Rectangular symmetries for coefficients of symmetric functions

TL;DR

This paper reveals that key structural constants in symmetric functions exhibit symmetries linked to rectangle complement and addition operations, enhancing understanding of their algebraic structure.

Contribution

It introduces new symmetry properties of Littlewood-Richardson, Kronecker, plethysm coefficients, and Kostka--Foulkes polynomials related to rectangle operations.

Findings

Identifies symmetries in structural constants of symmetric functions

Connects symmetries to rectangle complement and addition operations

Provides new insights into algebraic structure of symmetric functions

Abstract

We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka--Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.