a numbering 𝑑𝐶_𝑤 : ℬ𝑤 → Z on all

balls of 𝑤 by: 𝑑𝐶_𝑤(𝑏) = max 𝑘 (𝑏=𝑏max0,𝑏1...,𝑏𝑘) (︁ ˜_𝑑(𝑏 𝑘) + 𝑘 )︁

where the second maximum is taken over all path of length 𝑘 from 𝑏0 = 𝑏 to 𝑏𝑘 ∈ 𝐶 such

that 𝑏𝑖+1lies strictly northwest of 𝑏𝑖. It is shown in [9] that 𝑑𝐶𝑤 is well-defined and is a proper

numbering of 𝑤. Then 𝑑𝐶_𝑤 gives a collection of zig-zags {𝑍𝑖}𝑖∈Z where 𝑍𝑖 is the unique one

whose inner corner-posts are the balls of 𝑤 labelled by 𝑖. Let 𝑏𝑖 be the back corner-post of

𝑍𝑖, then st(𝑤) = {𝑏𝑖}𝑖∈Z forms a stream. Note the number of translation classes in st(𝑤)

equals the density of 𝐶. Then denote fw(𝑤) to be the partial permutation with balls being the outer-corner posts of the 𝑍𝑖’s.

Definition 3.1.4. For any box (𝑖, 𝑗), we define its block diagonal 𝐷(𝑐) by:

𝐷(𝑐) =⌈︂ 𝑗 𝑛 ⌉︂ −⌈︂ 𝑖 𝑛 ⌉︂ .

And for any (𝑛, 𝑛)-translation invariant set 𝑋 of boxes, we define 𝐷(𝑋) =∑︀

𝑥𝐷(𝑥), where

the sum runs over the subset of 𝑋 containing each translation representative exactly once. If 𝑋 is a stream, 𝐷(𝑋) is called the altitude of 𝑋.

Proposition 3.1.5 ( [9]). Given two subsets 𝐴, 𝐵 ⊂ [𝑛] and 𝑟 ∈ Z, there is a unique stream of altitude 𝑟 with row indices being 𝐴 and column indices being 𝐵, which will be denoted st𝑟(𝐴, 𝐵), and 𝐴, 𝐵, 𝜌 is called the defining data of the stream.

The algorithm of affine matrix ball construction Φ(𝑤) is given as follows: 1. Input 𝑤 ∈ ̃︁𝑆𝑛 and initialize (𝑃, 𝑄, 𝜌) to be all empties.

2. Repeat until 𝑤 is the empty partition:

(a) Record the defining data of st(𝑤) in the next rows of 𝑄, 𝑃, 𝜌 respectively; (b) Replace 𝑤 by fw(𝑤).

3. Output the triple (𝑃, 𝑄, 𝜌) ∈ Ωdom.

When restricting to the finite permutations, this algorithm gives identical result as the Robinson-Schensted correspondence, using either the classical row insertion algorithm, or the matrix ball construction, giving 𝑃 and 𝑄 as standard Young tableaux and 𝜌 = 0.

We refer to [9] for more details and examples and emphasize at the end of this section that similar to the finite Weyl group 𝑆𝑛, affine matrix ball construction gives the Kazhdan-Lusztig

cell structure by:

1. 𝑤 ∼𝐿𝑅 𝑤′ iff 𝜆(𝑤) = 𝜆(𝑤′);

2. 𝑤 ∼𝐿𝑤′ iff 𝑄(𝑤) = 𝑄(𝑤′);

3. 𝑤 ∼𝑅𝑤′ iff 𝑃 (𝑤) = 𝑃 (𝑤′);

3.2

Column-Type Permutations

We apply the affine matrix ball algorithm to column case partitions. 𝜆(𝑤) = (1𝑛) implies at each recursive step of the algorithm, the stream formed by the outer corner-posts has only 1 translation class, by Greene [16], this is equivalent to saying up to (𝑛, 𝑛)-translations, 𝑤 has a unique chain (under the northeast-southwest ordering) of length 𝑛, i.e., there exist indices 𝑖1, . . . , 𝑖𝑛, such that:

1. 𝑖1 < 𝑖1 < . . . < 𝑖𝑛;

2. 𝑖𝑗− 𝑖𝑗−1 ≤ 𝑛 − 1, for 𝑗 ∈ [2, 𝑛];

3. 𝑤(𝑖1) > 𝑤(𝑖2) > . . . > 𝑤(𝑖𝑛);

4. {︀𝑖1, . . . , 𝑖𝑛}︀ = [𝑛].

