(q\)-deformation of chromatic polynomials and graphical arrangements

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.