$q$-deformation of chromatic polynomials and graphical arrangements
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916675771695104 |
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| author | Nian, Tongyu Tsujie, Shuhei Uchiumi, Ryo Yoshinaga, Masahiko |
| author_facet | Nian, Tongyu Tsujie, Shuhei Uchiumi, Ryo Yoshinaga, Masahiko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $q$-deformation of chromatic polynomials and graphical arrangements Nian, Tongyu Tsujie, Shuhei Uchiumi, Ryo Yoshinaga, Masahiko Combinatorics Rings and Algebras 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. |
| title | $q$-deformation of chromatic polynomials and graphical arrangements |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2412.08290 |