From the algorithm, back corner-posts of each step are exactly

(𝑖1, 𝑤(𝑖𝑛)), (𝑖2, 𝑤(𝑖𝑛−1)), . . . , (𝑖𝑛, 𝑤(𝑖1)),

see Figure 3-2 and 3-3 for a step-by-step illustration, hence we know:

Φ(𝑤) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝑤(𝑖𝑛) 𝑤(𝑖𝑛−1) .. . 𝑤(𝑖1) , 𝑖1 𝑖2 .. . 𝑖𝑛 , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⌈︁

⌉︁

−

⌈︀

⌉︀

⌈︁

⌉︁

−

⌈︀

⌉︀

...

⌈︁

⌉︁

−

⌈︀

⌉︀

We will be studying mostly the affine permutations 𝑤 with 𝜆(𝑤) = (1𝑛_{), which we will call}

column-type or column-shape permutations for simplicity. The two-sided cell 𝐶1𝑛 formed by

all such permutations is the lowest two-sided cell, which is lowest in the sense of the preorder ≤𝐿𝑅 and the uniqueness is guaranteed by Lusztig’s 𝑎-function [33]. The lowest two-sided cell

has been extensively studied in [17, 43, 44, 56]. Since 𝑇 (1𝑛_{) = 𝑆}

𝑛, we will write the tabloids

𝑃, 𝑄 horizontally as a permutation in 𝑆𝑛 in order to save space. The right (resp. left) cells in

this two-sided cell are therefore labeled by 𝑆𝑛, where we denote them as {𝑅𝑃 | 𝑃 ∈ 𝑇 (1𝑛)}

(resp. {𝐿𝑄| 𝑄 ∈ 𝑇 (1𝑛)}),

⋆ (𝑖2, 𝑤(𝑖2)) (𝑖3, 𝑤(𝑖3)) (𝑖1, 𝑤(𝑖1)) (𝑖4, 𝑤(𝑖4)) (𝑖1, 𝑤(𝑖4)) ○ ○ ○ ○ ⋆ (𝑖2, 𝑤(𝑖1)) (𝑖3, 𝑤(𝑖2)) (𝑖4, 𝑤(𝑖3)) (𝑖2, 𝑤(𝑖3)) ○ ○ ○

Figure 3-2: Affine matrix ball construction for column-shape permutations - Step 1 and 2.

𝑆𝑛= ⟨𝑠1, . . . , 𝑠𝑛−1⟩, from the formula above, we have

Φ(𝑤0) = (𝑖𝑑, 𝑖𝑑, 0).

Note the 𝑖𝑑 in the triple refers to identity in 𝑆𝑛, where we identify it with the tabloid with 1

on the first row, 2 on the second row and so on.

Example 3.2.2. ̃︀𝑆2 is the disjoint union of two two-sided cells, one containing only the

⋆ (𝑖3, 𝑤(𝑖1)) (𝑖4, 𝑤(𝑖2)) (𝑖3, 𝑤(𝑖2)) ○ ○ ⋆ (𝑖4, 𝑤(𝑖1)) ○

Figure 3-3: Affine matrix ball construction for column-shape permutations - Step 3 and 4.

and there are no relations between 𝑠0 and 𝑠1. By computation, there is the following:

Φ(︀[2𝑘 + 2, 1 − 2𝑘] = 𝑠1(𝑠0𝑠𝑘1))︀ = ([1, 2], [1, 2], (−𝑘, 𝑘)), 𝑘 ≥ 0;

Φ(︀[−1 − 2𝑘, 2 + 2𝑘] = (𝑠1𝑠0)𝑘+1)︀ = ([1, 2], [2, 1], (−𝑘, 𝑘)), 𝑘 ≥ 0;

Φ(︀[2𝑘 + 1, 2 − 2𝑘] = (𝑠0𝑠1)𝑘)︀ = ([2, 1], [1, 2], (−𝑘, 𝑘)), 𝑘 ≥ 1;

Φ(︀[−2𝑘, 3 + 2𝑘] = 𝑠0(𝑠1𝑠0)𝑘)︀ = ([2, 1], [2, 1], (−𝑘, 𝑘)), 𝑘 ≥ 0.

