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 Fq. 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 qⁿ (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 Fq. 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-deformations" 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.