Diagrams and essential sets for signed permutations

TL;DR

This paper introduces diagrams and essential sets for signed permutations, providing a minimal, explicit, diagrammatic method to define associated Schubert varieties and degeneracy loci, extending classical notions for ordinary permutations.

Contribution

It extends the concepts of diagrams and essential sets to signed permutations, establishing a bijection with existing poset-theoretic definitions and enabling explicit computations.

Findings

Essential sets provide minimal rank conditions for signed permutations.

Bijection with Reiner, Woo, and Yong's poset-theoretic essential sets.

Diagrammatic methods facilitate computation of Schubert varieties.

Abstract

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.