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The 6 vertex model and Schubert polynomials. (English) Zbl 1138.05073

A staircase is a semi-standard Young tableau with column lengths \(k,k-1,\ldots,k-r\) in which the entries, in addition to being decreasing down columns and weekly increasing in rows, are weekly decreasing on NW-SE diagonals. Staircases arise in the Ehresmann-Bruhat order on the symmetric group [A. Lascoux and M. P. Schützenberger, Electron. J. Comb. 3, No. 2, Research paper R27, 35 p. (1996); printed version J. Comb. 3, No. 2, 633–667 (1996; Zbl 0885.05111)] The authors give a weighted enumeration of staircases with fixed left and right columns; these correspond to square-ice, or alternating sign matrices with fixed top and bottom parts. For staircases with first column 1 to \(n\), or last column empty (corresponding to alternating sign matrices fixed only at the top, or only at the bottom), the enumerator is the product of a monomial and a Schubert polynomial; in general, up to a normalization factor, it is the product of a monomial and a determinant of skew Schur functions.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E10 Combinatorial aspects of representation theory
82B23 Exactly solvable models; Bethe ansatz

Citations:

Zbl 0885.05111
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