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arXiv · 2412.08290

$q$-deformation of chromatic polynomials and graphical arrangements

Tongyu Nian·Shuhei Tsujie·Ryo Uchiumi·Masahiko Yoshinaga
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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.

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Tongyu Nian, Shuhei Tsujie, Ryo Uchiumi, Masahiko Yoshinaga. 2024-12-11. $q$-deformation of chromatic polynomials and graphical arrangements. https://arxiv.org/abs/2412.08290

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