We will use the rotation map

a lot afterwards, it is useful to see the image of rotation under the affine matrix ball con-struction: Lemma 3.2.3. For 𝑤 ∈ ̃︁𝑆𝑛, Φ(︀𝜑𝑘(𝑤))︀ =(︁𝑃 (𝑤) + 𝑘, 𝑄(𝑤) + 𝑘, 𝜌(𝑤) + 𝛿𝑘(𝑃 (𝑤)) − 𝛿𝑘(𝑄(𝑤)))︁ where 𝛿𝑘_𝑖 (𝑃 (𝑤)) = ⎧ ⎨ ⎩ 0, if 𝑃𝑖(𝑤) ∈ [𝑛 − 𝑘] ; 1, otherwise.

Proof. The matrix balls of 𝜑𝑘_{(𝑤) come from that of 𝑤 by shifting southwestwards by (𝑘, 𝑘).}

The relative positions of the balls does not change, so do the numberings at each step. Therefore the coordinate of the back corner posts in each step of the algorithm will shift

southeastwards by (𝑘, 𝑘) as well. By equation (3.5), 𝑃 (𝜑𝑘(𝑤)) = 𝑃 (𝑤) + 𝑘, 𝑄(𝜑𝑘(𝑤)) =

𝑄(𝑤) + 𝑘, and 𝜌𝑖(𝜑𝑘(𝑤)) = 𝜌𝑖(𝑤) + ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1, if 𝑄𝑖(𝑤) ∈ [𝑛 − 𝑘], 𝑃𝑖(𝑤) ∈ [𝑛 − 𝑘 + 1, 𝑛] ; −1, if 𝑃𝑖(𝑤) ∈ [𝑛 − 𝑘], 𝑄𝑖(𝑤) ∈ [𝑛 − 𝑘 + 1, 𝑛] ; 0, otherwise.

3.3

Inverses under Affine Matrix Ball Construction

In the finite case, Schützenberger [41] proved 𝑃 (𝑤−1) = 𝑄(𝑤), 𝑄(𝑤−1) = 𝑃 (𝑤). In the affine case, Φ(𝑤−1) behaves more complicated which is solved by affine matrix ball construction. We follow [8] to understand the fiber of the inverse map Ψ : Ω → ̃︁𝑆𝑛 as well as 𝜌(𝑤−1).

Since elements in tabloids can be understand also as [𝑛], we specify the broken order of the residue classes as:

1 < 2 < . . . < 𝑛 (3.6)

Definition 3.3.1 ( [8]). Let 𝑇 ∈ 𝑇 (𝜆) and 𝜆𝑖 = 𝜆𝑖+1, denote 𝑇𝑖, 𝑇𝑖+1 to be the 𝑖, 𝑖 + 1-th row

Fix any linear ordering of the elements in 𝑇𝑖: 𝑎1, . . . , 𝑎𝑚, and suppose 𝑇𝑖+1 has elements

𝑏1, . . . , 𝑏𝑚. For 𝑗 from 1 to 𝑚, match 𝑎𝑗 to the smallest (in broken order) unmatched 𝑏𝑘 > 𝑎𝑗

if such 𝑏𝑘 exists, otherwise match it to the smallest unmatched 𝑏𝑘. The local charge in row

𝑖 of 𝑇 is defined as:

lch𝑘(𝑇 ) = #{matched pairs (𝑎, 𝑏) ∈ 𝑇𝑖× 𝑇𝑖+1| 𝑎 > 𝑏}.

The definition is independent of the choice of linear ordering of 𝑇𝑖.

Definition 3.3.2. Given 𝑇 ∈ 𝑇 (𝜆), the symmetrized offset constant 𝑠(𝑇 ) of 𝑇 is a vector in

Zℓ(𝜆) _{defined as:} 𝑠𝑖(𝑇 ) = 𝑖−1 ∑︁ 𝑗=𝑖′ lch𝑗(𝑇 )

where 𝑖′ is the first row in 𝜆 with the same length as 𝜆𝑖, if 𝑖′ = 𝑖 − 1, the sum is considered

as 0. We also define the charge of 𝑇 by:

charge(𝑇 ) = ℓ(𝜆)−1 ∑︁ 𝑖=1 𝑖 · lch𝑖(𝑇 ) where lch𝑖(𝑇 ) = 0 if 𝜆𝑖 > 𝜆𝑖+1.

