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Dear GAP forum,

this is a 'short' note to announce a new GAP package for calculating

decomposition numbers of the Iwahori-Hecke algebras of symmetric groups.

I have just uploaded the package to samson.math.rwth-aachen.de where

it appears in the /pub/incoming directory under

specht.uu.gz

This file is a shar file; to extract the package proceed as follows:

1. gunzip specht.uu.gz

2. sh specht.uu

This will install SPECHT in your current directory (there are a large

number of library files, containing decomposition matrices, coming to

slightly over 1Mb). (non-UNIX users will have to gunzip the file, uudecode

it and use tar by hand, and then edit specht.g to set SPECHTHOME

accordingly).

The package can be used for

1. Calculating decomposition matrices of Hecke algebras of

the symmetric groups (mainly useful in characteristic 0;

but applicable more generally).

2. Calculating decomposition matrices of symmetric groups. The

decomposition matrices of S_n for the primes 2, 3, and 5

and n < 16 are included (and for "small" (n,p) are readily

calculated using the package).

3. Calculating decomposition matrices of q-Schur algebras,

although this is not fully supported as yet (no routines for

dealing with singular partitions yet exist)

Effectively, the package allows computations to be made in the Grothendiek

ring of these algebras. In addition, many of the combinatorial tools from

the representation theory of symmetric groups are implements in the

package; for example, routines for calculating quotients

and cores of partitions, the (inverse) Littlewood-Richard rule, listing

(semi-)standard tableaux (not documented at present; the commands are

SemistandardStableaux(nu,mu) and StandardTableaux(mu) ), inducing and

restricting specht modules, PIMs, and simple modules, "inducing"

decomposition matrices, the q-Schaper theorem, Mullineux's map, dimension

formulae for these modules,... are to be found in the package.

Documentation for the package is to be found in the file specht.ps; this can

be made available through GAP's online help by letting the GAP manual.tex

know about the included file specht.tex (you will need to add a

\Include{specht} to manual.tex and re-LaTeX the manual).

At this stage the package should be considered a beta release. Currently

I am rewriting part of the package so that it makes use of a recent

conjecture (now proved by Grojnowski) for calculating decomposition matrices

of Hecke algebras over fields of characteristic 0; the basic algorithm is

written, but I have yet to fully integrate this into the package; I hope to

do this by the end of January 1996.

I would very much like to hear any comments and criticisms on the package,

and suggestions for improvements or extensions. Ultimately I would like to

add full support for calculations for Hecke algebras over finite fields,

q-Schur algebras, and general linear groups (this will not happen for quite

some time however...). Any comments on the interface or regarding any

obviously 'missing' functions are particularly welcome.

As mentioned above, I hope to have the package finished, complete with the

LLT conjecture early next year. Until then, I would much appreciate

comments and criticisms.

Regards,

Andrew Mathas

Imperial College

a.mathas@ic.ac.uk

-- We are all of us in the gutter, but some of us are looking at the stars. Oscar Wilde

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