Computing with Coxeter groups and Hecke algebras (CHEVIE/PyCox)

• Applicant: Professor Dr. Meinolf Geck
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239429778

We will be concerned with the theory of Coxeter groups (or, more generally, finite groups generated by reflections in a real or complex space) and related algebraic structures, especially Hecke algebras. These form the combinatorial skeleton of various more complex structures arising in Lie theory. The consideration of algorithmic aspects has a long tradition in this area. Our objectives are:1. to contribute to the development of efficient algorithms for “high performance computations” with Coxeter groups and Hecke algebras;2. to integrate these algorithms (and related data) in freely available and user-friendly computer algebra systems;3. to perform specific experiments which help us clarify major open problems in the general theory. By “high performance computations” we mean computations which go far beyond what is currently available or possible. A specific goal is to develop general programs which are, finally, capable of computing the left cells and the corresponding W-graph representations of the biggest of the finite Coxeter groups of exceptional type (E8). This would settle a line of research in which the first standards were set by Alvis in the 1980s, followed by DuCloux in the 1990s. These, and related programs to be developed, will allow us to perform systematic experiments with a new version of Lusztig’s asymptotic algebra J (introduced by the applicant in 2009). The goal of these experiments will be to find patterns which help us clarify open questions about this algebra (e.g., the conjectured integrality of structure constants, or a description by generators and relations). It is hoped that this will lead to progress on Lusztig’s conjectures on Hecke algebras with unequal parameters, in particular, the combinatorial version of these conjectures in type Bn due to Bonnafe, Iancu, Lam and the applicant.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory
• International Connection: France, United Kingdom
• Participating Persons: Dr. Cédric Bonnafé; Dr. Max Neunhöffer