Now we are able to define the notion of dominant weights, which is different from that in representation theory of the special linear group.

Definition 3.3.3. The weight 𝜌 in the triple (𝑃, 𝑄, 𝜌) is called dominant, if 𝜌 − 𝑠(𝑃 ) + 𝑠(𝑄) is increasing segmentwise according to the part sizes of 𝜆, i.e., for each 𝑖, either 𝜆𝑖 > 𝜆𝑖+1,

or 𝜆𝑖 = 𝜆𝑖+1 and (𝜌 − 𝑠(𝑃 ) + 𝑠(𝑄))𝑖 < (𝜌 − 𝑠(𝑃 ) + 𝑠(𝑄))𝑖+1. And Ωdom= ⨆︁ 𝜆⊢𝑛 {(𝑃, 𝑄, 𝜌) ∈ 𝑇 (𝜆) × 𝑇 (𝜆) × Zℓ(𝜆) _|∑︁ 𝑖 𝜌𝑖 = 0, 𝜌 is dominant}.

The dominant representative 𝜌′ of 𝜌 in the triple (𝑃, 𝑄, 𝜌) can be computed by:

where (𝜌 − 𝑠(𝑃 ) + 𝑠(𝑄))↑ is a segmentwise increasing rearrangement of 𝜌 − 𝑠(𝑃 ) + 𝑠(𝑄) according to part sizes of 𝜆.

The inverse map Ψ : Ω → ̃︁𝑆𝑛 of the affine matrix ball construction involves a slightly

modified numbering process and we refer readers to [9] for a detailed description and only point out the following crucial result:

Theorem 3.3.4 ( [9]). For any 𝑤 ∈ ̃︁𝑆𝑛,

Ψ(Φ(𝑤)) = 𝑤,

and for any triple (𝑃, 𝑄, 𝜌) ∈ 𝑇 (𝜆) × 𝑇 (𝜆) × Zℓ(𝜆) _with ∑︀

𝑖𝜌𝑖 = 0, we have

Φ(Ψ(𝑃, 𝑄, 𝜌)) = (𝑃, 𝑄, 𝜌′)

where 𝜌′ is the dominant representative of 𝜌.

This tells us the image of Φ is a bijection between ̃︁𝑆𝑛 and Ωdom. Also Ψ(𝑃, 𝑄, 𝜌) = 𝑤

iff 𝑃 = 𝑃 (𝑤), 𝑄 = 𝑄(𝑤) and the dominant representative of 𝜌 is 𝜌(𝑤). The inverse of permutations behaves nicely under affine matrix ball construction:

Proposition 3.3.5. For 𝑤 ∈ ̃︁𝑆𝑛, Φ(𝑤−1) = (𝑄(𝑤), 𝑃 (𝑤), (−𝜌(𝑤))′) where (−𝜌(𝑤))′ is the

dominant representative of −𝜌(𝑤) in the fiber (of Ψ). If we define

̃︀

𝜌(𝑤) = 𝜌(𝑤) − 𝑠(𝑃 (𝑤)) + 𝑠(𝑄(𝑤)), (3.7)

to be the centralized version of 𝜌, then 𝜌(𝑤) is a segmentwise increasing vector, and the_̃︀ above proposition is equivalent to saying

̃︀

𝜌(𝑤) = −𝜌(𝑤_̃︀ −1)s. rev (3.8)

where 𝜌(𝑤_̃︀ −1)s. rev _{is the segmentwise reverse of the vector}

̃︀

𝜌(𝑤−1). In particular, when 𝑤 is column-type, 𝜌(𝑤) =_̃︀ 𝜌(𝑤_̃︀ −1)rev. Most importantly, for column-shape tabloids, 𝑠(𝑇 ) ∈ Z𝑛 where 𝑠𝑖(𝑇 ) = #{decents in [𝑇1, . . . , 𝑇𝑖]}, and charge(𝑇 ) = maj(𝑇 ) (the major index of 𝑇

3.4

Knuth Classes

First we define decent sets for both permutations and tabloids.

