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An alternating sign matrix is a square matrix whose entries are 1, 0, or −1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products \prod \limits _{i=1}^n (1-tx_i)\ \prod \limits_{1 \le i \lt j \le n}\ (1-t^2 x_i x_j)(1-t^2x_i x_j^{-1}) \quad \rm and and \prod \limits _{i=1}^n (1-tx_i)(1+t^2x_i)\ \prod \limits_{1 \le i \lt j \le n}\ (1-t^2 x_i x_j)(1-t^2x_i x_i^{-1}) \quad as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type Bn and Cn. A similar deformation of the denominator formula for type Dn is also given.

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas