Mathematical Research Letters

Volume 27 (2020)

Number 3

On the affine Schützenberger involution

Pages: 809 – 834



Dongkwan Kim (School of Mathematics, University of Minnesota Twin Cities, Minneapolis, Minn., U.S.A.)


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 $-1$.

Received 29 December 2018

Accepted 22 July 2019

Published 20 August 2020