Soichi Okada

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

AbstractAn 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) Littlewoods formulas to expand the products

Wreath products by the symmetric groups and product posets of Young's lattices

\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} )} }

An Application of Cauchy–Sylvester’s Theorem on Compound Determinants to a BC n -Type Jackson Integral

and

Applications of minor summation formulas to rectangular-shaped representations of classical groups

\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} )}

Applications of minor-summation formula I

2 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 Weyls denominator formulas for the root systems of type Bn and Cn. A similar deformation of the denominator formula for type Dn is also given.

Applications of Minor-Summation Formula I. Littlewood's Formulas

Abstract For a pair (A, B) of complementary subsets of positive integers, an (A, B)-partially strict shifted plane partition is a shifted plane partition σ = (aij)i⩽j (aij ⩾ ai,j+1and aij ⩾ ai + 1,j) such that (i) for any m ϵ A, m appears at most once in each row, and (ii) for any m ϵ B, m appears at most once in each column. In this paper we give a generaging function for them by using a “lattice path” method. As one application of this generating function, we can obtain a generating function for row-strict (resp. column-strict) shifted plane partitions, which is a determininantal formula in terms of elementary (resp. complete) symmetric polynomials. Another application is to derive a weighted generating function for monotone triangles.