Semisimple orbits of Lie algebras and card shuffling on Coxeter groups

1999/03/02 by Jason Fulman, Fulman, Jason
Mathematics · #20F55 #20G40 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:20F55 #msc:20G40

paper · pdf · doi:10.48550/arxiv.math/9903012

Added new section and example

arxiv created 1999/08/26 · arxiv updated 2009/11/30

Abstract

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 ≠ 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.

Related

• Semisimple orbits of Lie algebras and card shuffling measures on Coxeter groups
• Cellini's descent algebra and semisimple conjugacy classes of finite groups of Lie type
• Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems
• Markov chains on Weyl groups from the geometry of the flag variety
• Random Walks on the Symmetric Group: Cutoff for One-sided Transposition\n Shuffles