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

Okada, Soichi

Journal of Algebraic Combinatorics, Tome 2 (1993) no. 2, pp. 155-176.

Résumé

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. In this paper, we use the (generalized) Littlewood's formulas to expand the products Õ $_{ i = 1} ^{ n} (1 - tx _{ i} )$ Õ $_{1 \leqslant i j \leqslant n} (1 - t ^{2} x _{ i} x _{} )(1 - t ^{2} x _{ i} x _{ j} ^{ - 1} )$ prodlimits_i = 1^n (1 - tx_i )prodlimits_$1 \leqslant $i j $\leqslant n$ (1 - t^2 x_i x_ )(1 - t^2 x_i x_j^ - 1 ) and Õ $_{ i = 1} ^{ n} (1 = tx _{} ) (1 + t ^{2} x _{ i} )$ Õ $_{1 \leqslant i j \leqslant n} (1 - t ^{2} x _{ i} x _{ j} )(1 - t ^{2} x _{ i} x _{ j} ^{ - 1} )$ prodlimits_i = 1^n (1 = tx_ ) (1 + t^2 x_i )prodlimits_$1 \leqslant $i j $\leqslant n$ (1 - t^2 x_i x_j )(1 - t^2 x_i x_j^ - 1 ) 2 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.

Détail

Keywords: alternating sign matrix, monotone triangle, Weyl's denominator formula, Littlewood's formula