Specht 2.4
Decomposition matrices for the Hecke algebras of type A
A package for calculating decomposition numbers of
Hecke algebras of the symmetric groups and qSchur
algebras.
Andrew Mathas
University of Sydney
© Andrew Mathas October 1997.
Specht runs only under Gap 3.4; it is not
compatible with Gap 4 (which, unfortunately is not
backwardly compatible). When (if?) I have time I will release a
Gap 4 version of these programs.
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
The version of Specht available from this page is more recent
than the version included with Gap 3.4.4 at St. Andrews.
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:
 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.
 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.
 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.
 The qanalogue 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.
 Specht can be used to compute the decomposition numbers of
qSchur algebras (and the general linear groups), although there is
less direct support for these algebras. The decomposition matrices for the
qSchur algebras defined over fields of characteristic zero for $n<11$
and all e are included in Specht.
 The LittlewoodRichard 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.
 The decomposition matrices for the symmetric groups $\Sym_n$ are
included for $n<15$ and for all primes.
A complete description of what is available in the package can be found
in the manual.
First you must obtain the file specht2.4.tar.gz
(156 kB); either from here or via ftp from
http://wwwgroups.dcs.stand.ac.uk/~gap.
Once you have this file you can unpack it with the commands
gunzip specht2.4.tar.gz
tar xfv specht2.4.tar
When you do this you should find the following files in a directory called
``specht2.4'':
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.
Specht is now installed and ready to use:
gap> RequirePackage("specht");
gap> H:=Specht(3);
Specht(e=3, S(), P(), D(), Pq())
Installing Specht's documentation
The documentation for Specht can be found in the subdirectory 'doc'. The
more inportant files in this directory are:
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
To install the online documentation for Specht proceed as
follows:
 Copy
specht.tex into the GAP doc/
directory.
 Change directory to GAPs doc directory and add the line
\Include{specht} to manual.tex .
 LaTeX
manual.tex .
 Run makeindex (if available) and reLaTeX
manual.tex .
A (printed) copy of the Specht manual can be obtained by LaTeXing,
and printing, the file manual.tex in Specht's
doc directory.
