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1 Decomposition numbers of Hecke algebras of type A
 1.1 Description
 1.2 The modular representation theory of Hecke algebras
 1.3 Two small examples
 1.4 Overview over this manual
 1.5 Credits

1 Decomposition numbers of Hecke algebras of type A

1.1 Description

Hecke is a port of the GAP 3-package Specht \(2.4\) to GAP 4.

This package contains functions for computing the decomposition matrices for Iwahori-Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups (indeed, the latter is a special case of the former) many of the combinatorial tools from the representation theory of the symmetric group are included in the package.

These programs grew out of the attempts by Gordon James and Andrew Mathas [JM96] to understand the decomposition matrices of Hecke algebras of type A when \(q=-1\). The package is now much more general and its highlights include:

  1. Hecke provides a means of working in the Grothendieck ring of a Hecke algebra \(H\) using the three natural bases corresponding to the Specht modules, projective indecomposable modules, and simple modules.

  2. For Hecke algebras defined over fields of characteristic zero, the algorithm of Lascoux, Leclerc, and Thibon [LLT96] for computing decomposition numbers and "crystallized decomposition matrices" has been implemented. In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.

  3. Hecke provides a way of inducing and restricting modules. In addition, it is possible to "induce" decomposition matrices; this function is quite effective in calculating the decomposition matrices of Hecke algebras for small \(n\).

  4. The \(q\)-analogue of Schaper's theorem [JM97] is included, as is Kleshchev's [Kle96] algorithm of calculating the Mullineux map. Both are used extensively when inducing decomposition matrices.

  5. Hecke can be used to compute the decomposition numbers of \(q\)-Schur algebras (and the general linear groups), although there is less direct support for these algebras. The decomposition matrices for the \(q\)-Schur algebras defined over fields of characteristic zero for \(n<11\) and all \(e\) are included in Hecke.

  6. The Littlewood-Richard rule, its inverse, and functions for many of the standard operations on partitions (such as calculating cores, quotients, and adding and removing hooks), are included.

  7. The decomposition matrices for the symmetric groups \(S_n\) are included for \(n<15\) and for all primes.

1.2 The modular representation theory of Hecke algebras

The "modular" representation theory of the Iwahori-Hecke algebras of type A was pioneered by Dipper and James [DJ86] [DJ87]; here the theory is briefly outlined, referring the reader to the references for details.

Given a commutative integral domain \(R\) and a non-zero unit \(q\) in \(R\), let \(H=H_{R, q}\) be the Hecke algebra of the symmetric group \(S_n\) on \(n\) symbols defined over \(R\) and with parameter \(q\). For each partition \(\mu\) of \(n\), Dipper and James defined a Specht module \(S(\mu)\). Let \(rad~S(\mu)\) be the radical of \(S(\mu)\) and define \(D(\mu)=S(\mu)/rad~S(\mu)\). When \(R\) is a field, \(D(\mu)\) is either zero or absolutely irreducible. Henceforth, we will always assume that \(R\) is a field.

Given a non-negative integer \(i\), let \([i]_q=1+q+\ldots+q^{i-1}\). Define \(e\) to be the smallest non-negative integer such that \([e]_q=0\); if no such integer exists, we set \(e\) equal to \(0\). Many of the functions in this package depend upon e; the integer \(e\) is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.

A partition \(\mu=(\mu_1,\mu_2,\ldots)\) is \(e\)-singular if there exists an integer \(i\) such that \(\mu_i=\mu_{i+1}=\cdots= \mu_{i+e-1}>0\); otherwise, \(\mu\) is \(e\)-regular. Dipper and James [DJ86] showed that \(D(\nu)\neq 0\) if and only if \(\nu\) is \(e\)-regular and that the \(D(\nu)\) give a complete set of non-isomorphic irreducible \(H\)-modules as \(\nu\) runs over the \(e\)-regular partitions of \(n\). Further, \(S(\mu)\) and \(S(\nu)\) belong to the same block if and only if \(\mu\) and \(\nu\) have the same \(e\)-core [DJ87][JM97]. Note that these results depend only on \(e\) and not directly on \(R\) or \(q\).

Given two partitions \(\mu\) and \(\nu\), where \(\nu\) is \(e\) -regular, let \(d_{\mu,\nu}\) be the composition multiplicity of \(D(\nu)\) in \(S(\nu)\). The matrix \(D=(d_{\mu,\nu})\) is the decomposition matrix of \(H\). When the rows and columns are ordered in a way compatible with dominance, \(D\) is lower unitriangular.

The indecomposable \(H\)-modules \(P(\nu)\) are indexed by \(e\) -regular partitions \(\nu\). By general arguments, \(P(\nu)\) has the same composition factors as \(\sum_{\mu} d_{\mu,\nu} S(\mu)\); so these linear combinations of modules become identified in the Grothendieck ring of \(H\). Similarly, \(D(\nu) = \sum_{\mu} d_{\nu,\mu}^{-1} S(\mu)\) in the Grothendieck ring. These observations are the basis for many of the computations in Hecke.

1.3 Two small examples

Because of the algorithm of [LLT96], in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using Hecke. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:

gap> H:=Specht(4);   # e=4, 'R' a field of characteristic 0
<Hecke algebra with e = 4>
gap> RInducedModule(MakePIM(H,12,2));
<direct sum of 5 P-modules>
gap> Display(last);
P(13,2) + P(12,3) + P(12,2,1) + P(10,3,2) + P(9,6)

The [LLT96] algorithm was applied 24 times during this calculation.

For Hecke algebras defined over fields of positive characteristic the major tool provided by Hecke, apart from the decomposition matrices contained in the libraries, is a way of "inducing" decomposition matrices. This makes it fairly easy to calculate the associated decomposition matrices for "small" \(n\). For example, the Hecke libraries contain the decomposition matrices for the symmetric groups \(S_n\) over fields of characteristic \(3\) for \(n<15\). These matrices were calculated by Hecke using the following commands:

gap> H:=Specht(3,3);   # e=3, 'R' field of characteristic 3
<Hecke algebra with e = 3>
gap> d:=DecompositionMatrix(H,5);  # known for n<2e
<7x5 decomposition matrix>
gap> Display(last);
5    | 1
4,1  | . 1
3,2  | . 1 1
3,1^2| . . . 1
2^2,1| 1 . . . 1
2,1^3| . . . . 1
1^5  | . . 1 . .
gap> for n in [6..14] do
>       d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d);
>    od;

The function InducedDecompositionMatrix contains almost every trick for computing decomposition matrices (except using the spin groups).

Hecke can also be used to calculate the decomposition numbers of the \(q\)-Schur algebras; although, as yet, here no additional routines for calculating the projective indecomposables indexed by \(e\)-singular partitions. Such routines may be included in a future release, together with the (conjectural) algorithm [LT96] for computing the decomposition matrices of the \(q\)-Schur algebras over fields of characteristic zero.

1.4 Overview over this manual

Chapter 2 describes the installation of this package. Chapter 3 shows instructive examples for the usage of this package.

1.5 Credits

I would like to thank Anne Henke for offering me the interesting project of porting Specht \(2.4\) to the current GAP version, Max Neunhöffer for giving me an excellent introduction to the GAP 4-style of programming and Benjamin Wilson for supporting the project and helping me to understand the mathematics behind Hecke.

Also I thank Andrew Mathas for allowing me to use his GAP 3-code of the Specht \(2.4\) package.

The lastest version of Hecke can be obtained from

http://home.in.tum.de/~traytel/hecke/.

Dmitriy Traytel

traytel@in.tum.de

Technische Universität München, 2010.

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