An affine generalization of evacuation

TL;DR

This paper introduces an involution on tabloids similar to Schutzenberger's evacuation map, linking fixed points to Green's polynomial evaluations and revealing a domino-like recurrence relation.

Contribution

It establishes a new involution on tabloids, connecting combinatorial fixed points to algebraic polynomial evaluations and recurrence relations.

Findings

Number of fixed points equals Green's polynomial at q = -1

Fixed points satisfy a domino-like recurrence

Involution analogous to Schutzenberger's evacuation on tabloids

Abstract

We establish the existence of an involution on tabloids that is analogous to Schutzenberger's evacuation map on standard Young tableaux. We find that the number of its fixed points is given by evaluating a certain Green's polynomial at $q = -1$, and satisfies a "domino-like" recurrence relation.