This is a (slightly modified) copy of Fokko du Cloux's web page for the
coxeter software. The version of the software is the same as that
found on Fokko's home page
math.univ-lyon1.fr/homes-www/ducloux.
It also includes Fokko's positivity
and an enhanced 64 bit version of the
software (updated 10/19/2010).

`Coxeter`

*Coxeter* is a computer program for the study of combinatorial
aspects of Coxeter group theory, particularly those related to the Bruhat
ordering and Kazhdan-Lusztig polynomials. It is written in C++, and should
run on the main types of workstation under linux or unix. The current
version is now version 3.0, which is now available for
beta testing.

The main capabilities of the program are the following :

- Reduced expression and normal form computations;
- Bruhat ordering;
- Ordinary Kazhdan-Lusztig polynomials;
- Kazhdan-Lusztig polynomials with unequal parameters;
- Inverse Kazhdan-Lusztig polynomials;
- Cells and W-graphs;

The program doesn't have very much in common anymore with version 1, which has been in existence for many years. The main differences, from the user's viewpoint, are the following:

- The program now works for essentially
*any*Coxeter group (with some mild restrictions on the rank and the coefficients of the Coxeter matrix); in contrast, version 1 only worked for finite groups. Of course this doesn't mean that all computations are possible for all groups! - Kazhdan-Lusztig polynomials are computed "on demand", instead of all at once, as version 1 did;
- The modularity of the program is very much improved, and it should be possible for a user familiar with C++ to rather easily add his own extensions; unfortunately, the quality level of the code is still a far cry from what it should be to qualify as real library. A project for the future!
- As shown by the above list, the capabilites of the program are much extended;
- A number of output formats are available, and in particular there is a GAP output format, which produces files that are directly readable by GAP/CHEVIE. The TeX output mode has been dropped, however (for now, at least.)

**Positivity**: Also included is software for checking
non-negativity of the structure constants of the Hecke algebra in
the Kazhdan-Lusztig basis.
The main thrust is to do this verification in
type H4.
See the README file included in
the distribution, and the paper
Some positivity properties for the Hecke algebra
in type H4.
(J. Algebra 303 (2006), no. 2, 731--741).

Download the positivity software (a gzipped tar file).

**Coxeter 64** is a version of the
coxeter software designed to take advantage of 64 bit architecture. The
coefficients of Kazhdan-Luszztig polynomials are 32 bit unisgned
untegers, and the memory management has been improved for 64 bits.

Download coxeter 64 (tar file)

The author would like to stress that *Coxeter* was written primarily
with his own research interests in mind, and that it is made available
without any promise of technical or other assistance, nor of further
development. It is distributed under the
*copyleft*
license of the Free Software
Foundation; therefore it is not in the public domain, but may be
freely copied, distributed, modified, and incorporated into other software,
provided only that the modified versions be clearly identified as such, and
that programs incorporating all or part of this code be in turn distributed
under the copyleft license.