Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas
Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas
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交替符号矩阵和外尔分母公式的一些变形
DOI：
10.1023/a:1022463708817
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发表时间：
1993
影响因子：
0.8
通讯作者：
S. Okada
中科院分区：
文献类型：
--
作者：
S. Okada
交替符号矩阵是满足一定条件的方阵，其元素为1、0或- 1。置换矩阵是交替符号矩阵。本文利用（广义的）Littlewood公式，将乘积$$\prod\limits_{i = 1}^n {(1 - tx_i )\prod\limits_{1 \leqslant i < j \leqslant n} {(1 - t^2 x_i x_{} )(1 - t^2 x_i x_j^{ - 1} )} } $$和$$\prod\limits_{i = 1}^n {(1 = tx_{} )} (1 + t^2 x_i )\prod\limits_{1 \leqslant i < j \leqslant n} {(1 - t^2 x_i x_j )(1 - t^2 x_i x_j^{ - 1} )}$$ 2展开为180°旋转下不变的交替符号矩阵集所索引的和。如果代入= 1，这些展开式可简化为bnandcn型根系的Weyl分母式。也给出了类型dnis的分母公式的类似变形。
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 $$\prod\limits_{i = 1}^n {(1 - tx_i )\prod\limits_{1 \leqslant i < j \leqslant n} {(1 - t^2 x_i x_{} )(1 - t^2 x_i x_j^{ - 1} )} } $$ and $$\prod\limits_{i = 1}^n {(1 = tx_{} )} (1 + t^2 x_i )\prod\limits_{1 \leqslant i < j \leqslant n} {(1 - t^2 x_i x_j )(1 - t^2 x_i x_j^{ - 1} )}$$ 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we putt= 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of typeBnandCn. A similar deformation of the denominator formula for typeDnis also given.