Symmetric Function Theory at the Border of A_n and C_n

Graham Hawkes

Published 2018-09-10Version 1

We relate the type $A_n$ and type $C_n$ Stanley symmetric functions, by producing a new symmetric function: Our double Stanley symmetric function gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$ and is symmetric and Schur positive in general at $(\mathbf{x}, \mathbf{y})$ for $\omega \in A_n \subseteq C_n$. In order to produce a Schur expansion for our functions, we make two attempts to generalize Edelman-Greene to signed words. We successfully do this using a \emph{signed-recording Edelman-Greene} map. However, the dual notion of the \emph{singed-insertion Edelman-Greene} map appears a difficult task, although its possible existence leaves us with the interesting conjecture of the \emph{even} and \emph{odd} Stanley symmetric functions.