Torus orbits on homogeneous varieties and Kac polynomials of quivers

TL;DR

This paper establishes a connection between torus orbit counting polynomials in flag varieties, Kac polynomials of supernova quivers, and Tutte polynomials, revealing their combinatorial and geometric properties.

Contribution

It proves the equivalence of certain orbit counting polynomials with Kac polynomials and expresses them as Tutte polynomial specializations, linking geometry, algebra, and combinatorics.

Findings

Counting polynomials match Kac polynomials of supernova quivers.

Polynomials can be expressed as Tutte polynomial specializations.

Coefficients of these polynomials are non-negative, confirmed via combinatorial proof.

Abstract

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.