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\title[\texorpdfstring{$q$}{q}-deformation of graphic arrangements]{\texorpdfstring{$q$}{q}-deformation of graphic arrangements}

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\author[Tongyu Nian, Shuhei Tsujie, Ryo Uchiumi, \and Masahiko Yoshinaga]{\texorpdfstring{
Tongyu Nian\thanks{\href{mailto:u487971i@ecs.osaka-u.ac.jp}{u487971i@ecs.osaka-u.ac.jp}.}\addressmark{1},
Shuhei Tsujie\thanks{\href{mailto:tsujie.shuhei@a.hokkyodai.ac.jp}{tsujie.shuhei@a.hokkyodai.ac.jp}. S. T. was supported by JSPS KAKENHI, Grant Number JP22K13885.}\addressmark{2},
Ryo Uchiumi\thanks{\href{mailto:uchiumi.ryou.1xu@ecs.osaka-u.ac.jp}{ uchiumi.ryou.1xu@ecs.osaka-u.ac.jp}. R. U. was supported by JST SPRING, Grant Number JPMJSP2138.}\addressmark{1}, 
\and \\
Masahiko Yoshinaga\thanks{\href{mailto:yoshinaga@math.sci.osaka-u.ac.jp}{yoshinaga@math.sci.osaka-u.ac.jp}. M. Y. was partially supported by JSPS KAKENHI, Grant Number JP23H00081.}\addressmark{1}
}{
Tongyu Nian,
Shuhei Tsujie,
Ryo Uchiumi, 
and
Masahiko Yoshinaga
}}

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\address{\addressmark{1}Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan. \\ 
\addressmark{2}Department of Mathematics, Hokkaido University of Education, Asahikawa, Hokkaido 070-8621, Japan.}

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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 graphic 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 graphic arrangements. We show that many invariants of the ``$q$-deformation'' behave as ``$q$-deformation'' of invariants of the graphic arrangements. Such invariants include the characteristic (chromatic) polynomial, the Stirling number of the second kind, freeness, exponents, basis of logarithmic vector fields, etc. 
%
%, with the characteristic p
%
%In this paper, we first observe a mysterious similarity between the braid arrangements and the hyperplane arrangements consisting of all hyperplanes in the vector spaces over finite fields, with the characteristic polynomials being the determinants of Vandermonde matrices and Moore matrices, respectively. By using the facts of the representation of characteristic polynomials with falling factorials we try to extend this kind of similarity, called $q$-deformation, to other objects, such as graphic arrangements and Stirling numbers of the second kind. The $q$-deformation of graphic arrangement for a specific graph is represented by the set of hyperplanes whose variables are taken from the cliques in the graph and coefficients are taken over nonzero elements of the finite field GF(q). In this sense the ``all-hyperplane arrangements" are the $q$-deformations of the graphic arrangement of complete graphs. The characteristic polynomials of $q$-deformations of graphic arrangements are closely related to the chromatic polynomials as the characteristic polynomials of classical graphic arrangements do. The freeness of this novel structure behaves similar to the classical one, following supersolvability and we give the construction of the bases of free $q$-deformations of graphic arrangements.
}

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%\resume{


%}

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\keywords{Graphic arrangements, chromatic polynomial, $q$-analogue,  freeness, finite fields}

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\begin{document}

\maketitle
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\section{Introduction}
\subsection{Mysterious similarities}
A central arrangement $\mathcal{A}$ is a finite collection of linear hyperplanes in a finite dimensional vector space. 
Define the \textbf{intersection lattice} $L(\mathcal{A})$ and the \textbf{characteristic polynomial} $\chi(\mathcal{A},t)$ by 
\begin{align*}
L(\mathcal{A}) \coloneqq \Set{\bigcap_{H \in \mathcal{B}}H | \mathcal{B} \subseteq \mathcal{A}}, \qquad
\chi(\mathcal{A},t) \coloneqq \sum_{X \in L(\mathcal{A})}\mu(X)t^{\dim X}, 
\end{align*}
where $L(\mathcal{A})$ is ordered by the reverse inclusion and $\mu$ denotes the \textbf{Möbius function} on the lattice $L(\mathcal{A})$. 

A typical example of an arrangement is the \textbf{braid arrangement} $\mathcal{B}_{\ell}$ in $\mathbb{R}^{\ell}$ whose defining polynomial is the \textbf{Vandermonde determinant}, i.e., 
\begin{align*}
Q(\mathcal{B}_{\ell}) = \prod_{1 \leq i < j \leq \ell}(x_{j}-x_{i})
= \begin{vmatrix}
1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{\ell-1} \\
1 & x_{2} & x_{2}^{2} & \dots & x_{2}^{\ell-1} \\
\vdots & \vdots & \vdots & & \vdots \\
1 & x_{\ell} & x_{\ell}^{2} & \dots & x_{\ell}^{\ell-1} \\
\end{vmatrix}. 
\end{align*}
The characteristic polynomial of the braid arrangement $\mathcal{B}_{\ell}$ is 
\begin{align*}
\chi(\mathcal{B}_{\ell},t) = t(t-1)(t-2) \cdots (t-\ell+1). 
\end{align*}

