Multi-Symmetric Schur Functions

TL;DR

This paper introduces multi-symmetric Schur functions, generalizing classical Schur functions, and explores their combinatorial properties, expansions, and operators, revealing new structural insights and positive expansion formulas.

Contribution

It defines multi-symmetric Schur functions as stable limits of key polynomials and establishes their combinatorial, algebraic, and representation-theoretic properties, including a positive expansion into tensor products of Schur functions.

Findings

Established a diagrammatic combinatorial formula for monomial expansions.

Proved a triangularity result characterizing monomial supports.

Derived positive expansion coefficients as multiplicities of irreducible representations.

Abstract

We study a multi-symmetric generalization of the classical Schur functions called the multi-symmetric Schur functions. These functions form an integral basis for the ring of multi-symmetric functions indexed by tuples of partitions and are defined as certain stable-limits of key polynomials. We prove combinatorial results about the monomial expansions of the multi-symmetric Schur functions including a diagrammatic combinatorial formula and a triangularity result which completely characterizes their monomial multi-symmetric supports. The triangularity result involves a non-trivial generalization of the dominance order on partitions to tuples of partitions. We prove, using the Demazure character formula, that the multi-symmetric Schur functions expand positively into the basis of tensor products of ordinary Schur functions and describe the expansion coefficients as multiplicities of certain irreducible representations for Levi subgroups inside particular Demazure modules. Lastly, we find a family of multi-symmetric plethystic operators related to the classical Bernstein operators which act on the multi-symmetric Schur basis by a simple recurrence relation.