On the affine Sch\"utzenberger involution

TL;DR

This paper studies an involution on the affine Weyl group of type A, showing that fixed left cells correspond to evaluations of Green polynomials at -1, revealing a deep combinatorial symmetry.

Contribution

It establishes a precise link between fixed points of an affine Weyl group involution and Green polynomial evaluations, connecting algebraic and combinatorial structures.

Findings

Number of fixed left cells equals Green polynomial at -1

Involution preserves certain cell structures in affine Weyl groups

Provides new combinatorial interpretation of Green polynomials

Abstract

We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.