Combinatorics and Algorithms Associated with the

Theory of Kazhdan-Lusztig Cells

by

Tim Honeywill

Thesis

Submitted to The University of Warwick

for the degree of

Doctor of Philosophy

Mathematics Institute

September 2005

In Loving Memory Of

Bernard William Tier (a.k.a. ‘Chum’)

(1915-2004)

“Keep Your Eye On Your Goal”

Also Dedicated To The Memory Of

Dyfrig Williams

Dr Lyndon Woodward

Contents

Acknowledgments vi

Declarations viii

Abstract ix

Chapter 1 Preliminaries 1

1.1 Coxeter groups and affine Weyl groups . . . . . . . . . . . . . . . . . . 1

1.2 Hecke algebras and Kazhdan-Lusztig polynomials . . . . . . . . . . . . 3

1.3 Left cells, right cells and two-sided cells . . . . . . . . . . . . . . . . . 5

1.4 The a-function and distinguished involutions . . . . . . . . . . . . . . . 6

Chapter 2 A Combinatorial Approach to a Conjecture of Lusztig for Type à 9

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The Robinson-Schensted Correspondence . . . . . . . . . . . . 9

2.1.2 The Generalized Robinson-Schensted Algorithm . . . . . . . . . 10

2.2 The machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 The groups W ′, WGLn(C) and WSLn(C) . . . . . . . . . . . . . 12

2.2.2 Permutations of the integers . . . . . . . . . . . . . . . . . . . 12

2.2.3 Cells of W and the extended Generalized Robinson-Schensted

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.4 The partition belonging to an element of W . . . . . . . . . . . 16

2.2.5 W as a set of matrices . . . . . . . . . . . . . . . . . . . . . . 18

2.2.6 Left star operations . . . . . . . . . . . . . . . . . . . . . . . . 19

iii

2.3 The map W → Yn × Yn ×Dom(Fλ) . . . . . . . . . . . . . . . . . . 23

2.3.1 The adapted Fn procedure . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Obtaining an element which has Hλ Form . . . . . . . . . . . . 28

2.3.3 Mapping to Γλ ∩ Γ−1λ . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.4 The bijection from Γλ ∩ Γ−1λ to Dom(Fλ) and to Dom(F′λ) . . 36

2.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Inverting Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 From Dom(Fλ) or Dom(F ′λ) to Γλ ∩ Γ−1λ . . . . . . . . . . . . 41

2.4.2 Inverting the adapted Fn procedure . . . . . . . . . . . . . . . 43

2.4.3 Mapping back from an element with Hλ Form to an element with

FForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.4 Piecing it all together . . . . . . . . . . . . . . . . . . . . . . . 55

2.5 A remark on computing distinguished involutions from Young tableaux . 56

Chapter 3 The Decomposition into Left Cells of Finite Two-Sided Cells in

Affine Weyl Groups 62

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 The l.c.r. set Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.1 Tools for analysing Process A . . . . . . . . . . . . . . . . . . . 73

3.2.2 Tools for analysing Processes B and C . . . . . . . . . . . . . . 81

3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2.4 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 4 The Explicit Description of Finite Left Cells in Affine Weyl Groups103

4.1 Algorithm for computing all the elements in any given left cell . . . . . 104

4.1.1 Process α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.1.2 Tools for analysing Processes β and γ . . . . . . . . . . . . . . 107

4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.3 Left connectedness of left cells . . . . . . . . . . . . . . . . . . . . . . 116

iv

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.5 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Appendix A Slides Showing the Decomposition into Cells of the Three Rank

2 Affine Weyl Groups 128

Appendix B The Robinson-Schensted Correspondence 133

Appendix C The Generalized Robinson-Schensted Algorithm 137

Appendix D The Left Star Operations Underlying the Adapted Fn Proce-

dure: An Example 140

Appendix E An Example: Obtaining an Element with Hλ Form from an

Element with FForm 142

Appendix F An Example: Inverting the Bijection of Section 2.3 149

Appendix G The Distinguished Involutions in the Affine Weyl Group of Type

Ãk for k ≤ 4 154

Appendix H My Magma code for implementing the Lcrset Algorithm 169

Appendix I My Magma code for implementing my algorithm for computing

finite left cells 217

v

Acknowledgments

As I approach the end of my time as a Ph.D. student, I am wholeheartedly aware of so

many people without whom this last four years would not have been what they were.

Indeed, it is hard to see how I would have seen it through to the end without the support

of so many. There are many people to thank on a mathematical level - and these I do

my best to mention below - but I see as equally as valuable the friends and family who

have ‘lent me their shoulder’ in small ways or large. Even a smile at the moment when

I need one can be priceless! In this way, there are many people special to me for whom

this paragraph is intended as a thank you. Your input into my life is appreciated in a

way that cannot be put into words.

First and foremost my gratitude should go to my Supervisor, Dr Dmitriy Rumynin,

for all the time he has given me, combined with all the patience and enthusiasm which

he has shown me, over the past four years. I am also grateful to him for allowing me

the freedom to take the mathematical route in my research which I felt best suited me.

I am also indebted to Professor Roger Carter for many helpful insights and

suggestions, and for taking the time to conduct the very useful report on my progress

in my third and fourth years. He is an inspiration.

The work of Professor Shi has very clearly made a huge impact on me. I am

hugely grateful to him for his friendly correspondence via e-mail on a couple of matters.

I am grateful too to Professor Rui for answering a question by e-mail.

I would like to thank too my fellow Ph.D. student friends (now ex-Ph.D. stu-

dents), Paul Boddington and Iain Moffatt, who helped to keep me sane in my earlier

and more uncertain years of study.

vi

Computer programming has come to play a huge role in the second half of my

thesis and, given I had not even had a basic introduction to computer programming

beforehand, I simply could not have done this all without a guiding hand from others.

This guidance was given largely by Dr Gavin Brown who was good enough to have a

number of meetings with me early on. I have also had huge support from Professor John

Cannon and various members of the Magma team, for which I am extremely grateful.

Professor Derek Holt has also been of assistance, on one occasion suggesting two lines

of code which reduced the time taken by one of my programs by ninety percent!

I would like to thank Oliver Tearne for assistance in learning how to import colour

scans into my thesis.

Be it with LATEX-ing queries, programming queries, mathematical queries or just

offering support, I have good reason to thank the following: Stephen Coughlan, Billy

Donegan, Michael Kerber, Iuliana Marchis, Dr Robert Marsh, Dr Martyn Parker, David

Webdale, Dr Bruce Westbury, Mike Willis.

From the years leading up to the start of my Ph.D. studies there are many people

to thank too - I have been inspired by many teachers from my first days at secondary

school - but this acknowledgments section should not be a thesis in itself! I shall there-

fore mention just one man, Dr Vernon Armitage, who I see as a great inspiration to me,

a true gentleman and a lover of life who stil maintains a true passion for mathematics.

It is largely because of him that I am here today typing these acknowledgments.

Of course my thanks go to the EPSRC for funding me through my studies. I

would not have worked on this thesis but for their generous support.

My thanks also go to my two examiners for taking the time to read through my

thesis.

The final word must go to my family whom I love very much. Allowing me the

freedom to be me, they have been behind me all the way, with a guiding and sensitive

hand, and there is nothing more for which a son, brother and grandson could ask.

vii

Declarations

This thesis is my own work. I build on the ideas of other mathematicians but make it

clear when I am outlining their material as a motivation for and introduction to my own

work.

Although material contained herein may be submitted for publication at a later

date, I have not before used anything which forms a part of this thesis.

This thesis has not been submitted for a degree at another university.

viii

Abstract

This thesis contains two main topics, both of which relate to the Kahzdan-Lusztig

theory of Coxeter groups and Hecke algebras.

The first topic is a generalisation of the Robinson-Schensted bijection for the

symmetric group to two types of extended affine Weyl group of Type Ã. This is covered

in the work of Chapter 2.

The second topic concerns itself with finding as much information as possible

about finite two-sided cells in affine Weyl groups of arbitrary type. This is work in

progress and will be continued after the submission of this thesis. In Chapter 3, I

outline an algorithm of Jian Yi Shi for finding a representative of each left cell inside

an arbitrary two-sided cell in any Coxeter group that satisfies certain conditions (this

includes all affine Weyl groups). I then apply it to almost all of the finite cells in affine

Weyl groups of rank at most 8. I state a conjecture, modified from a suggestion by

Roger Carter, for the number of left cells inside the finite two-sided cells in affine Weyl

groups. This conjecture is verified by all results I have so far.

In Chapter 4, I describe an algorithm of my own which is derived from that of the

previous Chapter. This algorithm is used to compute explicitly all the elements inside a

finite left cell, given a single element belonging to that left cell. I discuss the important

role played by G-sets which allows us to understand the complete structure of a finite

two-sided cell without needing to explicitly compute all the left cells lying inside.

Although this thesis focuses just on affine Weyl groups and two extended affine

Weyl groups of type Ã, there is scope for some of the ideas within to be applied to other

types of Coxeter group.

ix

Chapter 1

Preliminaries

1.1 Coxeter groups and affine Weyl groups

A Coxeter group is any group W which is generated by a finite set of generators, S,

with a presentation of the following form (I take e to be the identity in the group).

W = 〈S|si2 = e, (sisj)msisj = e where si, sj ∈ S, i 6= j,msisj ∈ {2, 3, . . . ,∞}〉

Note that msisj = msjsi . The affine Weyl groups, which arise in Lie theory, are examples

of Coxeter groups. They are infinite groups which have a presentation described by the

Coxeter graphs shown in Figure 1.1. The Coxeter graph determines the group in the

following way. The nodes are labelled by the generators in S, a node being labelled by

i instead of si for neatness. Take each generator si to satisfy si2 = e. For the value of

msisj look at the nodes labelled by si and by sj . If there is no edge between these two

nodes, msisj = 2; if there is one, two or three edges, msisj = 3, 4 or 6 respectively; if

there is a single edge labelled by ∞, msisj =∞.

Observe that if msisj = 2 then si and sj commute; in general, if msisj = k then

sisjsi . . . = sjsisj . . . with k generators on each side of the equation.

There is no standard ordering of the generators of the affine Weyl group. I shall

maintain the labelling used in Figure 1.1 throughout this thesis except where clearly

stated otherwise. This is the labelling adopted by Magma, the Computer Algebra system

in which I have programmed [22]. The graphs I have given are not Coxeter graphs as

1

The affine Weyl groups of classical type

Ã1 d d1 2∞ Ãll ≥ 2

d d1 2 dl − 1 dldl + 1

B̃ll ≥ 3

dd d1 2l + 1

d3 dl − 1 dl C̃ll ≥ 2

d dl + 1 1 d2 dl − 1 dlD̃l

l ≥ 4dd d1

l + 1

2 d3 dl − 3 dl − 2 ddl − 1l

The affine Weyl groups of exceptional type

Ẽ6 d1 d3 d4 d5 d6dd

2

7

Ẽ7 d8 d1 d3 d4 d5 d6 d7d2

Ẽ8 d1 d3 d4 d5 d6 d7 d8 d9d2

F̃4 d5 d1 d2 d3 d4 G̃2 d1 d2 d3Figure 1.1: The affine Weyl groups.

2

they would usually be presented, but they carry the same information. I prefer to present

them as they are as it shows their similarity to Dynkin diagrams and their link to the

classification of Lie algebras.

For any Coxeter group W with generating set S, let l : W −→ N be the standard

length function given by l(w) = min{n|w = si1 . . . sin , sil ∈ S}. Then w = si1 . . . sinis called a reduced expression for w if n = l(w). For any s in S and w in W , l(sw) =

l(w)± 1.

In general, l(yw) ≤ l(y) + l(w) for any y, w in W . When l(yw) = l(y) + l(w),

I may highlight this fact by notating the product yw as y · w.

Let ≤ be the Bruhat partial order on W which can be defined as follows: y ≤ w

if and only if for any reduced expression w = si1 . . . sin there is a subsequence j1, . . . , jl

of i1, . . . , in such that y = sj1 . . . sjl is a reduced expression for y. One writes y < w

to mean that y ≤ w and y 6= w.

Using the Exchange Condition (see for example [15]) one can show that if

l(sw) = l(w) + 1 then w < sw and if l(sw) = l(w) − 1 then sw < w 1. Also

observe that y ≤ w if and only if y−1 ≤ w−1, for any y and w in W .

1.2 Hecke algebras and Kazhdan-Lusztig polynomials

Let u be an indeterminate and let A = Z[u, u−1] be the ring of Laurent polynomials in

u with integer coefficients.

To any Coxeter group W with generating set S one can associate the Hecke

algebra, H, which is the associative algebra over A with basis {T̃w|w ∈W} and multi-

plication defined by

T̃yT̃w = T̃yw if l(yw) = l(y) + l(w)

(T̃s − u−1)(T̃s + u) = 0 if s ∈ S

1A version of the Exchange Condition can be stated as follows: if l(sw) < l(w) for w = s1 . . . sr(not necessarily reduced) in W and s in S then there exists an index i such that sw = s1 . . . bsi . . . srwhere bsi denotes the omission of the generator si. This gives the latter result with the former resultingfrom replacing w with sw.

3

Kazhdan and Lusztig uniquely determined another basis ofH which they denoted

{Cw|w ∈W}.

The change of basis is described by the equation below:

Cw =∑

y∈W,y≤wul(w)−l(y)Py,w(u−2)T̃y

where Py,w is a polynomial in u with integer coefficients, of degree ≤ 12(l(w)− l(y)−1)

if y < w, and Pw,w = 1 [16]. When y 6≤ w one may choose to set Py,w = 0.

One writes y ≺ w if the degree of Py,w is 12(l(w) − l(y) − 1). Observe that if

y ≺ w then y < w and l(y) 6= l(w) mod 2. The notation y—w is used whenever either

y ≺ w or w ≺ y.

The Py,w are known as Kazhdan-Lusztig polynomials and play a huge role in Lie

theory.

It is in general not a trivial thing to compute a Kazhdan-Lusztig polynomial.

However, du Cloux has written some software which, in principle at least, will compute

‘on demand’ any Kazhdan-Lusztig polynomial of almost any Coxeter group [11] [12].

This software utilises a recursion formula of Kazhdan and Lusztig for the computation

by induction on the length of the longer element [16].

There is also the highly useful result below, which gives some Kazhdan-Lusztig

polynomials ‘for free’.

Lemma 1.2.1. [25, Lemma 1.4.5(ii)]

For any y, w in W with y < w and l(w) = l(y) + 1, Py,w = 1.

In particular, if either y = sw or y = ws for any y, w in W and s in S, then

y—w and w—y.

I shall later make use of the following Lemma as well.

Lemma 1.2.2. [25, Lemma 1.4.5(viii)]

For any y, w in W , Py,w = Py−1,w−1 .

In particular, y—w if and only if y−1—w−1.

4

1.3 Left cells, right cells and two-sided cells

The definition of cells, for any Coxeter group W , were given by Kazhdan and Lusztig and

motivated by a desire to construct representations of the Hecke algebra corresponding

to W by means of W -graphs [16]. In order to state their definition I must give the

following: for any w in a Coxeter group W with generating set S the sets given by the

below equations are called in order the left descent set and right descent set of w.

L(w) = {s ∈ S|sw < w}

R(w) = {s ∈ S|ws < w}

It can help to think of the left descent set as the set of generators s for which

there exists a reduced expression where s is on the far left-hand end. Similarly, the right

descent set gives those generators which can be placed on the far right-hand end of a

reduced expression for w. Of course R(w) = L(w−1) for any w.

For any y, w in a Coxeter group W write y ≤L w if there exists a sequence of

elements of W , y = x0, x1, . . . , xr = w, such that for each i, 1 ≤ i ≤ r, both xi−1—xi

and L(xi−1) 6⊆ L(xi). Then write y ∼L w if y ≤L w ≤L y. This is an equivalence

relation on W whose corresponding equivalence classes are called left cells.

One writes y ≤R w if there exists a sequence of elements of W , y = x0,

x1, . . . , xr = w, such that for each i, 1 ≤ i ≤ r, both xi−1—xi and R(xi−1) 6⊆ R(xi).

Then write y ∼R w if y ≤R w ≤R y, and the corresponding equivalence classes of W

are called right cells.

Finally, one writes y ≤LR w if there exists a sequence of elements of W , y =

x0, x1, . . . , xr = w, such that for each i, 1 ≤ i ≤ r, both xi−1—xi and either L(xi−1) 6⊆

L(xi) or R(xi−1) 6⊆ R(xi). The two-sided cells of W are the equivalence classes

belonging to the relation ∼LR, where y ∼LR w if y ≤LR w ≤LR y.

There is therefore a decomposition of any Coxeter group W into two-sided cells

and a further decomposition of two-sided cells both into left cells and into right cells. It

is also clear, by Lemma 1.2.2, that a right cell is simply the set of elements w−1 where w

runs through all the elements in some left cell. Hence understanding the decomposition

of two-sided cells into left cells immediately provides the decomposition into right cells.

5

When W is an affine Weyl group it is known that there are only finitely many left and

right cells [19, Theorem 2.2(a)].

As an illustration, I describe in Appendix A the decomposition of the affine Weyl

groups of types Ã2, B̃2 and G̃2 into both two-sided cells and left cells.

It is not in general straightforward to determine whether or not two elements of

a Coxeter group lie in the same left cell or in the same two-sided cell, although there are

some tools to assist with this task. Most of these will be introduced during the course

of this thesis but for now I shall mention a highly useful and well-known Lemma, which

allows one to quickly decide in some cases that two elements are not in the same left

cell or not in the same right cell.

Lemma 1.3.1. [16, Proposition 2.4]

For any y, w in a Coxeter group W :

(i) If y ∼L w then R(y) = R(w).

(ii) If y ∼R w then L(y) = L(w).

Also note that as a direct corollary to Lemma 1.2.1 it is clear that if y = sw

with L(y) 6⊆ L(w) 6⊆ L(y), for any y, w in W and s in S, then y and w must be in the

same left cell.This is the starting point for the algorithm which I construct in Chapter 4.

1.4 The a-function and distinguished involutions

For this final Section assume that W is an affine Weyl group.

For any y, w in W denote by µ(y, w) = µ(w, y) the coefficent of u12(l(w)−l(y)−1)

in Py,w. One can check that the multiplication in H is defined in terms of the {Cw|w ∈

W} basis by the following relations.

CsCw =

(u−1 + u)Cw if s ∈ L(w)∑y—w,sy

For any y, w, z in W let hy,w,z in A be defined by

CxCy =∑

z

hy,w,zCz.

By the multiplication relations above it is clear that hy,w,z(u) = hy,w,z(u−1).

Lusztig defined a function a : W → N ∪ {∞} by

a(z) = maxy,w∈W

deg hy,w,z

for any z in W for which the above is defined, and by a(z) =∞ otherwise [18]. However,

a(z) is finite for elements z in the affine Weyl groups and in all Coxeter groups on which

the algorithms of Chapters 3 and 4 are defined.

Although the a-function is in general non-trivial to compute, there are a number

of useful and well-known properties, a few of which I state here but most of which will

be mentioned in the main part of the thesis.

Lemma 1.4.1. [18, Proposition 2.2]

For any w in W , a(w) = a(w−1).

Lemma 1.4.2. [18, Proposition 2.3]

One has a(w) = 0 if and only if w = e, where e is the identity element in W .

Theorem 1.4.1. [18, Theorem 5.4]

Suppose that y, w are two elements in W such that y ≤LR w.

Then a(y) ≥ a(w).

In particular, if y ∼LR w then a(y) = a(w).

Define for any Coxeter group W and non-negative integer i

W(i) = {w ∈W |a(w) = i}.

Then by Theorem 1.4.1 W(i) is either empty or a union of two-sided cells of W .

7

Let δ(w) = deg Pe,w for any w in W where e is the identity element in W .

Lusztig showed that the inequality

l(w)− 2δ(w)− a(w) ≥ 0

holds for all w in W .

He defined the set

D = {w ∈W |l(w)− 2δ(w)− a(w) = 0}

and showed that when W is an affine Weyl group this is a finite set of involutions [19,

Theorem 2.2(b)]. In particular he showed that each left cell (and therefore each right cell

also) has a unique involution in D. This element is called the distinguished involution

in the given left cell.

For any Coxeter group W and any proper subset I of the generating set S which

generates a finite parabolic subgroup WI , let wI be the longest element in WI . When

W is an affine Weyl group, WI is finite for all proper subsets. I shall call wI a parabolic

element.

Lemma 1.4.3. One has a(wI) = l(wI).

Notice that, since δ(wI) is non-negative, the above Lemma forces wI to be a

distinguished involution inside its left cell. It follows that there is at most one parabolic

element in any given left cell.

8

Chapter 2

A Combinatorial Approach to a

Conjecture of Lusztig for Type Ã

The main results in this Chapter are Theorems 2.3.1 and 2.3.2 which describe a bijection

for two extended affine Weyl groups of type Ã. The existence of such a bijection was

already more or less known, thanks to a Conjecture of Lusztig [20, 10.5], but it has not

before been given an explicit description.

I also give an explicit description of the inverse of my bijection for both groups.

I finish the Chapter with some discussion for further work on the distinguished

involutions in type Ã.

2.1 Motivation

2.1.1 The Robinson-Schensted Correspondence

A standard Young tableau of rank n is n boxes, arranged in such a way that the number

of boxes on each row is not less than the number of boxes on the row below, labelled

by the numbers 1, . . . , n so that each number is used precisely once and the numbers

increase along each row and down each column. Two such tableaux are said to have

the same shape if they have the same number of boxes in each row. Below are three

standard Young tableaux of rank 8. The first and the last have the same shape.

9

1 3 4

2 5 7

6 8

1 4 6 8

2 7

3

5

1 2 7

3 5 8

4 6

In 1961, a beautiful explicit combinatorial bijection was described between the

symmetric group of rank n, Σn, and the set of pairs of standard Young tableaux of rank

n of the same shape. This is known as the Robinson-Schensted Correspondence [24]

and I give an outline of this in Appendix B.

The symmetric group Σn is a Coxeter group and arises in Lie theory as the

Weyl group of type An−1. Cells were defined on Σn eighteen years after the Robinson-

Schensted Correspondence. This remarkable link was then established.

Theorem 2.1.1. [16, first two equivalences] [2]

Suppose that via the Robinson-Schensted Correspondence y0 and w0 in Σn

are associated with the same-shape pairs of rank n standard Young tableaux labelled

(P (y0), Q(y0)) and (P (w0), Q(w0)) respectively. Then

y0 ∼L w0 ⇔ P (y0) = P (w0)

y0 ∼R w0 ⇔ Q(y0) = Q(w0)

y0 ∼LR w0 ⇔ P (y0) and P (w0) (and Q(y0) and Q(w0)) have the same shape

2.1.2 The Generalized Robinson-Schensted Algorithm

Following the result of Barbasch and Vogan, the natural question to ask was if tableaux

could similarly describe the cell structure in the affine Weyl group of type Ãn−1. This

question was answered by Shi [30].

Let Yn be the set of Young tableaux of rank n which are labelled by the numbers

1, . . . , n such that the numbers increase along each row but are not required to increase

down each column. The standard Young tableaux of rank n form a subset of Yn.

10

3 6 7

1 2 8

4 5

Young tableau

in Y8

Shi described a map, called the Generalized Robinson-Schensted Algorithm, from

an element w′ in the affine Weyl group of type Ãn−1 to a tableau which I denote P′(w′)

in Yn [30]. I give a brief outline of this map in Appendix C although the reader is

referred to the paper by Shi for the details.

Performing the same map on w′ −1 produces a second tableau, P ′(w′ −1), of

the same shape as P ′(w′). I shall call this Q′(w′).

Theorem 2.1.2. [29, 6.14]

Suppose that via the Generalized Robinson-Schensted Algorithm y′ and w′ in the

affine Weyl group of type Ãn−1 are associated with the same-shape pairs of tableaux in

Yn, (P ′(y′), Q′(y′)) and (P ′(w′), Q(w′)) respectively. Then

y′ ∼L w′ ⇔ P ′(y′) = P ′(w′)

y′ ∼R w′ ⇔ Q′(y′) = Q′(w′)

y′ ∼LR w′ ⇔ P ′(y′) and P ′(w′) (and Q′(y′) and Q′(w′)) have the same shape

Unlike the Robinson-Schensted Correspondence, the map from the affine Weyl

group of type Ãn−1 to same-shape pairs of tableaux in Yn is not a bijection, since the

affine Weyl group is an infinite group and Yn is a finite set.

11

2.2 The machinery

2.2.1 The groups W ′, WGLn(C) and WSLn(C)

I shall keep with the following notation for the remainder of this Chapter.

Let W ′ be the affine Weyl group of type Ãn−1. Let s1, . . . , sn be the generators

of W ′, ordered according to the Coxeter graph in Figure 1.1.

Consider Ω n W ′ where Ω is either the infinite cyclic group or the cyclic group

of order n.

When Ω is the infinite cyclic group, Ω n W ′ is the extended affine Weyl group

of type Ãn−1 associated with GLn(C). I shall denote this WGLn(C).

When Ω is the cyclic group of order n, ΩnW ′ is the extended affine Weyl group

of type Ãn−1 associated with SLn(C). I shall denote this WSLn(C).

The work of this Chapter is almost the same for WGLn(C) as for WSLn(C).

Therefore, by abusing notation for the sake of neatness, when what I write equally

applies to both WGLn(C) and WSLn(C) I shall refer to them both simultaneously via the

notation W . Similarly, both of the cyclic groups mentioned above are labelled Ω and

I shall denote the generator of each by ω. One simply needs to remember that when

dealing with WSLn(C), ωn is the identity, whereas in the case of WGLn(C), ω has infinite

order.

2.2.2 Permutations of the integers

For any fixed integer n > 0 and any integer i, let i be the representative of i in {1, . . . , n}

under the projection map from Z to Z/nZ. For example, when n = 15, 33 = 3 and

0 = 15. I shall use this notation throughout this Chapter.

Elements of W ′ and W may be identified with certain permutations as described

below. Throughout this Chapter I shall mostly describe an element by its permutation.

Lusztig described W ′ as permutations, w′, on the integers, such that

∑ni=1((i)w

′ − i) = 0

(i + n)w′ = (i)w′ + n

12

This is described by the action

(j)si =

j if j 6= i, i + 1

j + 1 if j = i

j − 1 if j = i + 1

for i = 1, . . . , n [17].

It is sufficient to know the image of n consecutive numbers under w′ in order to

know the action of w′ on the whole of the integers. Shi has described an element w′ of

W ′ as an n-tuple, [a1, . . . , an]n, which is the ordered image of n consecutive numbers

under the permutation w′ [30].

If∑n

i=1 ai =∑n

i=1 i then it follows that in particular a1, . . . , an are the images

of 1, . . . , n respectively. Infinitely many n-tuples for the same permutation arise via the

relation [a1, a2, . . . , an]n ∼ [a2, . . . , an, a1+n]n which, by the rule (i+n)w′ = (i)w′+n,

clearly corresponds to shifting forward by one the sequence of consecutive numbers whose

images are displayed in the n-tuple. In general,∑n

i=1 ai =∑n

i=1 i + Kn, for some

K ∈ Z, and moreover a1, . . . , an are the images under w′ of K + 1,K + 2, . . . ,K + n

respectively.

Similarly, W bijects with permutations, w, on the integers, such that∑ni=1((i)w − i) = n

(i + n)w = (i)w + n

For this, s1, . . . , sn act in the same way as for W′ and the action of ω is given by

(j)ω = j + 1

for all integers j.

Observe that siω = ωsi+1 for i = 1, . . . , n. When W is WSLn(C), ωn needs to

be identified with the identity permutation.

Since here∑n

i=1(i)w may take values other than∑n

i=1 i, given the image under

w of an arbitrarily chosen sequence of n consecutive numbers it is impossible to deter-

mine the permutation. I therefore choose to describe an element w of W as an n-tuple,

{b1, . . . , bn}n, which is strictly the ordered image of 1, . . . , n under the permutation w.

There is just one n-tuple representing each element of W .

13

Observe that∑n

i=1(i)w =∑n

i=1 i + Kn, where K is any integer when W is

WGLn(C) and where, without loss of generality, one can assume that K lies in {1, . . . , n}

when W is WSLn(C). The elements w for which K = 0 are precisely those which lie in

W ′.

I also extend the above notation to define bl to be the image under w of l, for

any integer l. Thus bl = bl + (l − l). For example, when n = 12, b13 = b1 + 12,

b−17 = b7 − 24.

2.2.3 Cells of W and the extended Generalized Robinson-Schensted Al-

gorithm

In Section 1.3 I gave the definition of cells of a Coxeter group. W ′ is a Coxeter group

but W is not.

One naturally extends the definition of cells on W ′ to W in the following way: for

any ω1, ω2 in Ω and y′, w′ in W ′ write ω1y′ ≤L ω2w′, y′ω1 ≤R w′ω2, ω1y′ ≤LR ω2w′

when y′ ≤L w′, y′ ≤R w′, y′ ≤LR w′ respectively. Then for any y, w in W , y ∼L w is

the equivalence relation given by y ≤L w ≤L y, y ∼R w by y ≤R w ≤R y and y ∼LR w

by y ≤LR w ≤LR y. The corresponding equivalence classes give us the left cells, right

cells and two-sided cells of W respectively1.

To each w′ in W ′ Shi has assigned a Young tableau P ′(w′) in Yn in such a

way that two elements in W ′ are assigned the same tableau if and only if they are in

the same left cell, and are assigned tableaux of the same shape if they are in the same

two-sided cell (Section 2.1.2). I wish to extend this process to obtain a Young tableau

P ′′(w) in Yn for any w in W .

If w′ is in W ′ and ω1 is in Ω, w′ and ω1w′ are in the same left cell of W . Also,

each left cell of W consists precisely of elements obtained in this manner, where w′ runs

through all the elements in a left cell of W ′ and ω1 runs through Ω. The same is also

true if ‘left cell’ is replaced everywhere by ‘two-sided cell’.

1It may help to think of these respectively as the sets {ω1w′|ω1 ∈ Ω, w′ ∈ left cell of W ′},{w′ω1|ω1 ∈ Ω, w′ ∈ right cell of W ′} and {ω1w′|ω1 ∈ Ω, w′ ∈ two-sided cell of W ′}.

14

Suppose that w′ = {a1, . . . , an}n is an element in W ′ and ω1w′ = {c1, . . . , cn}n

is an element in W with ω1 some element in Ω. Then [a1, . . . , an]n and [c1, . . . , cn]n

are two n-tuples representing the same element w′ in W ′.

The Generalized Robinson-Schensted Algorithm on W ′ is, as one would expect,

independent of the choice of n-tuple describing a given element [30]. It follows that

P ′([a1, . . . , an]n) and P ′([c1, . . . , cn]n) are identical tableaux in Yn.

Given an element w in W , I therefore define P ′′(w) as follows.

If w = {b1, . . . , bn}n then P ′′(w) = P ′([b1, . . . , bn]n)2.

For example, when n = 3, {3, 2, 1}3 and ω{3, 2, 1}3 = {2, 1, 6}3 represent two

different elements in W but which lie in the same left cell. On the other hand, [3, 2, 1]3

and [2, 1, 6]3 represent the same element in W ′ and so P ′′({3, 2, 1}3) = P ′′(ω{3, 2, 1}3).

Shi also assigned to each w′ in W ′ a Young tableau Q′(w′) = P ′(w′ −1) in Yn

such that two elements in W ′ are in the same right cell if and only if they have the same

Q′-tableau (Theorem 2.1.2).

I define Q′′(w) = P ′′(w−1) for any w in W . If w′ is in W ′ and ω1 is in Ω, w′

and w′ω1 are in the same right cell of W . Also, each right cell of W consists precisely

of elements obtained in this manner, where w′ runs through all the elements in a right

cell of W ′ and ω1 runs through Ω. One has

Q′′(w′ω1) = P ′′(ω−11 w′ −1) = P ′(w′ −1) = P ′′(w′ −1) = Q′′(w′).

I shall refer to the process by which I obtain a pair of tableaux (P ′′(w), Q′′(w)) from w

in W as the extended Generalized Robinson-Schensted Algorithm.

