Schubert polynomials and degeneracy locus formulas

TL;DR

This paper provides a new combinatorial proof for formulas representing degeneracy loci in classical groups, extending previous work on Schubert polynomials and revealing new formulas and connections.

Contribution

It introduces a purely combinatorial approach to prove general degeneracy locus formulas for isotropic partial flag varieties, building on and simplifying prior proofs.

Findings

New combinatorial proof of degeneracy locus formulas

Discovery of several new formulas in the context of isotropic partial flag varieties

Clarification of connections between earlier degeneracy locus formulas

Abstract

In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones.