$q$-deformation of chromatic polynomials and graphical arrangements

TL;DR

This paper introduces a new concept of q-deformed graphical arrangements, extending classical invariants like chromatic polynomials and Stirling numbers to a q-analog setting, revealing deep connections with hyperplane arrangements over finite fields.

Contribution

The paper defines q-deformation of graphical arrangements and demonstrates that key invariants extend naturally as q-analogs, linking combinatorics and hyperplane arrangements over finite fields.

Findings

q-deformed arrangements generalize classical invariants

Invariants like chromatic polynomial and Stirling numbers have q-analogues

Many properties extend to the q-deformed setting

Abstract

We first observe a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field $\mathbb{F}_q$. These two arrangements are defined by the determinants of the Vandermonde and the Moore matrix, respectively. These two matrices are transformed to each other by replacing a natural number $n$ with $q^n$ ($q$-deformation). In this paper, we introduce the notion of ``$q$-deformation of graphical arrangements'' as certain subarrangements of the arrangement of all hyperplanes over $\mathbb{F}_q$. This new class of arrangements extends the relationship between the Vandermonde and Moore matrices to graphical arrangements. We show that many invariants of the ``$q$-deformation'' behave as ``$q$-deformation'' of invariants of the graphical arrangements. Such invariants include the characteristic (chromatic) polynomial, the Stirling number of the second kind, freeness, exponents, basis of logarithmic vector fields, etc.