Matchings in Corona graph and classical symmetric varieties

TL;DR

This paper introduces a combinatorial approach to parametrizing Borel orbits in classical symmetric varieties using matchings in the Corona graph, leading to new results on orbit enumeration and conjecture resolution.

Contribution

It presents a novel combinatorial parametrization method for Borel orbits and proves properties like ultra log-concavity, unimodality, and non-integrality of certain polynomial coefficients.

Findings

Established ultra log-concavity and unimodality for orbit counts in Types AIII and CII.

Proved a conjecture on the non-integrality of polynomial coefficients in Type BI.

Provided a new combinatorial framework for studying Borel orbits in symmetric varieties.

Abstract

We introduce an alternative combinatorial parametrization of Borel orbits in classical symmetric varieties using matchings of the Corona graph. As an application, we obtain ultra log-concavity and unimodality for the number of Borel orbits in Types AIII and CII. Moreover, we prove a conjecture of Can and Ugurlu concerning the non-integrality of the coefficients of the polynomial that interpolates the number of orbits in Type BI.