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

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 $(M^{n-1})$.