I can now state

Corollary 2.2.1. Suppose that via the extended Generalized Robinson-Schensted Al-

gorithm y and w in W are associated with the same-shape pairs of tableaux in Yn,

(P ′′(y), Q′′(y)) and (P ′′(w), Q′′(w)) respectively. Then

y ∼L w ⇔ P ′′(y) = P ′′(w)

y ∼R w ⇔ Q′′(y) = Q′′(w)

y ∼LR w ⇔ P ′′(y) and P ′′(w) (and Q′′(y) and Q′′(w)) have the same shape2Note that w = [b1, . . . , bn]n only if w is in W

′, but that that what matters is that w and [b1, . . . , bn]nlie in the same left cell of W .

15

2.2.4 The partition belonging to an element of W

Suppose P is a finite partially ordered set.

A chain is a subset of P which is totally ordered by the induced order of P . An

antichain is a subset such that on no two elements is the order of P defined.

A k-chain family is a subset of P which is a union of k chains and a k-antichain

family is a subset which is a union of k antichains.

Let ck(P ) be the maximal cardinality of all possible k-chain families and define

λi(P ) = ci(P ) − ci−1(P ), setting c0(P ) = 0. Similarly, let ak(P ) be the maximal

cardinality of all possible k-antichain families and define µi(P ) = ai(P ) − ai−1(P ),

setting a0(P ) = 0.

By a Theorem of Greene [14], λ = {λ1(P ) ≥ λ2(P ) ≥ . . . ≥ λr(P )}, where r

is the size of the largest antichain, is a partition of the cardinality of P . Also,

µ = {µ1(P ) ≥ µ2(P ) ≥ . . . ≥ µt(P )} is a also a partition of the cardinality of P but

instead t is the size of the largest chain. Moreover, µ is the partition dual to λ.

The following definition is due to Shi for w′ in W ′ [30, 1.3, 1.4, 1.7]. This defini-

tion is independent of choice of n-tuple representing w′ and therefore by the discussion

near the end of Section 2.2.3 the definition can be applied sensibly to all elements of

W .

Given w = {b1, . . . , bn}n in W , write bi ≤w bj if bj − bi ≥ 0 and 1 ≤ j ≤ i ≤ n,

or if bj − bi > n and 1 ≤ i < j ≤ n. This defines a partial order ≤w on the set

{b1, . . . , bn} and therefore one can obtain two partitions of n, λ and µ, as described

above.

There is a bijection between the set of two-sided cells of W and the set of

partitions of n (Corollary 2.2.1). I choose to identify a two-sided cell of W with the

partition λ, where λ is the partition arising as described from any element in that two-

sided cell.

The shape of the Young tableaux P ′′(w) and Q′′(w) in Yn, arising from the

extended Generalized Robinson-Schensted Algorithm on an element w in W , happens

to be the partition µ.

16

Example 2.2.1. Given w = {2,−24, 24, 7, 13,−8, 26,−4, 9}9, one has

2 ≤w 24, 13, 26

−24 ≤w 2, 24, 7, 13,−8, 26,−4, 9

24 ≤w −

7 ≤w 24, 26

13 ≤w 24, 26

−8 ≤w 2, 24, 7, 13, 26, 9

26 ≤w −

−4 ≤w 2, 24, 7, 13, 26, 9

9 ≤w 24, 13, 26

This gives the following Hasse diagram; the existence of a downward path from

bj to bi is interpreted as bi ≤w bj .24 26

7 13

2 9

−8 −4

−24

Maximal k-chain families are {26, 13, 9,−4,−24}, {26, 13, 9,−4,−24, 24, 7,−8}

and {26, 13, 9,−4,−24, 24, 7,−8, 2} for k = 1, 2, 3 respectively. Therefore w lies in the

two-sided cell of W which is identified with the partition λ = {5 ≥ 3 ≥ 1}.

17

Maximal k-antichain families are {2, 7, 9}, {2, 7, 9, 24, 26}, {2, 7, 9, 24, 26,−8,−4},

{2, 7, 9, 24, 26,−8,−4, 13} and {2, 7, 9, 24, 26,−8,−4, 13,−24} for k = 1, 2, 3, 4, 5 re-

spectively. Therefore, on running the extended Generalized Robinson-Schensted Algo-

rithm on w and its inverse one would obtain two Young tableaux in Yn, each of shape

µ = {3 ≥ 2 ≥ 2 ≥ 1 ≥ 1}. Note that µ is dual to λ.

Remark 2.1. I was able in this example to form a maximal (k + 1)-chain family by

taking the union of a chain and a maximal k-chain family, and similarly for antichains,

but this need not always be possible. When this is possible with the antichains, the

element is said to have a complete antichain family (Section 2.3.4).

2.2.5 W as a set of matrices

Given an element w in W and having identified it with a permutation w on the integers,

it can equally be viewed as an infinite affine matrix, with rows and columns labelled by

the integers (column labels increasing as you move left to right and row labels increasing

as you move down) and with a one in the (i, (i)w)-position, for all integers i, and zeros

everywhere else. Each row and column therefore has precisely one non-zero entry.

An IC (Increasing Chain) Block is a collection of consecutive rows whose non-

zero entries move from left to right as one moves down from row to row. If the property

is instead that the non-zero entries move from right to left then I have a DC (Decreasing

Chain) Block.

An IC Block is called a Maximal IC Block if both adding the row immediately

above and adding the row immediately below no longer results in another IC Block.A

Maximal DC Block is defined in a simlar way.

The size of a Block is the number of rows in it. For any integer i, I say that a

Block is at i when the row labelled by i + 1 is the least-numbered row of the Block. By

the property that (k + n)w = (k)w + n for all integers k, if there is a Maximal IC or

Maximal DC Block at i, the same is true at any j for which i = j.

18

Example 2.2.2. The element w = {8, 15, 11, 28, 6,−7,−3, 10}8 can be viewed as a

matrix.

11

11

11

11

11

11

11

row 1

There is a Maximal IC Block of size 5 at −3 (and at 5,−11, . . .) and a Maximal

DC Block of size 3 at 3 (and at 11,−5, . . .).

2.2.6 Left star operations

The concept of a left star operation is due to Kazhdan and Lusztig [16]. Although this

was defined for Coxeter groups, I can extend to W in a natural way.

Let ≤ be the Bruhat partial order on W ′ (Section 1.1). Then I define this partial

order on W via ω1y′ ≤ ω2w′ if and only if ω1 = ω2 and y′ ≤ w′ where ω1, ω2 are in Ω

and y′, w′ are in W ′.

Then the definition of a left descent set, originally given for a Coxeter group

(Section 1.3), can also be given for W . For any w in W the left descent set of w is the

set L(w) = {si | i ∈ {1, . . . , n}, siw ≤ w}.

For any integer i let

DL(si, si+1) = {w ∈W | L(w) ∩ {si, si+1} consists of one element}

I shall abbreviate DL(si, si+1) to DL(si).

19

Since the order of sisi+1 is three, if w lies in DL(si) then so does precisely one

of siw and si+1w. I denote {siw, si+1w} ∩DL(si) by ?w, where ? = {si, si+1}, or just?w when there is no ambiguity.

Note that ?(?w) = w. The map from w to ?w is an involution known as the left

star operation on w in DL(si). This operation is only defined when w lies in DL(si).

To explain left star operations in terms of matrices I first need this result of Shi.

Lemma 2.2.1. [25, Lemma 4.2.4]

For any w′ in W ′,

L(w′) = {si|i ∈ {1, . . . , n}, (i + 1)w′ < (i)w′}.

Consider ωlw′ in W where w′ is in W ′ and l is some integer. Suppose si is in

L(w′).

Then si−lωlw′ = ωlsiw′ ≤ ωlw′ so si−l is in L(ω

lw′).

Also (i − l + 1)ωlw′ = (i + 1)w′ < (i)w′ = (i − l)ωlw′ and if (i − l) is not in

{1, . . . , n}, it is still clear that (i− l + 1)ωlw′ < (i− l)ωlw′.

Corollary 2.2.2. For any w in W ,

L(w) = {si|i ∈ {1, . . . , n}, (i + 1)w < (i)w}.

Viewing w as a matrix, I can now deduce that the left star operation in DL(si) is defined

on w if and only if w has one of the following forms

(i) (ii)

(iii) (iv)

11

1

11

1

11

1

11

1

where the ith row is the first row shown in each of the above.

20

Moreover, when this is the case, one has

{form of w, form of ?w where ? = {si, si+1}} = {(i), (iv)} or {(ii), (iii)}.

For example, if w has the form labelled by (iv), ?w is obtained from w by

interchanging the (i + 1)st and (i + 2)nd rows. This corresponds to left multiplying by

si+1. Remember that, due to the periodic nature of the action of si+1, the (l+1)st and

(l + 2)nd rows of the matrix are also interchanged for any l with l = i.

The left star operation in DL(si) can very easily be described when w is expressed

in the form w = {b1, . . . , bn}n.

Consider bi, bi+1, bi+2. The left star operation in DL(si) is defined precisely

when bi, bi+1, bi+2 is neither an increasing nor a decreasing sequence, in which case the

largest and smallest numbers are interchanged.

Realising that the left star operation is left multiplication by si or si+1 and that

therefore the interchange is periodic, precisely two numbers of the n-tuple for w will

change and this will simply be an interchange of the two numbers except when the left

star operation results in left multiplication by sn.

Example 2.2.3. If w = {1, 4, 3, 8, 6, 5}6, ?w in DL(si, si+1) is

{4, 1, 3, 8, 6, 5}6 if i = 1

{1, 4, 8, 3, 6, 5}6 if i = 2

{1, 4, 8, 3, 6, 5}6 if i = 3

undefined if i = 4

{−1, 4, 3, 8, 6, 7}6 if i = 5

undefined if i = 6

Notice that a left star operation in DL(si) and a left star operation in DL(si+1),

if both defined, may give the same element, namely the result of left multiplication by

si+1.

The left star operation is an extremely useful tool, thanks to the Lemma below.

21

Lemma 2.2.2. [25, 1.6.3]

Given w′ in W ′, suppose that ?w′ in DL(si) is defined.

Then w′∼L?w′.

Any w in W can be expressed as ωlw′ for some integer l and some w′ in W ′.

Left multiplying w′ by ωl has the effect of shifting the affine matrix w′ up l rows.

Thus, if ?1(ωlw′) is defined for ?1 = {si−l, si−l+1} it follows that?2w′ is defined for

?2 = {si, si+1}, and ?1(ωlw′) = ωl(?2w′) ∼L?2w′ ∼L w′ ∼L ωlw′.

Corollary 2.2.3. Given w in W , suppose that ?w in DL(si) is defined.

Then w∼L?w.

So in Example 2.2.3 I have found three new elements in the same left cell of W

as w. Applying sequences of left star operations in different DL(si) will continue to find

more.

Recall Lemma 1.3.1(i). This result is true on W as well: for any w′ in W ′ and

any integer l, R(ωlw′) = R(w′) by Corollary 2.2.2 and since L(w) = R(w−1) for all w

in W .

Corollary 2.2.4. For any y, w in W :

If y ∼L w then R(y) = R(w).

