Alternating sign matrices and some deformations of Weyl's denominator formulas (Q686976)

From the author's summary: 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. The author uses the (generalized) Littlewood's formulas to expand the products \[ \prod^ n_{i=1}(1-tx_ i)\prod_{1\leq i<j\leq n}(1-t^ 2x_ ix_ j)(1-t^ 2x_ ix_ j^{-1});\;\prod^ n_{i=1}(1-tx_ i)(1+t^ 2x_ i)\prod_{1\leq i<j\leq n}(1-t^ 2x_ ix_ j)(1-t^ 2x_ ix_ j^{- 1}) \] as sums indexed by sets of alternating sign matrices invariant under a \(180^ \circ\) rotation. If we put \(t=1\), these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type \(B_ n\) and \(C_ n\). A similar deformation of the denominator formula for type \(D_ n\) is also given.