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

TL;DR

This paper explores measures on Coxeter groups derived from Lie algebra orbits and card shuffling, providing formulas, eigenvalues, and conjectures about their properties, with proofs for specific cases and types.

Contribution

It introduces a new measure on Coxeter groups from semisimple orbits, relates it to existing card shuffling measures, and proves conjectures for certain types and classes.

Findings

Formulas for measures in special cases

Eigenvalues of associated Markov chains computed

Conjecture on measure equality and non-negativity proved for specific types

Abstract

Solomon's descent algebra is used to define a family of signed measures M(W,x) for a finite Coxeter group W and non-zero x. The measures corresponding to W of types A and B are known to arise from the theory of card shuffling and to be related to the Poincare-Birkhoff-Witt theorem and splittings of Hochschild homology. Formulas for these measures are obtained in special cases. The eigenvalues of the associated Markov chains are computed. By elementary algebraic group theory, choosing a random semisimple orbit on a Lie algebra corresponding to a finite group of Lie type G^F induces a measure on the conjugacy classes of the Weyl group W of G^F. It is conjectured that this measure on conjugacy classes is equal to the measure arising from M(W,q) (and further that M(W,q) is non-negative on all elements of W). This conjecture is proved for all types for the identity conjugacy class of W, and is confirmed for all conjugacy classes for W of types A, B, and C.