Semisimple orbits of Lie algebras and card shuffling on Coxeter groups

Details Semisimple orbits of Lie algebras and card shuffling on Coxeter groups Semisimple orbits of Lie algebras and card shuffling on Coxeter groups

1999-03-02

arxiv.org/abs/math/9903012v2

Mathematics

Group Theory

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Scientific paper

Random walk on the chambers of hyperplanes arrangements is used to define a family of card shuffling measures $H_{W,x}$ for a finite Coxeter group W and real $x \neq 0$. By algebraic group theory, there is a map from the semisimple orbits of the adjoint action of a finite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the Weyl group. For types A, B, and the identity conjugacy class of W for all types, it is proved that for q very good, this measure on conjugacy classes is equal to the measure arising from $H_{W,q}$.

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