Virtual Permutations

Affine transitions for involution Stanley symmetric functions. We study a family of symmetric functions F indexed by involutions z in the affine symmetric group. These power series are analogues of Lam’s affine Stanley symmetric functions and generalizations of the involution Stanley symmetric functions introduced by Hamaker, Pawlowski, and the first author. Our main result is to prove a transition formula for F which can be used to define an affine involution analogue of the Lascoux-Schützenberger tree. Our proof of this formula relies on Lam and Shimozono’s transition formula for affine Stanley symmetric functions and some new technical properties of the strong Bruhat order on affine permutations.