On a Bruhat-like poset

TL;DR

This paper studies a poset structure arising from a stratification of a compactified space of Hermitian matrices, revealing its modular ortholattice nature, computing its Möbius function, and describing the topology of its intervals.

Contribution

It introduces a new poset structure related to Hermitian matrices, proves it is a modular ortholattice, and analyzes its combinatorial and topological properties.

Findings

The poset is a modular ortholattice.

The Möbius function of the poset is computed.

The topology of order intervals is characterized.

Abstract

We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its M\"{o}bius function and we describe the topology of its order intervals.