Definition 3.4.1. Given an affine permutation 𝑤, we define its right descent set 𝑅(𝑤) and left descent set 𝐿(𝑤) as:

𝑅(𝑤) ={𝑖 ∈ [𝑛] | 𝑤(𝑖) > 𝑤(𝑖 + 1)} (3.9)

𝐿(𝑤) ={𝑖 ∈ [𝑛] | 𝑤−1(𝑖) > 𝑤−1(𝑖 + 1)} (3.10)

Given a tabloid 𝑇 , the 𝜏 -invariant of 𝑇 is defined as:

𝜏 (𝑇 ) = {𝑖 ∈ [𝑛] | 𝑖 lies in a strictly higher row than 𝑖 + 1}. (3.11)

Decent sets interacts nicely with affine matrix ball construction:

Proposition 3.4.2 ( [8]). For 𝑤 ∈ ̃︁𝑆𝑛, 𝐿(𝑤) = 𝜏 (𝑃 (𝑤)) and 𝑅(𝑤) = 𝜏 (𝑄(𝑤)).

Now we are able to define Knuth moves.

Definition 3.4.3. Two affine permutations 𝑤 and 𝑤𝑠𝑖 are connected by a right Knuth move

at position 𝑖 if at least one of 𝑤(𝑖 + 2) and 𝑤(𝑖 − 1) is numerically between 𝑤(𝑖) and 𝑤(𝑖 + 1).

In other words, 𝑅(𝑤) and 𝑅(𝑤𝑠𝑖) are incomparable under the containment partial ordering.

We therefore have a equivalence class named right Knuth class the generated by right Knuth

moves 𝑤 ∼RKC 𝑤𝑠𝑖, and denote by RKC𝑤 to be the right Knuth class containing 𝑤.

Two affine permutations 𝑤 and 𝑠𝑖𝑤 are connected by a left Knuth move at position 𝑖 if

𝑤−1 and 𝑤′−1 = 𝑤−1𝑠𝑖 are connected by a right Knuth move at position 𝑖, in other words,

𝐿(𝑤) and 𝐿(𝑠𝑖𝑤) are incomparable. We therefore have a equivalence class named left Knuth

class the generated by left Knuth moves 𝑤 ∼LKC 𝑠𝑖𝑤, and denote by LKC𝑤 to be the left

Knuth class containing 𝑤.

Two tabloids 𝑇 and 𝑇′ are connected by a Knuth move if 𝑇 is obtained from 𝑇′ by

interchanging 𝑖 and 𝑖 + 1 and 𝜏 (𝑇 ) and 𝜏 (𝑇′) are incomparable.

There is the following remarkable theorem about how the image under affine matrix ball construction of permutations looks after a Knuth move.

Theorem 3.4.4 ( [8]). 1. Suppose 𝑤 and 𝑤𝑠𝑖 differs by a right Knuth move. Then:

(a) 𝑃 (𝑤) = 𝑃 (𝑤𝑠𝑖);

(b) 𝑄(𝑤𝑠𝑖) differs from 𝑄(𝑤) by exchanging 𝑖 and 𝑖 + 1;

(c) 𝜌(𝑤𝑠𝑖) = 𝜌(𝑤) if 𝑖 ̸= 𝑛, otherwise 𝜌(𝑤𝑠𝑖) differs by 𝜌(𝑤) by subtracting 1 from

row 𝑘 and adding 1 to row 𝑘′, where 𝑖 = 𝑛 lies in row 𝑘 in 𝑄(𝑤) and 𝑖 + 1 = 1

lies in row 𝑘′ in 𝑄(𝑤).

2. Suppose 𝑤 and 𝑠𝑖𝑤 differs by a right Knuth move. Then:

(a) 𝑄(𝑤) = 𝑄(𝑠𝑖𝑤);

(b) 𝑃 (𝑠𝑖𝑤) differs from 𝑃 (𝑤) by exchanging 𝑖 and 𝑖 + 1;

(c) 𝜌(𝑠𝑖𝑤) = 𝜌(𝑤) if 𝑖 ̸= 𝑛, otherwise 𝜌(𝑠𝑖𝑤) differs by 𝜌(𝑤) by subtracting 1 from

row 𝑘′ and adding 1 to row 𝑘, where 𝑖 = 𝑛 lies in row 𝑘 in 𝑃 (𝑤) and 𝑖 + 1 = 1

lies in row 𝑘′ in 𝑃 (𝑤).

