Coxeter and Schubert combinatorics of $\mu$-Involutions

TL;DR

This paper explores the combinatorial and Coxeter-theoretic properties of $d$-involutions within the context of the variety of complete quadrics, introducing new operators, descriptions, and polynomial expansions.

Contribution

It provides a combinatorial description of $d$-involutions, develops Coxeter-theoretic tools, and extends Schubert polynomial expansions for these objects.

Findings

Described atoms of $d$-involutions.

Established an exchange lemma and transposition operators.

Expanded $d$-involution Schubert polynomials as multiplicity-free sums.

Abstract

The variety of complete quadrics is the wonderful compactification of $GL_n/O_n$ and admits a cell decomposition into Borel orbits indexed by combinatorial objects called $\mu$-involutions. We study Coxeter-theoretic properties of $\mu$-involutions with results including a combinatorial description for their atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. The corresponding orbit closures can be realized inside the flag variety. In this setting, we study the cohomology representatives of these orbits, which are, up to a scalar, the $\mu$-involution Schubert polynomials. We expand $\mu$-involution Schubert polynomials as a multiplicity-free sum of $\nu$-involution Schubert polynomials when $\nu$ refines $\mu$ and provide recurrences analogous to Monk's rule for Schubert polynomials.