It follows from this result that DL(si) is a union of right cells. Hence any left

star operation is defined on all or no elements of any given right cell.

Lemma 2.2.3. [16, Corollary 4.3(i)]

Suppose that ?y′ and ?w′ are defined in DL(si) in W′.

If y′ ∼R w′ then ?y′ ∼R?w′.

Any element w in W can be expressed as w′ωl where w′ is an element in W ′

in the same right cell of W as w and l is some integer. Since right multiplying w′ by

ωl effectively shifts the affine matrix of w′ l columns to the right, it is obvious that

?(w′ωl) = (?w′)ωl ∼R?w′. It follows from this that Lemma 2.2.3 also holds for W .

Corollary 2.2.5. Suppose that ?y and ?w are defined in DL(si) in W .

If y ∼R w then ?y ∼R?w.

22

So if a left star operation is defined on a right cell, it maps all elements in that

right cell to elements which are all themselves in some right cell. Using the fact that any

left star operation is an involution, one establishes a bijective map between two right

cells.

Also, by Corollary 2.2.3, the two right cells must belong to the same two-sided

cell and the result of restricting to any left cell in that two-sided cell must also be

bijective. This establishes the following Theorem, which had been realised by Shi for

W ′ [25, Lemma 18.3.2].

Theorem 2.2.1. Let L and R be a left cell and right cell respectively in the same

two-sided cell of W .

Any left star operation is either not defined on any element of L ∩R or defined

on every element of L ∩ R. In the latter case, the left star operation gives a bijective

map from L ∩R to L ∩R1, for some right cell R1 in the same two-sided cell.

Note that L(?w) 6= L(w) for any w in W by Corollary 2.2.2 and so R1 will

always be a different right cell to R in the above Theorem, by Corollary 2.2.4.

2.3 The map W → Yn × Yn ×Dom(Fλ)

2.3.1 The adapted Fn procedure

An element w1 in W has FForm at c if it has an IC Block of size n at c (Section 2.2.5).

Alternatively, w1 = {b1, . . . , bn}n has FForm at c if bc+1 < bc+2 < . . . < bc+n.

Of course, w1 also has FForm at any d for which c = d. Two FForms, at c and

at d, say, are said to be located at the same place whenever c = d.

Shi showed how any element w′ of W ′ can, via a sequence of left star operations,

be mapped to an element w′1, in the same left cell of W′, which has FForm at some

c [25, 9.2.1]. This is called the Fn procedure [30, 4.3]. The same process can be applied

to any w in W to obtain some w1 which has FForm with w ∼L w1.

I now describe an algorithm which I call the adapted Fn procedure and which

performs the above process. At each stage of the Fn procedure of Shi there is a shift

23

forwards which is allowed with the [. . .]n notation of W ′ but not with the {. . .}n notation

of W (Section 2.2.2). Therefore my adapted Fn procedure runs the Fn procedure, but

on strictly the image of 1, . . . , n (namely b1, . . . , bn where w = {b1, . . . , bn}n), and then

reverses all the shifts forward at the end. It is only the first and the final elements of

this procedure which have meaning as the n-tuple in the {. . .}n notation and I therefore

use rounded parentheses throughout to reflect this fact. Below I describe the adapted

Fn procedure on W , for n ≥ 3 (else the procedure is trivial).

i Suppose w = {b1, . . . , bn}n. The procedure begins with

(b01, . . . , b0n) = (b1, . . . , bn).

ii If b01 < b02 < . . . < b0n then there is nothing to do and the procedure

ends. Otherwise let i > 1 be the smallest integer with b0i < b0,i−1 and j,

1 ≤ j < i, be the smallest integer with b0i < b0j . Then one writes

(b01, b02, . . . , b0n)(1)7−→ (b11, b12, . . . , b1n)

where (b11, . . . , b1n) is obtained from (b01, . . . , b0n) by moving the term b0i

to the position of b0j , and also by moving the term b0j to the right-hand

end and adding n at the same time.

(b11, . . . , b1n) = (b01, . . . , b0,j−1, b0i, b0,j+1, . . .

. . . , b0,i−1, b0,i+1, . . . , b0n, b0j + n).

iii If b11 < b12 < . . . < b1n then the procedure finishes. Otherwise apply ii in

a similar fashion to get

(b11, b12, . . . , b1n)(2)7−→ (b21, b22, . . . , b2n).

In the same way, if needed, the procedure continues by obtaining

(b31, . . . , b3n), (b41, . . . , b4n) and so on. It is known, however, that there

exists some k ∈ Z≥0 such that bk1 < bk2 < . . . < bkn [25, 9.2.1].

24

iv I call the map (c1, . . . , cn) 7→ (cn − n, c1, . . . , cn−1) a shift back.

Write (bk1, . . . , bkn) 7→ (b′1, . . . , b′n) where (b′1, . . . , b′n) is obtained from

(bk1, . . . , bkn) by applying the shift back k times.

v Then w1 = {b′1, . . . , b′n}n is the result of applying the adapted Fn procedure

to w = {b1, . . . , bn}n.

Example 2.3.1. Applying the adapted F10 procedure to

w = {7, 9, 14, 3, 5, 2, 8, 11, 16, 10}10 gives w1 = {0, 3, 7, 9, 11, 16, 24, 2, 5, 8}10, an ele-

ment in the same left cell as w and which has FForm at 7.

(7, 9, 14, 3, 5, 2, 8, 11, 16, 10)(1)7−→ (3, 9, 14, 5, 2, 8, 11, 16, 10, 17)(2)7−→ (3, 5, 14, 2, 8, 11, 16, 10, 17, 19)(3)7−→ (2, 5, 14, 8, 11, 16, 10, 17, 19, 13)(4)7−→ (2, 5, 8, 11, 16, 10, 17, 19, 13, 24)(5)7−→ (2, 5, 8, 10, 16, 17, 19, 13, 24, 21)(6)7−→ (2, 5, 8, 10, 13, 17, 19, 24, 21, 26)(7)7−→ (2, 5, 8, 10, 13, 17, 19, 21, 26, 34)

7−→ (0, 3, 7, 9, 11, 16, 24, 2, 5, 8)

The FForm is clear if w1 is viewed in matrix form.

25

11

11

11

11

11

11

11

row 7

w1 as a matrix

It seems algorithm-friendly to apply the adapted Fn procedure as I have stated

it. It is, however, useful to know what the element of W at each stage of the procedure

is in the {. . .}n notation. This can be calculated just by performing the appropriate

number of shifts back.

Example 2.3.2. The elements of W represented at each stage in Example 2.3.1 are

shown below.

{| 7, 9, 14, 3, 5, 2, 8, 11, 16, 10}10(1)7−→ {7, | 3, 9, 14, 5, 2, 8, 11, 16, 10}10(2)7−→ {7, 9, | 3, 5, 14, 2, 8, 11, 16, 10}10(3)7−→ {7, 9, 3, | 2, 5, 14, 8, 11, 16, 10}10(4)7−→ {7, 9, 3, 14, | 2, 5, 8, 11, 16, 10}10(5)7−→ {7, 9, 3, 14, 11, | 2, 5, 8, 10, 16}10(6)7−→ {3, 7, 9, 14, 11, 16, | 2, 5, 8, 10}10(7)7−→ {0, 3, 7, 9, 11, 16, 24, | 2, 5, 8}10

The corresponding n-tuples in rounded parentheses are simply the n consecutive

numbers to the right of the vertical lines drawn above, recalling the property that

(i + n)w = (i)w + n for all integers i.

26

The above Example should make it clear that, if the adapted Fn procedure on

a given element has k stages (not counting the shifts back as a stage) the resulting

element will have its FForm located at k. In Example 2.3.1, I obtain an element with

FForm at 7.

For 1 ≤ l ≤ k, suppose the lth stage of the adapted Fn procedure is about

to be performed on (bl−1,1, . . . , bl−1,n). Let il be the least integer with il > 1 and

bl−1,il < bl−1,il−1. It is clear that i1, i2, . . . , ik is a non-decreasing sequence.

If il = 2, no left star operation is applied and the stage is trivial (think of the

behaviour in terms of the {. . .}n notation). Otherwise the lth stage describes left star

operations in DL(sl+il−3), DL(sl+il−4), . . . , DL(sl), in that order.

Hence the k-tuple, (i1, . . . , ik), completely determines the sequence of left star

operations underlying the adapted Fn procedure. This is illustrated by an example in

the Appendix (Section D).

Lemma 2.3.1. The sequence of left star operations underlying the adapted Fn proce-

dure on w in W depends only on the right cell containing w.

Proof. By Lemma 1.3.1(ii) and Corollary 2.2.2, the value of i1 in the k-tuple is the same

for any element in a given right cell. The left star operations underlying the first stage

of the adapted Fn procedure are determined, however, by i1. By Theorem 2.2.1, it is

clear therefore that the first stage gives a bijective map between right cells. Thus all

the elements in the image lie in the same right cell.

Applying the same argument again shows that the left star operations underlying

the second stage are the same. This process is repeated a finite number of times to give

the required result.

Given an arbitrary right cell, Lemma 1.3.1(ii) and Corollary 2.2.2 show that either

no elements in that cell have FForm or all elements have FForm and, moreover, have

FForm located at the same place. By Lemma 2.3.1 and Theorem 2.2.1 I can conclude

the following.

27

Lemma 2.3.2. For any element w in W , in the left cell L and the right cell R, the

adapted Fn procedure establishes a bijective map from w in L ∩ R to w1 in L ∩ R1,

where R1 is some right cell all of whose elements have FForm at c, where c is equal to

the number of stages of the procedure (excluding shifts back).

Remark 2.2. The above map is not necessarily the identity if R is a right cell whose

elements already have FForm. For example, {1, 3, 2}3 has FForm at 2 but is mapped to

{3, 1, 2}3, an element in a different right cell but which also has FForm.

2.3.2 Obtaining an element which has Hλ Form

Suppose that the partition λ = {λ1 ≥ . . . ≥ λr} has been computed for the two-sided

cell containing some element, which I label wr′+1, in W (Section 2.2.4). I say that wr′+1

has Hλ Form at c if, when viewing wr′+1 as a matrix, it has a Maximal DC Block of size

λ1 at c+λr +λr−1+ . . .+λ2, a Maximal DC Block of size λ2 at c+λr +λr−1+ . . .+λ3,

..., a Maximal DC Block of size λr−1 at c+λr and a Maximal DC Block of size λr at c.

As with the FForm, if an element has Hλ Form at c then it also has Hλ Form

at d where c = d, and I say that two Hλ Forms, at c and at d, say, are located at the

same place if c = d.

Example 2.3.3. Let {9,−14,−16, 35, 17, 11,−8,−12, 12, 13, 14,−11, 38, 15,−6, 39}16

be the element wr′+1. One can check that this lies in the two-sided cell associated with

λ = {5 ≥ 4 ≥ 3 ≥ 2 ≥ 1 ≥ 1}. It has Hλ Form at −8.

