GAP 3 Share Package "specht"

Specht: Decomposition matrices for the Hecke algebras of type A

Share package since release 3.4, about July 1996, communicated by Herbert Pahlings, Aachen.
This package has been transferred to GAP 4 by Dmitriy Traytel. The GAP 4 version is called Hecke.

Author

Andrew Mathas.

Implementation

Language: GAP 3 (Note: Specht is not compatible with GAP 4.)
Operating system: Any
Current version: 2.4

Description

A package for calculating decomposition numbers of Hecke algebras of the symmetric groups and q-Schur algebras.

This package contains functions for computing the decomposition matrices for 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 later is a special case of the former - many of the combinatorial tools from the representation theory of the symmetric group are included in this package.

These programs grew out of the attempts by Gordon James and myself 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. Specht 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 we have implemented the algorithm of Lascoux, Leclerc, and Thibon for computing decomposition numbers and "crystallized decomposition matrices". In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.
3. We provide 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 is included, as is Kleshchev's [K] algorithm of calculating the Mullineux map. Both are used extensively when inducing decomposition matrices.
5. Specht 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 Specht.
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 Sym_n are included for n < 15 and for all primes.

Home Page

Specht

Manual

You can reach the HTML version of the Specht 2.4 manual given on the Specht Home Page. Note that chapter 71 of the GAP 3 manual describes an earlier version.

Contact address

Andrew Mathas
University of Sydney
Sydney
Australia
email: mathas@maths.usyd.edu.au