2022/03/22 by Soohyun Park, Park, Soohyun
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2203.11930
openalex publication_date 2022/03/22 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
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 Kn 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 GLn-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.