Cyclic sieving for longest reduced words in the hyperoctahedral group

TL;DR

This paper demonstrates that the set of reduced expressions for the longest element in the hyperoctahedral group exhibits cyclic sieving, with a natural cyclic action linked to the major index generating function.

Contribution

It introduces a cyclic action on reduced words in the hyperoctahedral group and connects it to the major index generating function, revealing a cyclic sieving phenomenon.

Findings

The set R(w_0) has a cyclic action by moving the first letter to the end.

The orbit structure is encoded by the major index generating function.

The cyclic sieving phenomenon is established for R(w_0).

Abstract

We show that the set R(w_0) of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, R(w_0) possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on R(w_0).