Overlap Identities for Littlewood-Schur Functions

TL;DR

This paper introduces a new operation called overlap on partitions and proves two identities for Littlewood-Schur functions, generalizing classical symmetric functions and providing new combinatorial and algebraic insights.

Contribution

It presents the first overlap identities for Littlewood-Schur functions and offers visual characterizations for partition pairs with a given overlap, expanding the theoretical framework.

Findings

Proved two new overlap identities for Littlewood-Schur functions.

Derived identities using Laplace expansion of determinantal formulas.

Provided visual characterizations of partition pairs with a fixed overlap.

Abstract

Our results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. Littlewood-Schur functions are a generalization of Schur functions, whose study was introduced by Littlewood. More concretely, the Littlewood-Schur function $LS_\lambda(X; Y)$ indexed by the partition $\lambda$ is a polynomial in the variables $X \cup Y$ that is symmetric in both $X$ and $Y$ separately. The first overlap identity represents $LS _\lambda(X; Y)$ as a sum over subsets of $X$, while the second overlap identity essentially represents $LS_\lambda(X; Y)$ as a sum over pairs of partitions whose overlap equals $\lambda$. Both identities are derived by applying Laplace expansion to a determinantal formula for Littlewood-Schur functions due to Moens and Van der Jeugt. In addition, we give two visual characterizations for the set of all pairs of partitions whose overlap is equal to a partition $\lambda$.