Mind your P and Q-symbols: Why the Kazhdan-Lusztig basis of the Hecke algebra of type A is cellular

82 Pages·2003·0.526 MB·English

by  Geordie Williamson

Mind your P and Q-symbols:
Why the Kazhdan-Lusztig basis of the
Hecke algebra of type A is cellular
Geordie Williamson
An essay submitted in partial fulfillment of
of the requirements for the degree of
B.A. (Honours)
Pure Mathematics
University of Sydney
SIDERE·MENS·EADEM·MUTATO
October 2003

To my mother

Acknowledgements
FirstandforemostIwouldliketothankmysupervisorDr. GusLehrer. Iwouldnothaveprogressed
very far without his inspiration and support. I would also like to thank Anthony Henderson, Bob
Howlett and Andrew Mathas for giving up their time for me during the year. On a more personal
note, I would like to thank my family and friends (both on the fourth floor of Carslaw and at 8
Northumberland Ave.) for their support and friendship this year. Lastly, I would like to thank
John Lascelles, Alan Stapledon and Steve Ward for proof reading sections of the essay.
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Contents
Acknowledgements i
Introduction v
Preliminaries vii
1 The Symmetric Group 1
1.1 The Length Function and Exchange Condition . . . . . . . . . . . . . . . . . . . . . 1
1.2 Generators and Relations for Sym . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
n
1.3 The Bruhat Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Descent Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Young Tableaux 10
2.1 Diagrams, Shapes and Standard Tableaux . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The Row Bumping Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The Robinson-Schensted Correspondence . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Partitions Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Growth Diagrams and the Symmetry Theorem . . . . . . . . . . . . . . . . . . . . . 14
2.6 Knuth Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Tableau Descent Sets and Superstandard Tableaux . . . . . . . . . . . . . . . . . . . 20
2.8 The Dominance Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 The Hecke Algebra and Kazhdan-Lusztig Basis 26
3.1 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 The Linear Representations of H (q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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3.3 Inversion in H (q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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3.4 An Involution and an anti-Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 The Kazhdan-Lusztig Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Multiplication Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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4 Cells 39
4.1 Cell Orders and Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Representations Associated to Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Some Properties of the Kazhdan-Lusztig Polynomials. . . . . . . . . . . . . . . . . . 42
4.4 New Multiplication Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 New Definitions of the Cell Preorders. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 The Kazhdan-Lusztig Basis as a Cellular Basis 48
5.1 Cellular Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Elementary Knuth Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 The Change of Label Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Left Cells and Q-Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 Property A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.6 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A Appendix 65
A.1 An Alternative Proof of Kazhdan and Lusztig’s Basis Theorem . . . . . . . . . . . . 65
A.2 Two-Sided Cells and the Dominance Order . . . . . . . . . . . . . . . . . . . . . . . 67
Bibliography 69
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Introduction
The Hecke algebras emerge when one attempts to decompose an induced representation of certain
finite matrix groups. In particular, the Hecke algebra of the symmetric group is isomorphic to the
algebra of intertwining operators of the representation of GL(n,q) obtained by inducing the trivial
representation from the subgroup of upper triangular matrices. By Schur’s Lemma, the restriction
of any intertwining operator to an irreducible representation must be scalar. Hence the problem of
decomposingtheinducedrepresentationisequivalenttodecomposingtheassociatedHeckealgebra.
However, the representation theory of the Hecke algebra is difficult. In 1979 Kazhdan and Lusztig
[17] introduced a special basis upon which a generating set for the Hecke algebra acts in an easily
describedmanner. Inordertoconstructtherepresentationsaffordedbythisnewbasisthenotionof
cells was introduced and it was shown that each cell corresponds to a representation of the algebra.
In general these representations are not irreducible. However Kazhdan and Lusztig showed that,
