Decomposition matrices for the Hecke algebras of type A
A package for calculating decomposition numbers of
Hecke algebras of the symmetric groups and q-Schur
© Andrew Mathas October 1997.
University of Sydney
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
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
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
A complete description of what is available in the package can be found
in the manual.
First you must obtain the file specht-2.4.tar.gz
(156 kB); either from here or via ftp from
Once you have this file you can unpack it with the commands
- 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 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.
- 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.
- 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.
- The decomposition matrices for the symmetric groups $\Sym_n$ are
included for $n<15$ and for all primes.
Specht is now installed and ready to use:
tar -xfv specht-2.4.tar
When you do this you should find the following files in a directory called
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
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(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
A (printed) copy of the Specht manual can be obtained by LaTeXing,
and printing, the file
specht.tex into the GAP
- Change directory to GAPs doc directory and add the line
- Run makeindex (if available) and re-LaTeX
manual.tex in Specht's