Now we define a directed graph 𝒢_𝜆𝑅 (resp. 𝒢_𝜆𝐿) on permutations of shape 𝜆 and 𝑤 → 𝑤𝑠𝑖

(resp. 𝑤 → 𝑠𝑖𝑤) if 𝑤 and 𝑤𝑠𝑖 (resp. 𝑠𝑖𝑤) differs by a right (resp. left) Knuth move and

ℓ(𝑤𝑠𝑖) > ℓ(𝑤) (resp. ℓ(𝑠𝑖𝑤) > ℓ(𝑤)). The connected components of 𝒢𝜆𝑅 (resp. 𝒢𝜆𝐿) are

the right (resp. left) Knuth classes. Recall we have a rotation map 𝜑(𝑤) = 𝑠𝑤𝑠−1 where

𝑠 = [2, 3, . . . , 𝑛 + 1] ∈ 𝑆𝑛. The matrix balls of 𝜑(𝑤) are exactly those of 𝑤 shifted by (1, 1).

Hence 𝜑 commutes with Knuth moves and is a graph automorphism on 𝒢𝑅

𝜆 (resp. 𝒢𝜆𝐿).

Theorem 3.4.4 tells a right (resp. left) cell is a disjoint union of right (resp. left) Knuth classes. And [8] gives a complete characterization of Knuth classes by specifying what 𝑄 and 𝜌 could be like in each Knuth class.

Definition 3.4.5. For 𝜆 ⊢ 𝑛, define 𝑑𝜆 = gcd(𝜆T1, 𝜆T2, . . .).

Theorem 3.4.6 ( [8], Theorem 8.6). Let 𝑇, 𝑇′ ∈ 𝑇 (𝜆), then 𝑇 and 𝑇′ _{differs by a Knuth}

move iff

charge(𝑇 ) ≡ charge(𝑇′) (mod 𝑑𝜆).

Definition 3.4.7. Given 𝑤 ∈ ̃︁𝑆𝑛, we define the monodromy group 𝐺𝑅𝑤 based at 𝑤 to be

Theorem 3.4.8 ( [8], Theorem 7.28). For 𝑤 of shape 𝜆, we have 𝐺𝑅_𝑤 = {︃ _𝑘 ∑︁ 𝑖=1 𝑎𝑖1𝑚𝑖 ⃒ ⃒ ⃒ ⃒ ⃒ 𝑎𝑖 ∈ Z, 𝑘 ∑︁ 𝑖=1 𝑎𝑖𝑚𝑖 = 0 }︃

where 𝑚1 > 𝑚2 > . . . > 𝑚𝑘 are distinct part sizes of 𝜆T and 1𝑚𝑖 ∈ Z

ℓ(𝜆) _{with 1’s in the first}

𝑚𝑖 rows and 0 after on.

Again we are applying those results to understand Knuth classes of column-shape par-titions. Theorem 3.4.6 and 3.4.8 indicates permutations in a Knuth class of column-shape have the same 𝑃 and the charge of 𝑄 must have the same residue mod 𝑛, and finally there are no monodromy, i.e. if Ψ(𝑃, 𝑄, 𝜌) ̸= Ψ(𝑃′, 𝑄′, 𝜂) both belong to this Knuth class, then 𝜌 ̸= 𝜂.

3.5

Structure of Lowest Two-sided Cell

The right (resp. left) cell structure of the lowest two-sided cell has been extensively studied. Shi [43, 44], Xi [56] and Guilhot [17] gave descriptions of the left cell decomposition of the lowest two-sided cell for general type extended affine Weyl group in different settings. In this section, we gave a more explicit and simple description of this decomposition using the affine matrix ball construction for ̃︁𝑆𝑛. Moreover, we study the right (resp. left) Knuth class

structure for type A where we elaborate as follows. Denote

𝒜 = {𝜎 ∈ ̃︁𝑆𝑛| 𝜎(1) > 𝜎(2) > . . . > 𝜎(𝑛), 𝜎(𝑖) − 𝜎(𝑖 + 1) ≤ 𝑛 − 1, 𝑖 ∈ [𝑛 − 1]}. (3.12)

Proposition 3.5.1.

