Fixed points of reverse Hessenberg convolution varieties

TL;DR

This paper reanalyzes fixed points of certain varieties associated with unit interval graphs to provide a new proof of Kato's result linking geometric structures to chromatic quasisymmetric polynomials.

Contribution

It offers a novel proof of Kato's theorem by examining fixed points, connecting geometric and combinatorial aspects of unit interval graphs.

Findings

Reproves Kato's result using fixed point analysis

Establishes a geometric interpretation of chromatic quasisymmetric polynomials

Links varieties to graph coloring invariants

Abstract

Associated to any unit interval graph, Syu Kato introduced a variety which gives (via the geometric Satake correspondence) a graded $GL_m$ representation whose character is the chromatic quasisymmetric polynomial of the graph. In this short note, we reprove Kato's result by analyzing the fixed points of his varieties.