A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings

TL;DR

This paper proves a factorization theorem for classical group characters, linking Schur functions to orthogonal and symplectic characters, with applications to rhombus tilings and plane partitions.

Contribution

It introduces a new factorization theorem for Schur functions of rectangular shape, connecting them to orthogonal and symplectic characters, and applies this to tiling and partition problems.

Findings

Factorization of Schur functions into orthogonal and symplectic characters based on parity of M.

Application of factorization to rhombus tilings of hexagons.

Extension of the factorization to sums of Schur functions of different shapes.

Abstract

We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A similar factorization theorem is proven for the sum of two Schur functions of respective rectangular shapes $(M^n)$ and $(M^{n-1})$.