PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras

@article{Geck2012PyCoxCW,
  title={PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras},
  author={Meinolf Geck},
  journal={Lms Journal of Computation and Mathematics},
  year={2012},
  volume={15},
  pages={231-256}
}
  • M. Geck
  • Published 2012
  • Mathematics
  • Lms Journal of Computation and Mathematics
We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older 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 ≤8 (except E 8 ). We also discuss the computation of Lusztig’s leading coefficients of character values and… Expand

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References

SHOWING 1-10 OF 32 REFERENCES
Computing Kazhdan–Lusztig cells for unequal parameters
Abstract Following Lusztig, we consider a Coxeter group W together with a weight function L. This gives rise to the pre-order relation ⩽ L and the corresponding partition of W into left cells. WeExpand
Leading coefficients and cellular bases of Hecke algebras
  • M. Geck
  • Mathematics
  • Proceedings of the Edinburgh Mathematical Society
  • 2009
Abstract Let H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer,Expand
Representations of Coxeter groups and Hecke algebras
here l(w) is the length of w. In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functionsExpand
Computational construction of irreducible W-graphs for types E6 and E7
Abstract The concept of W-graph was introduced in the influential paper [David Kazhdan, George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] ofExpand
ON THE INDUCTION OF KAZHDAN–LUSZTIG CELLS
Barbasch and Vogan showed that the Kazhdan–Lusztig cells of a finite Weyl group are compatible with parabolic subgroups. Their proof uses the known bridge between the theory of cells and the theoryExpand
Cells in affine Weyl groups, II
This paper is a continuation of [S]. The main theme of [S] was the study of a numerical function w --f a(w) on a Coxeter group W which in the case of Weyl groups is closely related to theExpand
CHEVIE — A system for computing and processing generic character tables
CHEVIE is a computer algebra package which collects data and programs for the representation theory of finite groups of Lie type and associated structures. We explain the theoretical and conceptualExpand
Positivity results for the Hecke algebras of non-crystallographic finite Coxeter groups
Abstract This paper is a report on a computer check of some important positivity properties of the Hecke algebra in type H 4 , including the nonnegativity of the structure constants in theExpand
Inducing W-graphs
Abstract. Let ℋ be the Hecke algebra associated with a Coxeter group W. Many interesting ℋ-modules can be described using the concept of a W-graph, as introduced in the influential paper [4] ofExpand
Kazhdan--Lusztig cells and the Frobenius--Schur indicator
Let $W$ be a finite Coxeter group. It is well-known that the number of involutions in $W$ is equal to the sum of the degrees of the irreducible characters of $W$. Following a suggestion of Lusztig,Expand
...
1
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3
4
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