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 $x_{1},x_{1}^{-1},\dots,x_{n},x_{n}^{-1}$ 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).