Forgotten characters

Abstract

A partial permutation of $[n]:=\{1,\dots ,n\}$ is a bijection $g:I\to J$ between two subsets $I,J\subseteq [n]$. Given a partial permutation $g$ of $[n]$, let $a_g\in \mathbb{C}[{\mathfrak{S}}_n]$ be the group algebra sum of those permutations $w\in {\mathfrak{S}}_n$ which extend $g$. Informally, a partial permutation $g$ is obtained by forgetting some data in a genuine permutation. The forgotten symmetric functions are the least-studied of the six `standard' bases for the ring of symmetric functions. We show that forgotten symmetric functions arise naturally in class function evaluations on partial permutations.

13 pages