We study families of symmetric functions F̂_z indexed by affine involutions z and F̂_z^FPF indexed by affine fixed-point-free involutions z. These power series are analogues of Lam's affine Stanley symmetric functions and generalizations of the (fixed-point-free) involution Stanley symmetric functions introduced by Hamaker, Marberg, and Pawlowski. Our main result is to prove transition formulas for these two families of symmetric functions which can be used to define an affine (fixed-point-free) involution analogues of the Lascoux-Schützenberger tree. Our proof of these two formulas is based on Lam and Shimozono's transition formula for affine Stanley symmetric functions and the property that affine fixed-point-free involutions form a quasiparabolic set as introduced by Rains and...[ Read more ]

We study families of symmetric functions F̂_z indexed by affine involutions z and F̂_z^FPF indexed by affine fixed-point-free involutions z. These power series are analogues of Lam's affine Stanley symmetric functions and generalizations of the (fixed-point-free) involution Stanley symmetric functions introduced by Hamaker, Marberg, and Pawlowski. Our main result is to prove transition formulas for these two families of symmetric functions which can be used to define an affine (fixed-point-free) involution analogues of the Lascoux-Schützenberger tree. Our proof of these two formulas is based on Lam and Shimozono's transition formula for affine Stanley symmetric functions and the property that affine fixed-point-free involutions form a quasiparabolic set as introduced by Rains and Vazirani.