28

11

111

11

11

11

11

11

11

1111

11

11

1

row 1

I can also observe the Hλ Form just by looking at the n-tuple representing

wr′+1. Recalling the notation that bi = (i)wr′+1 for all integers i, decreasing sequences

bl > bl+1 > . . . > bl+m are shown by a curved line under bl, bl+1, . . . , bl+m in the

n-tuple. The start of the Hλ Form is indicated by a vertical line (wr′+1 also has Hλ

Form at 8, 24, −24 and so on).

{ 9 ,−14,−16, 35 , 17 , 11 ,−8 ,−12, 12 , 13 , 14 ,−11, 38 , 15 ,−6 , 39 }16|

Shi has described an algorithm for producing an element w′r′+1 in W′ which has

Hλ Form, given some element w′1 which has FForm at some c [25, 9.3]. This algorithm

describes an explicit sequence of left star operations. Left star operations have now also

29

been defined on W (Section 2.2.6). Viewing elements as affine matrices (Section 2.2.5),

it is clear that Shi’s algorithm has the same effect when applied to the whole of W .

This algorithm depends only on λ and on c.

In order to state this algorithm, I need to first introduce iterated left star oper-

ations on W [25, 8.1: for W ′].

Definition 2.3.1. Write ŵ?(c+1,s)←−−−−− w for 1 ≤ s < n and any integer c, if

i w has a DC Block at c of size s.

ii There exists a sequence of elements x0 = w, x1, . . . , xs−1 = ŵ such that

for every h, 1 ≤ h ≤ s− 1, we have xh =?xh−1 in DL(sc+h−1).

Notice that ŵ will exist if w does not have a DC Block at c− 1 of size s + 1.

Definition 2.3.2. Write ŵ?(c+1,s,m)←−−−−−−− w for 1 ≤ s < s + m ≤ n and any integer c

if there exists a sequence of elements x0 = w, x1, . . . , xm = ŵ such that for every h,

1 ≤ h ≤ m one has xh?(c+2−h,s)←−−−−−−− xh−1. If m = 0, we set ŵ = w.

Definition 2.3.3. Write w?(c+1,s,m)−−−−−−−→ ŵ if w

?(c+1+m,s,m)←−−−−−−−−− ŵ.

I say that wm+1 has Hλm Form at c (1 ≤ m ≤ r) if wm+1 has a Maximal DC

Block of size λ1 at c+λr +λr−1 + . . .+λ2, a Maximal DC Block of size λ2 at c+λr +

λr−1 + . . .+λ3, ..., and a Maximal DC Block of size λm at c+λr +λr−1 + . . .+λm+1,

and a Maximal IC Block of size n + 1− (λ1 + λ2 + . . . + λm) at c.

Shi’s algorithm works by first mapping to an element w2 which has Hλ1 Form and

then to an element w3 which has Hλ2 Form, an element w4 which has Hλ3 Form, and

so on. Since a Maximal IC Block of size l can be described as precisely l consecutive

Maximal DC Blocks of size one, it follows that the algorithm will end when one has

obtained an element wr′+1 which has Hλr′ Form, where r′ is the number of parts λl of

λ with λl > 1.

30

Lemma 2.3.3. [25, 9.3.1]

Suppose that w1 in W has FForm at c. Then there exists a sequence

w1 = x0, x1, . . . , xλ1−2 = w2 such that for each h, 0 ≤ h ≤ λ1 − 3,

xh?(c+1−

Ph+1t=1 t,h+2,n−h−2)−−−−−−−−−−−−−−−−−→ xh+1.

In particular, w2 has Hλ1 Form at c−∑λ1−2

t=1 t + 1.

Lemma 2.3.4. [25, 9.3.6]

Suppose that wm+1 in W has Hλm Form at c for some m, 1 ≤ m < r′. Then

there exists a sequence wm+1 = x10, x11, . . . , x1,m+1 = x20, x21, . . . , x2,m+1 = . . . =

xλm+1−1,0, xλm+1−1,1, . . . , xλm+1−1,m+1 = wm+2 such that for each h,

0 ≤ h ≤ λm+1 − 2,

xh+1,l?(c+n+1−

Pht=1 t−

Pm+1−lt=1 λt,λm+1−l,h+1)←−−−−−−−−−−−−−−−−−−−−−−−−−−−− xh+1,l−1

for 1 ≤ l ≤ m, and

xh+1,m?(c+1−

Ph+1t=1 t,h+2,n−

Pmt=1 λt−h−2)−−−−−−−−−−−−−−−−−−−−−−−−→ xh+1,m+1.

In particular, wm+2 has Hλm+1 Form at c−∑λm+1−1

t=1 t.

One should be aware that iterated left star operations are not defined on arbitrary

elements. Some remarkable work by Shi [25, 9.1] was required to ensure that this

algorithm is defined on any element which has FForm.

Starting with an element w1 which has FForm at c, therefore, by applying

Lemma 2.3.3 once and Lemma 2.3.4 (r′ − 1) times I obtain an element wr′+1 which

has Hλ Form at c −∑λ1−2

t=1 t −∑r′−1

m=1(∑λm+1−1

t=1 t) + 1. Since the algorithm consists

entirely of left star operations, wr′+1 is in the same left cell as w1 (Section 2.2.6).

Recall that I have applied the adapted Fn procedure to an element w in the left

cell L and right cell R to obtain an element w1 in L∩R1, where R1 is a right cell all of

whose elements have FForm at some c (Section 2.3.1). Since the partition assigned to

each element in a left or right cell is the same (Section 2.2.4), Shi’s algorithm determines

the same sequence of left star operations for all elements in L ∩ R1. I can therefore

apply Theorem 2.2.1 to obtain the following Lemma.

31

Lemma 2.3.5. Shi’s algorithm provides a bijective map from L∩R1 to L∩Rr′+1 where

R1 is a right cell all of whose elements have FForm at c and Rr′+1 is a right cell all of

whose elements have Hλ Form at d = c−∑λ1−2

t=1 t−∑r′−1

m=1(∑λm+1−1

t=1 t) + 1.

A lengthy example of this algorithm applied to an element which has FForm is

given in Appendix E.

2.3.3 Mapping to Γλ ∩ Γ−1λ

For any partition of n, λ = {λ1 ≥ . . . ≥ λr}, I now define uλ to be the longest

element of the subgroup of W generated by all sj for which j 6=∑l

i=1 λr+1−i for any

l, 1 ≤ l ≤ r. Then by construction uλ is an element which lies in the two-sided cell of

W associated with λ.

By the rule siω = ωsi+1, the element ωiuλω

−i is in the same two-sided cell as

uλ for any integer i (Section 2.2.3). Let ∆λ,i be the unique left cell of W containing

ωiuλω−i. I abbreviate ∆λ,0 to ∆λ. By Section 1.3 I observe that ∆−1λ,i is a right cell for

each i. Note that uλ = u−1λ also lies in ∆−1λ . Also observe that uλ has Hλ Form at 0.

Similarly, ωiuλω−i lies in ∆−1λ,i and has Hλ Form at −i (and at −i).

Therefore, by Lemma 1.3.1(ii) and Corollary 2.2.2, all elements in ∆−1λ,i have Hλ

Form at −i, for all integers i. One can check, for both W = WGLn(C) and

W = WSLn(C), that ∆−1λ,i = ∆

−1λ,i

for any integer i and hence there are at most n

distinct right cells of the form ∆−1λ,i (in general, there are not always exactly n distinct

right cells: for example, for n = 4 and λ = {2 ≥ 2}, ωuλω−1 = ω3uλω−3).

It is known that any element which has Hλ Form lies inside a right cell of the

form ∆−1λ,i [38, 2.2]. In Section 2.3.2 I obtained an element wr′+1 in L ∩ Rr′+1 which

has Hλ Form at d, where d = c−∑λ1−2

t=1 t−∑r′−1

m=1(∑λm+1−1

t=1 t)+1, so Rr′+1 = ∆−1λ,−d.

Then the element ωdwr′+1 has Hλ Form at 0 and therefore lies inside ∆−1λ . Recall that

ωdwr′+1 lies in the same left cell as wr′+1 (Section 2.2.3). Hence left multiplication by

ωd establishes a map from L∩Rr′+1 to L∩∆−1λ which is obviously bijective and which

only depends on d, the location of the Hλ Form of the element wr′+1.

Lemma 2.3.6. Left multiplication by ωd is a bijective map from L∩Rr′+1 to L∩∆−1λ .

32

By applying the adapted Fn procedure (Section 2.3.1), applying Shi’s algorithm

for obtaining an element with Hλ Form (Section 2.3.2) and left multiplying by a power

of ω, I have described a map, which I shall denote θ, from an arbitrary right cell to the

right cell ∆−1λ which preserves the left cell which I have labelled L. Remember that

taking the inverse of all elements in a left cell forms a right cell and taking the inverse

of all elements in a right cell forms a left cell (Section 1.3).

Lemma 2.3.7. Calculating the inverse element gives a bijective map from L ∩∆−1λ to

∆λ ∩ L−1, where L−1 is the right cell formed by taking the inverse of every element in

L.

Since the map θ is defined on arbitrary left cell-right cell intersection, it is defined

on ∆λ ∩ L−1.

Lemma 2.3.8. The map θ gives a bijective map from ∆λ ∩ L−1 to ∆λ ∩∆−1λ .

For any partition λ = {λ1 ≥ . . . ≥ λr} of n, let wλ be the longest element of

the subgroup of W generated by all sj for which j 6=∑l

i=1 λi for any l, 1 ≤ l ≤ r. Like

uλ, wλ also lies in the two-sided cell associated with λ. Let Γλ be the unique left cell

containing wλ. Note that Γλ = ∆λ if and only if all parts of λ are the same.

Lemma 2.3.9. Consider the map, ρ, on W which is defined by the below action on the

generators.

Wρ−→ W

sn 7−→ sn

si 7−→ sn−i (1 ≤ i ≤ n− 1)

ω 7−→ ω−1

Then ρ is an involution. Moreover, if I restrict ρ to ∆λ ∩∆−1λ , I have a bijective map

to Γλ ∩ Γ−1λ .

In terms of the {. . .}n notation of W , ρ can be expressed as follows: if w =

{b1, . . . , bn}n, then

ρ({b1, . . . , bn}n) = {c1, . . . , cn}n

where ci = n + 1− bn+1−i for 1 ≤ i ≤ n.

33

Proof. That ρ is an involution on W is clear. It is well-defined since the relations in W ′

are preserved.

I show that ρ(w) lies in Γλ ∩ Γ−1λ for any w in ∆λ ∩∆−1λ . Since ρ(uλ) = wλ,

it is sufficient to show that ρ(w) ∼L ρ(u) and ρ(w) ∼R ρ(u) whenever w ∼L u and

w ∼R u respectively, for any w, u in W .

