Pieri Rule for GQs Computed via Strict Decomposition Tableaux

TL;DR

This paper introduces a new family of shifted tableaux to compute the Pieri rule for GQ functions, advancing understanding of their combinatorial structure and confirming a conjecture related to trapezoid shapes.

Contribution

It identifies an alternative family of shifted tableaux that enumerates the Pieri rule for GQ functions, partially resolving a previous conjecture.

Findings

New shifted tableaux family enumerates Pieri rule for GQ functions

Provides combinatorial interpretation for GQ expansion involving trapezoid shapes

Advances understanding of K-theoretic Schubert calculus in Lagrangian Grassmannian

Abstract

$GQ$ functions are symmetric functions indexed by strict partitions that represent $K$-theoretic Schubert classes in the Lagrangian Grassmannian. Buch and Ravikumar proved a Pieri rule for expanding $GQ_{\lambda}\cdot GQ_p$ in terms of $GQ$s via certain shifted skew tableaux. In this paper we identify an alternative family of shifted tableaux that enumerates this Pieri rule. This partially resolves a conjecture from previous work that these tableaux enumerate the expansion of $GQ_{\lambda} \cdot GQ_{\tau}$ in terms of $GQ$s where $\tau$ is a trapezoid shape.