Specht 2.4

Decomposition matrices for the Hecke algebras of type A

Andrew Mathas
University of Sydney

For installation notes see below. What follows is a brief description of the package; more details can be found in the manual, which can be downloaded separately from this page as either a dvi file (43kB) or a postscript file (88kB).

Specht is included as a share package in Gap, version 3.4.4. It is made available under the usual terms and conditions of Gap.

If you have any questions or problems please email me.

Andrew Mathas

Description

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 [JM1] 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 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 [LLT] 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 [JM] 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.

Installation

1. gunzip specht-2.4.tar.gz
2. tar -xfv specht-2.4.tar
   When you do this you should find the following files in a directory called ``specht-2.4'':

   README       -this file
   doc/         -Specht documentation (see below)
   gap/         -Gap source
   init.g       -initialization file
   lib/         -Specht library files 
   

   Ideally, Specht should be installed in the Gap packages directory, however, it can be installed anywhere. If Specht is not installed in the GAP packages directory then include suitably edited versions of the following lines

   Add(PKGNAME, "/path/to/directory/containing/specht/"); PKGNAME:=Reversed(PKGNAME);

   in your .gaprc file in your home directory (create such a file with these lines if you don't already have one). The second line ensures that this version of Specht is used in preference to any other version lying around on your system.

gap> RequirePackage("specht");
gap> H:=Specht(3);
Specht(e=3, S(), P(), D(), Pq())


Installing Specht's documentation

  specht.tex  -LaTeX source for the manual
  specht.html  -an HTML version of the manaul
  manual.tex  -header file for LaTeXing specht.tex
  install.tex -these installation notes

1. Copy specht.tex into the GAP doc/ directory.
2. Change directory to GAPs doc directory and add the line \Include{specht} to manual.tex.
3. LaTeX manual.tex.
4. Run makeindex (if available) and re-LaTeX manual.tex.