Write w = ωiw′, u = ωju′ such that w′, u′ lie in W ′. Then w ≤L u if and only

if w′ ≤L u′ (Section 2.2.3). Also ρ(w) = ω−iw′ρ and ρ(u) = ω−ju′ρ, where w′ρ and u′ρare respectively the result of replacing si with sn−i (1 ≤ i ≤ n− 1) and fixing sn in any

expression for w′ and u′. Therefore ρ(w) ≤L ρ(u) if and only if w′ρ ≤L u′ρ. I show that

w′ ≤L u′ implies w′ρ ≤L u′ρ.

If w′ ≤L u′ there exists a sequence of elements in W ′, w′ = w′0, w′1, . . . , w′n = u′

such that for each l, 1 ≤ l ≤ n, w′l−1—w′l and L(w′l−1) 6⊆ L(w′l) (Section 1.3). Consider

the sequence w′ρ = w′ρ,0, w

′ρ,1, . . . , w

′ρ,n = u

′ where, for 0 ≤ l ≤ n, w′ρ,l is obtained from

w′l in the same way as w′ρ and u

′ρ are obtained from w

′ and u′.

It is clear that L(w′ρ,l−1) 6⊆ L(w′ρ,l) if L(w′l−1) 6⊆ L(w′l) since

L(w′ρ) = {ρ(si) | si ∈ L(w′)}

for any w′ in W ′.

Since ρ just relabels the generators of W ′ in such a way that the relations in

W ′ are preserved, it is obvious that w′ρ,l−1—w′ρ,l if and only if w

′l−1—w

′l. Therefore

w′ρ ≤L u′ρ if w′ ≤L u′. The same argument shows that u′ ≤L w′ implies u′ρ ≤L w′ρ. It

follows that ρ(w) ∼L ρ(u) if w ∼L u.

In a similar fashion one can show that ρ(w) ∼R ρ(u) if w ∼R u where w and u

are instead expressed as w′ωi and u′ωj , for w′ and u′ in W ′.

Thus ρ(w) is in ∆λ ∩∆−1λ for any w in Γλ ∩ Γ−1λ and, since ρ is an involution,

ρ restricts to ∆λ ∩∆−1λ to give a bijective map.

I finally show that the description of ρ in terms of the {. . .}n notation agrees

with the action of ρ on the generators.

34

Writing sn, si (1 ≤ i ≤ n − 1) and ω in the {. . .}n notation and applying

ρ({b1, . . . , bn}n) = {c1, . . . , cn}n as described in the Lemma I get respectively the

following.

ρ({0, 2, 3, . . . , n− 2, n− 1, n + 1}n) = {0, 2, 3, . . . , n− 2, n− 1, n + 1}n

ρ({1, 2, . . . , i− 1, i + 1, i, i + 2, . . . , n− 1, n}n) = {1, 2, . . . , n− i− 1,

n− i + 1, n− i, n− i + 2, . . . , n− 1, n}n for 1 ≤ i ≤ n− 1

ρ({2, 3, . . . , n + 1}n) = {0, 1, . . . , n− 1}n

The right-hand side of the equations above give sn, sn−i and ω−1 respectively in the

{. . .}n notation which is what I require.

This agreement on the generators is sufficient.

In this Section I have completed the description of a bijective map from L ∩ R

to Γλ ∩ Γ−1λ for any left cell L and right cell R. To recollect, the map is established as

follows (at each stage the left cell-right cell intersection to which I map is expressed in

the form ‘left cell ∩ right cell’):

L∩R (1)−→ L∩R1(2)−→ L∩Rr′+1

(3)−→ L∩∆−1λ(4)−→ ∆λ∩L−1

(5)−→ ∆λ∩∆−1λ(6)−→ Γλ∩Γ−1λ

Map (1) is described in Section 2.3.1 and is a bijection by Lemma 2.3.2. Map

(2) is described in Section 2.3.2 and is a bijection by Lemma 2.3.5. Maps (3), (4), (5)

and (6) are described in this Section and are bijections by Lemmas 2.3.6, 2.3.7, 2.3.8

and 2.3.9 respectively. So far I have only described explicitly the maps in the direction

shown above. In Section 2.4 I explicitly show the maps in the reverse direction.

35

2.3.4 The bijection from Γλ ∩ Γ−1λ to Dom(Fλ) and to Dom(F ′λ)

Suppose that w is in Γλ ∩ Γ−1λ and that λ = {λ1 ≥ . . . ≥ λr} is the partition of n

associated to w (Section 2.2.4). Let

0 = g0 < g1, . . . < gl = r

satisfy λgi+1 = . . . = λgi+1 > λgi+1+1 = . . . = λgi+2

for i = 0, . . . , l − 2.

Then define hi = λgi−λgi+1 for i = 1, . . . , l, where one sets λgl+1 = 0. Observe

that h1 + . . . + hl = λg1 = λ1.

For example, when λ = {5 ≥ 5 ≥ 4 ≥ 3 ≥ 3 ≥ 3 ≥ 1 ≥ 1}, g0 = 0, g1 = 2,

g2 = 3, g3 = 6, g4 = 8 and h1 = 1, h2 = 1, h3 = 2, h4 = 1.

Then define Fλ to be a connected reductive group satisfying

Fλ ' GLh1(C)×GLh2(C)× . . .×GLhl(C)

and let F ′λ = Fλ ∩ SLn(C).

Xi defined a bijection from Γλ∩Γ−1λ in WGLn(C) to the set IrrFλ of isomorphism

classes of irreducible rational representations of Fλ over the complex numbers, and a

similar bijection from Γλ ∩ Γ−1λ in WSLn(C) to the set IrrF′λ of isomorphism classes

of irreducible rational representations of F ′λ over the complex numbers. The set of

isomorphism classes for Fλ and F′λ are labelled Dom(Fλ) and Dom(F

′λ) respectively.

It is well known that the set IrrGLm(C) of irreducible rational representations

of GLm(C) is in one-to-one correspondence with the set Zmdom, where

Zmdom = {(t1, t2, . . . , tm) ∈ Zm | t1 ≥ t2 ≥ . . . ≥ tm}

It follows that the set IrrFλ is in one-to-one correspondence with

Dom(Fλ) = Zh1dom × Zh2dom × . . .× Z

hldom

and the dominant weights, α, in Dom(Fλ) will be given as λ1-tuples of the form

α = (α1,1, . . . , α1,h1 , . . . , αl,1, . . . , αl,hl).

Given any α = (αi,j) and β = (βi,j) in Dom(Fλ), I say that α and β are

equivalent if there exists an integer f such that αi,j−βi,j = fgi for all i, j. This clearly

36

is an equivalence relation and Dom(F ′λ) can be thought of as the corresponding set of

equivalence classes in Dom(Fλ).

I shall outline the bijection of Xi from Γλ ∩ Γ−1λ in WGLn(C) to Dom(Fλ). The

bijection for WSLn(C) is almost the same and shall be discussed separately at the end

of this Section.

For any element w = {b1, . . . , bn}n in W , let µ = {µ1 ≥ . . . ≥ µt} be the

partition which is associated to the antichains of the poset with partial order ≤w (Sec-

tion 2.2.4) and which is dual to λ.

Definition 2.3.4. I say that B = {B1, . . . , Bm} is a complete antichain family of w if

i For every i, 1 ≤ i ≤ m, Bi is an antichain with respect to ≤w.

ii For any 1 ≤ i, j ≤ m, Bi ∩Bj = { } if and only if i 6= j.

iii I have m = t.

iv The cardinality of each Bi is µi, 1 ≤ i ≤ m.

The below Example shows that not all elements in W have a complete antichain

family.

37

Example 2.3.4. Let w = {2, 6, 7, 8, 11, 12, 13, 17}8. Below is the Hasse diagram with

respect to ≤w.17 11 12 13

2 6 7 8

See that µ = {6 ≥ 2}. There is only one antichain of size six, {11, 12, 13, 6, 7, 8},

but the remaining two numbers form a chain under ≤w. Hence w does not have a

complete antichain family.

This is not a problem, however, for the elements with which I will be dealing.

Lemma 2.3.10. [38, 5.1.12]

Each element of Γλ ∩ Γ−1λ has a complete antichain family.

This is a delicate result, since it is not true even that all elements of Γλ have

this property.

By the work of Xi [38] I know that the complete antichain family belonging to

any w in Γλ ∩ Γ−1λ contains hi antichains of size gi, 1 ≤ i ≤ l. Observe that hi and gi

depend only on the two-sided cell containing w, and also that∑l

i=1 higi = n.

Let Bi,1, . . . , Bi,hi be the hi antichains of size gi. Express the elements in each

antichain Bi,j in the following way:

bgi,j + cgi,jn > bgi−1,j + cgi−1,jn > . . . > b1,j + c1,jn

where 1 ≤ bk,j ≤ n and ck,j are integers for all 1 ≤ k ≤ gi and 1 ≤ j ≤ hi.

Order the antichains, Bi,1, . . . , Bi,hi , in such a way that

bgi,1 + cgi,1n > bgi,2 + cgi,2n > . . . > bgi,hi + cgi,hin

Thus the antichains of size gi are ordered so that the largest number taken from

each antichain in turn forms a decreasing sequence.

38

Then define, for 1 ≤ i ≤ l, 1 ≤ j ≤ hi,

αi,j(B) =∑

1≤k≤gi

ckj

By this setup, and a Lemma of Xi [38, 5.1.4], it is true that αi,j(B) ≥ αi,j+1(B)

for 1 < j + 1 ≤ hi. It is clear therefore that

α(B) = (α1,1(B), . . . , α1,h1(B), . . . , αl,1(B), . . . , αl,hl(B))

does lie inside Dom(Fλ). Moreover, Xi showed that α(B) is independent of the choice

of complete antichain family B of w [38, 5.2.4]. Therefore the map from w to α(w),

where α(w) is α(B) for any complete antichain family B of w, is a well-defined map on

Γλ ∩ Γ−1λ .

Lemma 2.3.11. [38, 5.2.6(b)]

The map α, from w in WGLn(C) to α(w), defines a bijection between Γλ ∩ Γ−1λ

and Dom(Fλ).

I now discuss the bijection between Γλ ∩Γ−1λ in WSLn(C) and Dom(F′λ). Recall

the defintion of Dom(F ′λ) earlier in this Section. One can see now that this is sensible

since if α and β belong to the same equivalence class in Dom(F ′λ) and w is an element

in WGLn(C) such that α(w) = α then α(ω−fnw) = β for some integer f . In fact, for

each integer M there is precisely one representative, β = (βi,j), of the equivalence class

containing a given α, such that

Mn ≤∑i,j

βi,j ≤ (M + 1)n− 1

Hence, for the map from Γλ ∩ Γ−1λ in WSLn(C) to Dom(F′λ), if one utilises the iden-

tification ωn = e (where e is the identity) to always ensure that the elements, w =

{b1, . . . , bn}n, satisfy∑n

i=1 bi =∑n

i=1 i + Kn with 0 ≤ K ≤ n− 1, one just computes

α(w) as one would for WGLn(C) and one will automatically obtain the representative of

the equivalence class in Dom(F ′λ) for which M = 0.

