## Homepage of the developement version of the GAP part of CHEVIE

CHEVIE is a joint project of

CHEVIE is a computer algebra package for symbolic calculations with generic character tables of groups of Lie type, and related mathematical structures: Weyl groups and Iwahori-Hecke algebras. It has been extended to deal with some infinite Coxeter groups such as the affine Weyl groups, and also to deal with groups generated by pseudo-reflections in a complex vector space and the associated `cyclotomic Hecke algebras' and braid groups. It is written partly in MAPLE, and partly in GAP3. This page concerns the latest developements of the GAP part of CHEVIE. You should look at the CHEVIE homepage for other information on CHEVIE.

## The GAP part of CHEVIE

The purpose of this page is to always give access to the latest developement version of the GAP part of CHEVIE; if you find some bugs, bad designs, etc... don't complain about them, or rather if you do, do it by sending Email to me.

Here is brief summary of what's new compared to the version CHEVIE 3.1 in the official version GAP 3.4.4 of 1997.

• Affine Weyl groups, general Coxeter groups, and the corresponding Hecke algebras (with the Kazhdan-Lusztig bases $C$ and $C\text{'}$$D$ and $D\text{'}$ are not yet implemented for infinite groups).
• Now any group isomorphic to a Coxeter group can be made into a Coxeter group (e.g. the symmetric group in its natural permutation representation).
• More support for generic programming with Coxeter groups: most routines are now written in terms of primitives FirstLeftDescending, LeftDescentSet, etc.. and work generically for arbitrary Coxeter groups whatever the representation (e.g. Affine Weyl groups elements are represented as matrices, instead of the permutations used for finite Coxeter groups).
• Some support for unequal-parameter Kazhdan-Lusztig polynomials and bases.
• Hecke modules on Hecke algebras of general Coxeter groups, including the Kazhdan-Lusztig bases defined by Deodhar and Soergel for these modules.
• The possibility of defining Coxeter Cosets corresponding to the `very twisted' Ree and Suzuki groups of Lie type. Together with affine Weyl groups, this corresponds to the possibility of starting with more general Cartan matrices than before.
• Reflection cosets for arbitrary complex reflection groups have also been implemented, together with their automatic classification.
• More Hecke algebras for complex reflexion groups (there are complete character tables excepted for groups in the range G31 to G34 where there are only partial tables). Quite a few methods work now for arbitrary finite complex reflection groups, such as type recognition (decomposition into a product of irreducible groups), so routines for e.g. character tables have become fast and accurate using such decompositions.
• The actual representing matrices for representations of Hecke algebras. Thank to data from various people (Alvis, Naruse, Howlett, Yin) all representations of Hecke algebras for finite coxeter groups are in CHEVIE, and thanks to work with Gunter Malle there are representing matrices for cyclotomic algebras of all complex reflection groups, excepted that there are only partial lists for the exceptional complex groups in the range G29—G34.
• Braid monoids for general Coxeter groups, dual braid monoids for finite Coxeter groups and well-generated complex reflection groups, and general Garside and locally Garside monoids. The algorithms to determine conjugacy and compute centralizers in Garside groups of Franco, Gonzalez-Meneses and Gebhardt have been implemented.
• Semisimple elements of reductive groups, including the computation of centralizers, and determining the list of isolated and quasi-isolated classes.
• Unipotent characters for reductive groups and Spetses. Lusztig induction of unipotent characters and Lusztig's fourier trans has been implemented.
• Unipotent classes of reductive groups (including the bad characteristic case) and the generalized Springer correspondence.