Graph coloring-related properties of (generating functions of) Hodge-Deligne polynomials

TL;DR

This paper explores the deep connections between the topology of configuration spaces, Hodge-Deligne polynomials, and graph colorings, revealing new combinatorial interpretations and relationships with character varieties and symmetries.

Contribution

It introduces a novel link between generating functions of Hodge-Deligne polynomials and graph colorings, combining plethystic exponentials with chromatic symmetric polynomials.

Findings

Connection between Hodge-Deligne polynomials and acyclic graph colorings

Representation of Hodge-Deligne polynomials via plethystic exponentials

Symmetries of Hodge-Deligne polynomials related to graph colorings

Abstract

Motivated by a connection between the topology of (generalized) configuration spaces and chromatic polynomials, we show that generating functions of Hodge-Deligne polynomials of quasiprojective varieties and colorings of acyclic directed graphs with the complete graph $K_n$ as the underlying undirected graph. In order to do this, we combine an interpretation of Crew-Spirkl for plethysms involving chromatic symmetric polynomials using colorings of directed acyclic graphs to plethystic exponentials often appearing Hodge-Deligne polynomials of varieties. Applying this to a recent result of Florentino-Nozad-Zamora, we find a connection between $GL_n$-character varieties of finitely presented groups and colorings of these "complete" directed graphs by (signed) variables used in the Hodge-Deligne polynomials of the irreducible representations. As a consequence of the constructions used, generators of Hodge-Deligne polynomials carry over to generators of the plethystic exponentials as a power series in a way that is compatible with this graph coloring interpretation and gives interesting combinatorial information that simplifies when we consider Hodge-Deligne polynomials of birationally equivalent varieties to be equivalent. Finally, we give an application of methods used to symmetries of Hodge-Deligne polynomials of varieties and their configuration spaces and their relation to chromatic symmetric polynomials.