Odd Shifted Parking Functions

TL;DR

This paper introduces odd shifted parking functions to combinatorially realize the $V$-expansion of Stanley's shifted parking function symmetric functions, connecting them to spin characters and representation theory.

Contribution

It provides the first combinatorial and representation-theoretic realizations of the $V$-expansion of $SH_n$, resolving an open problem and linking to spin characters.

Findings

Introduces odd shifted parking functions for combinatorial realization.

Provides two representation-theoretic models interpreting $SH_n$ as spin characters.

Establishes connections between $SH_n$ and Haglund's $(q,t)$-Schr"oder theorem.

Abstract

Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand $SH_n$ into the $V-$basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the $V-$expansion of $SH_n$, resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret $SH_n$ as the spin character of a projective representation. We conclude with further directions, including a relationship between $SH_n$ and Haglund's $(q,t)-$Schr\"oder theorem.