JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 2, No 2 (1993)

Soichi Okada

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} ) \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 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} ) \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^\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.

1. D.E. Littlewood, The Theory of Group Characters, Oxford University Press, London, 2nd edition, 1950.
2. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
3. W.H. Mills, D.P. Robbins, and H. Rumsey, Jr, "Alternating sign matrices and descending plane partitions," J. Combin. Theory Ser. A 34 (1983) 340-359.
4. W.H. Mills, D.P. Robbins, and H. Rumsey, Jr, "Self-complementary totally symmetric plane partitions," J. Combin. Theory Ser. A 42 (1986) 277-292.
5. S. Okada, "Partially strict shifted plane partitions," J. Combin. Theory Ser. A S3 (1990) 143-156.
6. D.P. Robbins and H. Rumsey, Jr, "Determinants and alternating sign matrices," Adv. Math.62 (1986) 169-184.
7. T. Tokuyama, "A generating function of strict Gelfand patterns and some formulas on characters of general linear group," J. Math. Soc. Japan 40 (1988) 671-685.