Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D

TL;DR

This paper introduces vexillary signed permutations in types B, C, and D, linking degeneracy loci to Pfaffian formulas and connecting with existing Grassmannian and Schubert polynomial theories.

Contribution

It defines vexillary signed permutations for types B, C, and D and establishes Pfaffian formulas for their degeneracy loci classes, unifying several existing theories.

Findings

Pfaffian formulas explicitly describe degeneracy loci classes.

Vexillary signed permutations generalize classical notions to types B, C, and D.

Connections made between these formulas and double Schubert polynomials.

Abstract

We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these loci are given by explicit Pfaffian formulas. The Grassmannian formulas of Kazarian are important special cases, and the corresponding double Schubert polynomials of Ikeda, Mihalcea, and Naruse are shown to be equal to these Pfaffians.