Abstract
We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Schützenberger. We give several equivalent characterizations of vexillary signed permutations, including descriptions in terms of essential sets and pattern avoidance, and we relate them to the vexillary elements introduced by Billey and Lam.
Highlights
The class of vexillary permutations in Sn, first identified by Lascoux and Schützenberger [17, 18], plays a central role in the combinatorics of the symmetric group and the corresponding geometry of Schubert varieties and degeneracy loci
The name derives from the fact that the Schubert polynomial of a vexillary permutation is equal to a flagged Schur polynomial
This was given a geometric explanation in [11]: vexillary permutations correspond to degeneracy loci defined by simple rank conditions, whose classes are computed by variations of the Kempf–Laksov determinantal formula
Summary
The class of vexillary permutations in Sn, first identified by Lascoux and Schützenberger [17, 18], plays a central role in the combinatorics of the symmetric group and the corresponding geometry of Schubert varieties and degeneracy loci. The name derives from the fact that the Schubert polynomial of a vexillary permutation is equal to a flagged Schur polynomial This was given a geometric explanation in [11]: vexillary permutations correspond to degeneracy loci defined by simple rank conditions, whose classes are computed by variations of the Kempf–Laksov determinantal formula. The vexillary signed permutations considered in [2, 3] correspond to degeneracy loci defined by rank conditions of a simple kind, and whose Schubert polynomials can be written as flagged Pfaffians. Given a triple τ = (k, p, q), the vexillary signed permutation w(τ ) is defined so that the rank conditions for the corresponding degeneracy locus are dim(Epi ∩ Fqi ) ki, for 1 i s, and so that w(τ ) is minimal (in Bruhat order) with this property.
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