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

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We prove that a Schur function of rectangular shape (Mⁿ) whose variables are specialized to x₁,x₁⁻¹,...,xₙ,xₙ⁻¹ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at -x₁,...,-xₙ, 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 (Mⁿ) and (Mⁿ⁻¹).