𝒜 = LKC𝑤0, (3.13)

𝒜−1 = RKC𝑤0, (3.14)

Proof. From equation (3.5), we know for any 𝜎 ∈ 𝒜, 𝑃 (𝜎) =[︁𝜎(𝑛), 𝜎(𝑛 − 1), . . . , 𝜎(1)]︁ and 𝑄(𝜎) = [1, . . . , 𝑛].

Claim: There is a bijection 𝒜 → {𝜏 ∈ 𝑆𝑛| maj(𝜏 ) ≡ 0 (mod 𝑛)} by 𝜎 ↦→ 𝑃 (𝜎).

First we show this map is well-defined. For any 𝜎 ∈ 𝒜, we know by definition of 𝒜, if 𝜎(𝑖) > 𝜎(𝑖 + 1), then ⌈︁ 𝜎(𝑖) 𝑛 ⌉︁ = ⌈︁ 𝜎(𝑖+1) 𝑛 ⌉︁ , otherwise ⌈︁ 𝜎(𝑖) 𝑛 ⌉︁ = ⌈︁ 𝜎(𝑖+1) 𝑛 ⌉︁ + 1. Then let 𝑗1 < 𝑗2 <

. . . < 𝑗𝑘be the indices where

⌈︁

𝜎(𝑛+1−𝑗) 𝑛

⌉︁

= 𝑎 for some 𝑎 and 1 ≤ 𝑗 ≤ 𝑗1and

⌈︁

𝜎(𝑛+1−𝑗) 𝑛

⌉︁

= 𝑎+1 for 𝑗1+ 1 ≤ 𝑗 ≤ 𝑗2 and so on. Then we have the following:

𝑛 ∑︁ 𝑖=1 ⌈︂ 𝜎(𝑖) 𝑛 ⌉︂ =𝑎𝑗1+ (𝑎 + 1)(𝑗2− 𝑗1) + . . . + (𝑎 + 𝑘 − 1)(𝑗𝑘− 𝑗𝑘−1) + (𝑎 + 𝑘)(𝑛 − 𝑖𝑘) =𝑎𝑛 + 𝑘𝑛 − 𝑗1− . . . − 𝑗𝑘 We also know: 𝑛 ∑︁ 𝑖=1 ⌈︂ 𝜎(𝑖) 𝑛 ⌉︂ = 𝑛 ∑︁ 𝑖=1 (︂⌈︂ 𝜎(𝑖) 𝑛 ⌉︂ − 𝜎(𝑖) 𝑛 )︂ + 𝑛 ∑︁ 𝑖=1 𝜎(𝑖) 𝑛 = (︂ 0 𝑛 + . . . + 𝑛 − 1 𝑛 )︂ +𝑛(𝑛 + 1) 2𝑛 = 𝑛.

Hence maj 𝑃 (𝜎) ≡ −(𝑗1+ . . . + 𝑗𝑘) ≡ 0 (mod 𝑛).

Next we define a inverse as follows:

For any 𝜏 ∈ 𝑆𝑛, with maj(𝜏 ) ≡ 0 (mod 𝑛), we have:

∑︁ 𝑖∈[𝑛] 𝜏 𝑤0(𝑖)<𝜏 𝑤0(𝑖+1) 𝑖 ≡ − ∑︁ 𝑖∈[𝑛] 𝜏 (𝑖)>𝜏 (𝑖+1) 𝑖 ≡ 0 (mod 𝑛)

Let Ascent(𝜏 𝑤0) = {𝑖1 < . . . < 𝑖𝑘} and we do the following procedure to the entries in the

window notation of 𝜏 𝑤0:

For 𝑙 = 1, . . . , 𝑘: add 𝑛 to the first 𝑖𝑙 entries; then because of 𝑛 |

∑︀

𝑙𝑖𝑙, we could subtract

a common multiple of 𝑛 from each entry to make the sum 𝑛(𝑛+1)₂ , this gives an element in

𝒜. So we constructed a inverse to the map (𝜎 ∈ 𝒜) ↦→ 𝑃 (𝜎). Hence we proved the claim.

