Lexicographic Shellability of Partial Involutions

TL;DR

This paper investigates the structure of posets related to Borel orbit closures on matrices, demonstrating lexicographic shellability of the Bruhat poset of partial involutions and introducing new EL-labelings for permutations and involutions.

Contribution

It establishes the lexicographic shellability of the Bruhat poset of partial involutions and develops new EL-labelings for permutations and involutions, advancing combinatorial understanding.

Findings

Bruhat poset of partial involutions is lexicographically shellable

New EL-labelings for permutations and involutions are introduced

Enhanced understanding of Borel orbit closure posets on matrices

Abstract

In this manuscript we study inclusion posets of Borel orbit closures on (symmetric) matrices. In particular, we show that the Bruhat poset of partial involutions is a lexicographiically shellable poset. Also, studying the embeddings of symmetric groups and involutions into rooks and partial involutions, respectively, we find new $EL$-labelings on permutations as well as on involutions.