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$$\mathsf{PyCox}$$: computing with (finite) Coxeter groups and Iwahori-Hecke algebras. (English) Zbl 1296.20009
Summary: We introduce the computer algebra package \mathsf{PyCox}, written entirely in the \mathsf{Python} language. It implements a set of algorithms, in a spirit similar to the older \mathsf{CHEVIE} system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan-Lusztig cells and $$W$$-graphs, which works efficiently for all finite groups of rank $$\leqslant 8$$ (except $$E_8$$). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works for $$E_8$$ as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case.

MSC:
 20C40 Computational methods (representations of groups) (MSC2010) 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects) 68W30 Symbolic computation and algebraic computation
Software:
PyCox; CHEVIE; GAP
Full Text:
References:
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