Now we know that 𝒜 is a subset of column-shape permutations containing 𝑤0, with major

index of the insertion tabloid 𝑃 all having the same residue 0, and there are no monodromy (since there is the inverse map of the claim gives a unique way to recover 𝜎 from 𝜎 = 𝑃 (𝜎)𝑤0),

and these indicates 𝒜 = LKC𝑤0.

To show the number of 𝒜 is (𝑛 − 1)!, it suffices to show {𝜏 ∈ 𝑆𝑛 | maj(𝜏 ) ≡ 0 (mod 𝑛)}

has size (𝑛 − 1)!. In fact the residue class (mod 𝑛) of the major index is equi-distributed

in 𝑆𝑛. This is because for any 𝜏 ∈ 𝑆𝑛, denote by 𝜏 + 1 to be the permutation obtained

from 𝜏 by adding 1 (mod 𝑛) to its window notation, then maj(𝜏 ) − 1 ≡ maj(𝜏 + 1) (mod 𝑛). Hence |𝒜| = (𝑛 − 1)!. Finally, 𝒜−1 = RKC𝑤0 is clear by definition of right and left Knuth

moves.

Example 3.5.2. When 𝑛 = 3, 𝒜 = {𝑤0, 𝑠0𝑤0}; when 𝑛 = 4,

𝒜 = {𝑤0, 𝑠0𝑤0, 𝑠1𝑠0𝑤0, 𝑠3𝑠0𝑤0, 𝑠1𝑠3𝑠0𝑤0, 𝑠2𝑠1𝑠3𝑠0𝑤0} .

See Figure A-1, A-3, A-4 in Appendix A for Hasse diagrams of elements in 𝒜 for 𝑛 = 3, 4, 5 respectively.

In fact, when considering column-shape affine permutations, Theorem 3.4.4 can be gen-eralized to the following:

Theorem 3.5.3. For 𝑤 ∈ ̃︁𝑆𝑛 of column-shape, if ℓ(𝑤𝑠𝑖) > ℓ(𝑤) (resp. ℓ(𝑠𝑖𝑤) > ℓ(𝑤)), then

𝑤𝑠𝑖 (resp. 𝑠𝑖𝑤) is also of column-shape and 𝑃 (𝑤𝑠𝑖) and 𝑄(𝑤𝑠𝑖) can be computed from Φ(𝑤)

using the rules exactly same as in Theorem 3.4.4 (but 𝜌(𝑤𝑠𝑖) may be different).

Proof. We only prove the right multiplication version. Result of the left multiplication can be obtained by taking inverses.

First of all ℓ(𝑤𝑠𝑖) > ℓ(𝑤) iff 𝑤(𝑖) < 𝑤(𝑖 + 1). By equation (3.5), we have a decreasing

chain (𝑖1, 𝑤(𝑖1)), . . . , (𝑖𝑛, 𝑤(𝑖𝑛))) with 𝑖1 < 𝑖1 < . . . < 𝑖𝑛, 𝑖𝑗 − 𝑖𝑗−1 ≤ 𝑛 − 1, for all 𝑗 ∈ [2, 𝑛],

𝑤(𝑖1) > 𝑤(𝑖2) > . . . > 𝑤(𝑖𝑛), {𝑖1, . . . , 𝑖𝑛} = [𝑛].

Without loss of generality, suppose that 𝑖 = 𝑖𝑓, since 𝑤(𝑖) < 𝑤(𝑖 + 1), there exists 𝑘 > 0,

𝑖 + 1 − 𝑘𝑛 = 𝑖𝑓′ and 𝑓′ < 𝑓 . Hence 𝑤(𝑖 + 1) − 𝑛𝑘 < 𝑤(𝑖). Right multiplication by 𝑠_𝑖 amounts

to shifting the ball (𝑖, 𝑤(𝑖)) southwards by 1 position and the ball (𝑖 + 1 − 𝑘𝑛, 𝑤(𝑖 + 1) − 𝑘𝑛) northwards by 1 position. The decreasing chain remains decreasing as previously it does not occupy any balls in row 𝑖 + 1 and 𝑖 − 𝑛𝑘. So 𝑤𝑠𝑖 is still of column-shape. If 𝑓′ = 𝑓 − 1 and