Lemma 2.3.12. [38, 8.4]

The map α, from w in WSLn(C) to α(w), defines a bijection between Γλ ∩ Γ−1λ

and Dom(F ′λ).

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2.3.5 Summary

I have established a map from an arbitrary element w in W to a dominant weight in

Dom(Fλ) when W = WGLn(C) and in Dom(F′λ) when W = WSLn(C). I denote this

map Φ.

It is summarised by the following stages.

1. Compute λ, the partition associated to w as described in Section 2.2.2.

2. Apply the adapted Fn procedure to w and obtain an element w1 which has FForm

located, say, at c (Section 2.3.1).

3. Apply Section 2.3.2 to obtain from w1 an element wr′+1 which has Hλ Form

located at d = c−∑λ1−2

t=1 t−∑r′−1

m=1(∑λm+1−1

t=1 t) + 1, where r′ is the number of

parts λl of λ with λl > 1.

4. Compute x = ωdwr′+1.

5. Compute x−1.

6. Apply 2. followed by 3. then 4. to x−1 and obtain an element y, say.

7. Compute ρ(y).

8. Compute α(ρ(y)). This is Φ(w).

I have shown the following two Theorems.

Theorem 2.3.1. (a) The map Φ : WGLn(C) −→ Dom(Fλ) is an explicit bijection

when restricted to the intersection of any left cell and any right cell in the same two-

sided cell in WGLn(C).

(b) The map WGLn(C) −→ Yn × Yn ×Dom(Fλ) given by

w 7→ (P ′′(w), Q′′(w),Φ(w))

where w 7→ (P ′′(w), Q′′(w)) is the extended Generalized Robinson-Schensted Algorithm

applied to w, is an explicit bijection.

40

Proof. Lemmas 2.3.2, 2.3.5, 2.3.6, 2.3.7, 2.3.9 and 2.3.11 show that I have created an

explicit bijection in part (a). Part (b) is a direct Corollary of part (a).

Theorem 2.3.2. (a) The map Φ : WSLn(C) −→ Dom(F ′λ) is an explicit bijection when

restricted to the intersection of any left cell and any right cell in the same two-sided cell

in WSLn(C).

(b) The map WSLn(C) −→ Yn × Yn ×Dom(F ′λ) given by

w 7→ (P ′′(w), Q′′(w),Φ(w))

where w 7→ (P ′′(w), Q′′(w)) is the extended Generalized Robinson-Schensted Algorithm

applied to w, is an explicit bijection.

Proof. Lemmas 2.3.2, 2.3.5, 2.3.6, 2.3.7, 2.3.9 and 2.3.12 show that I have created an

explicit bijection in part (a). Part (b) is a direct Corollary of part (a).

2.4 Inverting Φ

Given any (P ′′(w), Q′′(w),Φ(w)) in Yn × Yn ×Dom(Fλ) I set out to describe in this

Section an explicit algorithm for obtaining the element w in WGLn(C) which is associated

with this triple via Theorem 2.3.1(b). The map Φ is a bijection when restricted to the

intersection of the left cell and right cell containing w. The idea is to invert this map

using just the tableaux P ′′(w) and Q′′(w) as information.

I achieve the same goal with the bijection for WSLn(C).

2.4.1 From Dom(Fλ) or Dom(F ′λ) to Γλ ∩ Γ−1λ

The contents of this Section were described by Xi [38, 5.3].

First assume that W = WGLn(C). Given a dominant weight

α = (α1,1, . . . , α1,h1 , . . . , αl,1, . . . , αl,hl)

in Dom(Fλ) I show how to construct an element wα in Γλ∩Γ−1λ such that α(wα) = α.

41

Fortunately I know the partition λ = {λ1 ≥ . . . ≥ λr}. It is the dual of

the shape of the pair of Young tableaux, (P ′′(w), Q′′(w)), associated to w under the

extended Generalized Robinson-Schensted Algorithm. Notice that I therefore know the

partition λ of w and the pair of Young tableaux even though I do not at present know

w itself: it is w I am trying to compute.

Let ei = λ1 +λ2 + . . .+λi for 1 ≤ i ≤ r and e0 = 0. Then for all 1 ≤ j ≤ r and

1 ≤ k ≤ λj let dj,k = ej−1 + k. Recall also the definitions of g1, . . . , gl (Section 2.3.4)

and wλ (Section 2.3.3).

First assume that all αi,j are non-negative.

If α = (0, . . . , 0) in Dom(Fλ), wα is the unique element in Γλ ∩Γ−1λ which also

lies in the symmetric group generated by s1, . . . , sn−1. This is wλ which I know since I

know λ.

If α = (1, 0, . . . , 0) in Dom(Fλ), then wα is defined by

(a)wα =

(i + 1)λ1 if a = di,1, 1 ≤ i ≤ g1 − 1

λ1 + n if a = dg1,1

(a)wλ if a 6= di,1, 1 ≤ i ≤ g1

For any α with all αi,j non-negative, one builds up wα inductively as follows.

Suppose that αi,j ≥ 1 and suppose I have wα′ such that α(wα′) = α′where

α′ = (α1,1, . . . , αi,1, . . . , αi,j−1, αi,j − 1, 0, . . . , 0).

Set jgi = j and for k = gi, gi − 1, . . . , 2 choose 1 ≤ jk−1 ≤ λk−1 such that

(dk−1,jk−1−1)wα′ > (dk,jk)wα′ > (dk−1,jk−1)wα′

where I set (dk−1,0)wα′ =∞.

Then, if α′′ = (α1,1, . . . , αi,1, . . . , αi,j−1, αi,j , 0, . . . , 0), wα′′ such that

α(wα′′) = α′′ is defined by the action below.

(a)wα′′ =

(dk,jk)wα′ if a = dk−1,jk−1 , 2 ≤ k ≤ gi

(d1,j1)wα′ + n if a = dgi,j

(a)wα′ if a 6= dk,jk , 1 ≤ k ≤ gi

42

Suppose now that not all αi,j are non-negative. Then choose a natural number

q such that αi,j + qgi ≥ 0 for all 1 ≤ i ≤ q, 1 ≤ j ≤ hi. Above shows how to find w in

Γλ ∩ Γ−1λ such that

α(w) = (α1,1 + qg1, . . . , α1,h1 + qg1, . . . , αl,1 + qgl, . . . , αl,hl + qgl)

in Dom(Fλ).

Then w ∼L ω−lnw and ω−lnw = wω−ln so w ∼R ω−lnw (Section 2.2.3), so

ω−lnw also lies in Γλ ∩ Γ−1λ . Furthermore,

α(ω−lnw) = (α1,1, . . . , α1,h1 , . . . , αl,1, . . . , αl,hl).

Now suppose that W = WSLn(C).

Given a dominant weight α = (α1,1, . . . , α1,h1 , . . . , αl,1, . . . , αl,hl) in Dom(F′λ).

I wish to find an element wα in Γλ ∩ Γ−1λ in WSLn(C) such that α(wα) = α. Re-

call that Dom(F ′λ) can also be described as a set of equivalence classes in Dom(Fλ)

(Section 2.3.4).

Suppose that α and β are two weights in Dom(Fλ) which belong to the same

equivalence class in Dom(F−1λ ). The map just described from Dom(Fλ) to Γλ ∩ Γ−1λ

in WGLn(C) gives two different elements when applied to α and β. However, it gives

two elements that are equivalent in WSLn(C) under the identification ωn = e (where e

is the identity element).

Hence if I do for WSLn(C) exactly as I do for WGLn(C) only I remember to

identify ωn with the identity in WSLn(C) then I have a well-defined map from Dom(F′λ)

to Γλ ∩ Γ−1λ in WSLn(C).

2.4.2 Inverting the adapted Fn procedure

Mapping back from Γλ∩Γ−1λ is simple once I have established how to invert the adapted

Fn procedure (Section 2.3.1) and so the purpose of this Section is to address this issue.

Recall that I am able to describe the explicit sequence of left star operations

underlying this procedure as soon as I know the k-tuple (i1, . . . , ik). Moreover the k-

tuple is the same for all elements in any given right cell. However, at this stage, I can

43

only find the k-tuple belonging to a right cell by explicitly computing the adapted Fn

procedure on an element in that right cell. Since I am inverting, I instead need to find

this data just from the Young tableau in Yn which represents that right cell. Hence

I shall construct an algorithm for associating a k-tuple to each Young tableau in Yn,

where k is the number of stages of the adapted Fn procedure when computed on any

element in the right cell given by that tableau. In order to do this, I need to create an

element lying inside any right cell, given just the tableau in Yn labelling it.

Given any Young diagram, count the boxes from left to right along rows, starting

on the top row and moving down a row each time, and count the rows from top to

bottom. Then, given any Young tableau, I define pi and qi (1 ≤ i ≤ n) to be such that

the pthi box and the qthi row contain the entry i.

Example 2.4.1. Given the Young tableau below

2 4 7 8 10

1 5 12

6 9 13

3 14

11

I have

(p1, . . . , p14) = (6, 1, 12, 2, 7, 9, 3, 4, 10, 5, 14, 8, 11, 13)

(q1, . . . , q14) = (2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 5, 2, 3, 4)

Recall that t is the number of parts of the partition µ associated with w in W

(Section 2.2.4). Alternatively, it is the number of rows in the Young tableau representing

the right cell containing w and µ is the shape of that tableau.

Lemma 2.4.1. Suppose that (p1, . . . , pn) and (q1, . . . , qn) have been computed for

some Young tableau in Yn.

Consider w = {b1, . . . , bn}n where bi = pi +(C(t+1)−2Cqi)n for some integer

C ≥ 1. This defines an infinite collection of elements lying in the right cell given by

that tableau.

44

Proof. Suppose the Young tableau describing the right cell is TR.

I show that running the extended Generalized Robinson-Schensted Algorithm on

w−1, for any C ≥ 1, gives the Young tableau TR. A brief outline of this Algorithm is

found in Appendix C.

Observe that, if w−1 = {c1, . . . , cn}n, ci is calculated by searching for bj with

bj = pj = i. Since j is mapped by w to bj = i + (C(t + 1) − 2Cql)n, it follows that

i is mapped by w−1 to ci = j − (C(t + 1) − 2Cqj)n. Also, pj = i means that the ith

box of the Young tableau TR contains the entry j. Hence, for 1 ≤ i ≤ n, ci = j where

pj = i, in other words j is the entry in the ith box of TR.

In this way, c1, c2, . . . , cn will be the entries in the 1st, 2nd, . . . , nth boxes of TR,

respectively. Moreso, qi+1 = qi + 1 whenever ci+1 is an entry on the row below ci

(1 ≤ i ≤ n − 1). Since the entries of any tableau in Yn increase along rows, and

since C ≥ 1, it is clear that c1 < c2 < . . . < cn, so w−1 already has FForm. Also,

by construction, {cµ1+µ2+...+