in the case of the Hecke algebra of the symmetric group, their construction does yield irreducible
representations. In order to prove this Kazhdan and Lusztig introduced and studied the so-called
‘star operations’ on elements of the symmetric group.
ItisimplicitinKazhdanandLusztig’spaperthatthesestaroperations(andtheequivalenceclasses
which they generate) have a deep combinatorial significance. This significance can be explained
in terms of the Robinson-Schensted correspondence. In 1938 Robinson [25] showed that, to every
permutation, one can associate a pair of ‘standard tableaux’ of the same shape. Then in 1961
Schensted [26] showed that this map was a bijection. In 1970 Knuth [20] studied the equivalence
classes of the symmetric group corresponding to a fixed left or right tableau and showed that two
permutationsareequivalentifandonlyiftheycanberelatedbycertainbasicrearrangementsknown
as ‘Knuth transformations’. The amazing thing is that these Knuth transformations are precisely
the star operations of Kazhdan and Lusztig. Hence, much of the algebraic theory developed by
Kazhdan and Lusztig can be described combinatorially, using the language of standard tableaux.
SinceamultiplicationtablefortheHeckealgebraofaWeylgroupwasfirstwrittendownbyIwahori
[16] in 1964, many algebras similar to the Hecke algebra have been discovered. In almost all cases
their representation theory is approached by using analogous techniques to the original methods
of Kazhdan and Lusztig. The similarities between these algebras, in particular the multiplicative
propertiesofadistinguishedbasis,ledGrahamandLehrer[12],in1996,todefinea‘cellularalgebra’.
This definition provides an axiomatic framework for a unified treatment of algebras which possess
a ‘cellular basis’.
In the paper in which Graham and Lehrer defined a cellular algebra, the motivating example was
the Hecke algebra of the symmetric group. However, there is no one source that explains why the
Hecke algebra of the symmetric group is a cellular algebra. The goal of this essay is to fill this gap.
However, we are not entirely successful in this goal. We must appeal to a deep theorem of Kazhdan
and Lusztig to show that a certain degenerate situation within a cell cannot occur (Kazhdan and
Lusztig prove this using geometric machinery beyond the scope of this essay).
Thestructureofthisessayisasfollows. InChapter1wegathertogethersomefundamentalconcepts
associatedtothesymmetricgroupincludingthelengthfunction, theBruhatorderanddescentsets.
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We also discover a set of generators and relations for the symmetric group. This is intended to
motivate the introduction of the Hecke algebra by generators and relations in Chapter 3.
Chapter 2 provides a self-contained introduction to the calculus of tableaux. Most of the results,
including the Robinson-Schensted correspondence, the Symmetry Theorem and Knuth equivalence
are fundamental to the subject and are present in any text on standard tableaux (possibly with
different proofs). Towards the end of the chapter we introduce the tableau descent set and super-
standard tableau. These are less well-known but are fundamental to our arguments in Chapter 5.
Chapters 3 and 4 are an introduction to Kazhdan-Lusztig theory in the special case of the Hecke
algebra of the symmetric group. In Chapter 3 we define the Hecke algebra and then prove the
existence and uniqueness of the Kazhdan-Lusztig basis. In Chapter 4 we introduce the cells as-
sociated to the basis of an algebra and show how they lead to representations. The rest of the
chapter is then dedicated to deriving the original formulation of the cell preorders due to Kazhdan
and Lusztig. Although we will not make it explicit, all of the material of Chapters 3 and 4 can be
proved without alteration in the more general case of the Hecke algebra of a Coxeter group.
In Chapter 5 we define a cellular algebra and then combine the results of Chapters 2, 3 and 4 with
the aim of showing that the Kazhdan-Lusztig basis is cellular. We first show how the elementary
Knuth transformations can be realised algebraically and then work towards a complete description
ofthecellsintermsoftheleftandrighttableauoftheRobinson-Schenstedcorrespondence. Wealso
show that the representations afforded by left cells within a given two-sided cell are isomorphic.
However, as mentioned above, we cannot complete our proof that the Kazhdan-Lusztig basis is
cellular entirely via elementary means: we must appeal to a theorem of Kazhdan and Lusztig to
show that all left cells within a two-sided cell are incomparable in the left cell preorder.
The appendix contains two sections. The first gives an elegant alternative proof of the existence
and uniqueness of the Kazhdan-Lusztig basis due to Lusztig [19]. In the second section we discuss
the relationship between the dominance order and two-sided cell order.
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