Flip actions and Gelfand pairs for affine Weyl groups

TL;DR

This paper explores combinatorial actions of affine Weyl groups, demonstrating their multiplicity-free nature through Gelfand pairs, and compares various representations on combinatorial objects.

Contribution

It introduces a new construction of Gelfand subgroups in affine Weyl groups and addresses a question about permutation representations being multiplicity-free.

Findings

Permutation representations are multiplicity-free modulo an involution.

Constructs Gelfand subgroups in affine Weyl groups of types C and B.

Provides a comparison of combinatorial actions on various objects.

Abstract

Several combinatorial actions of the affine Weyl group of type $\widetilde{C}_{n}$ on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types $\widetilde{C}_{n}$ and $\widetilde{B}_n$.