On Sjöstrand’s Skew Sign-imbalance Identity

The aim of this note is to give a quick derivation of Theorem 1 using the techniques developed in [1] and the skew domino Cauchy identity. Let Gλ/μ(X; q) = ∑ D q spin(D)xweight(D) be the spin-weight generating function of domino tableaux with shape λ/μ; see for example [1]. Here we will use the convention that spin(D) is equal to half the number of vertical dominoes in D. Though not stated explicitly, the following identity is a straightforward generalization of the “domino Cauchy identity” proved in any of [1, 2, 4].