Symplectic Q-functions

TL;DR

This paper introduces symplectic Q-functions as a new class related to Hall--Littlewood functions, establishing their properties, conjecturing positivity, and providing combinatorial tableau descriptions.

Contribution

It defines symplectic Q-functions, proves their key properties, and explores positivity conjectures, extending the theory of symmetric functions to the symplectic setting.

Findings

Symplectic Q-functions share properties with Schur Q-functions.

A tableau description and Pieri-type rule are established.

Positivity conjectures for structure constants are proposed.

Abstract

Symplectic $Q$-functions are a symplectic analogue of Schur $Q$-functions and defined as the $t=-1$ specialization of Hall--Littlewood functions associated with the root system of type $C$. In this paper we prove that symplectic $Q$-functions share many of the properties of Schur $Q$-functions, such as a tableau description and a Pieri-type rule. And we present some positivity conjectures, including the positivity conjecture of structure constants for symplectic $P$-functions. We conclude by giving a tableau description of factorial symplectic $Q$-functions.