There are mysterious similarities between the braid arrangements and the arrangements consisting of all hyperplanes in vector spaces over finite fields.  
Let $q$ be a prime power and $\mathbb{F}_{q}$ the finite field of order $q$. 
Define the arrangement $\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$ as the set of all hyperplanes in $\mathbb{F}_{q}^{\ell}$. 
Its defining polynomial is the determinant of \textbf{Moore matrix}, i.e., 
\begin{align*}
Q\left(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})\right) 
= \prod_{i = 1}^{\ell}\prod_{c_{1}, \dots, c_{i-1} \in \mathbb{F}_{q}}(c_{1}x_{1} + \dots + c_{i-1}x_{i-1} + x_{i})
=\begin{vmatrix}
x_{1} & x_{1}^{q} & x_{1}^{q^{2}} & \dots & x_{1}^{q^{\ell-1}} \\
x_{2} & x_{2}^{q} & x_{2}^{q^{2}} &  \dots & x_{2}^{q^{\ell-1}} \\
\vdots & \vdots & \vdots &  & \vdots \\
x_{\ell} & x_{\ell}^{q} & x_{\ell}^{q^{2}} & \dots & x_{\ell}^{q^{\ell-1}}
\end{vmatrix}. 
\end{align*}
The characteristic polynomial of $\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$ is 
\begin{align*}
\chi\left(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell}),t\right) = (t-1)(t-q)(t-q^{2}) \cdots (t-q^{\ell-1}). 
\end{align*}

By formally replacing $q^{k}$ in the expressions of $Q\left(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})\right)$ and $\chi\left(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell}),t\right)$ with $k$, we obtain the expressions for $Q(\mathcal{B}_{\ell})$ and $\chi(\mathcal{B}_{\ell},t)$. 
A similar phenomenon can be observed in the context of freeness of arrangements. 
An arrangement $\mathcal{A}$ is said to be \textbf{free} if the module of logarithmic polynomial vector fields $D(\mathcal{A})$ is a free module over the polynomial ring (See \cite{orlik1992arrangements} for details). 
Both of $Q(\mathcal{B}_{\ell})$ and $\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$ are free with bases 
\begin{align*}
&\sum_{i=1}^{\ell}x_{i}^{k}\partial_{i} \, (0 \leq k \leq \ell-1) \text{ for } D(\mathcal{B}_{\ell}) \text{ and }
\sum_{i=1}^{\ell}x_{i}^{q^{k}}\partial_{i} \, (0 \leq k \leq \ell-1) \text{ for } D\left(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})\right), 
\end{align*}
where $\partial_{i}$ denotes $\frac{\partial}{\partial x_{i}}$ for each $i \in \{1, \dots, \ell\}$ (See \cite[Example 4.22 and 4.24]{orlik1992arrangements}). 

Moreover, there are mysterious similarities for subarrangements. 
Let $G$ be a simple graph on $[\ell] = \{1, \dots, \ell\}$. 
Define the \textbf{graphic arrangement} $\mathcal{A}_{G}$ in $\mathbb{R}^{\ell}$ by 
\begin{align*}
\mathcal{A}_{G} \coloneqq \Set{ \{x_{i}-x_{j} = 0\} | \{i,j\} \in E_{G}}. 
\end{align*}
Note that every subarrangement of $\mathcal{B}_{\ell}$ is of the form $\mathcal{A}_{G}$ and it is well known that the chromatic polynomial $\chi(G,t)$ coincides with the characteristic polynomial $\chi(\mathcal{A}_{G},t)$. 
The following proposition for the chromatic polynomial is trivial by definition. 
\begin{proposition}\label{If k colors are not enough then fewer colors are not enough}
Suppose $\chi(G,k) = 0$ for some $k \in \mathbb{Z}_{\geq 0}$. 
Then $\chi(G,j) = 0$ for $0 \leq j \leq k$. 
\end{proposition}

There is a $q$-version of \Cref{If k colors are not enough then fewer colors are not enough}. 

\begin{proposition}[{\cite[Lemma 7]{yoshinaga2007free-potjasams}}]
Let $\mathcal{A}$ be an arrangement in $\mathbb{F}_{q}^{\ell}$. 
If $\chi(\mathcal{A},q^{k}) = 0$ for some $k \in \mathbb{Z}_{\geq 0}$, then $\chi(\mathcal{A},q^{j}) = 0$ for any $0 \leq j \leq k$. 
\end{proposition}
\begin{proof}
Let $\mathcal{A} \otimes \mathbb{F}_{q}^{k}$ denote the subspace arrangement in $(\mathbb{F}_{q}^{k})^{\ell}$ defined by 
\begin{align*}
\mathcal{A} \otimes \mathbb{F}_{q}^{k} \coloneqq \Set{ H \otimes_{\mathbb{F}_{q}} \mathbb{F}_{q}^{k} | H \in \mathcal{A}}. 
\end{align*}
Then the intersection lattices $L(\mathcal{A})$ and $L(\mathcal{A}\otimes\mathbb{F}_{q}^{k}) = \Set{X \otimes_{\mathbb{F}_{q}} \mathbb{F}_{q}^{k} | X \in L(\mathcal{A})}$ are naturally isomorphic. 
By \cite[Proposition 3.1]{bjorner1997subspace-aim}, 
\begin{align*}
\chi(\mathcal{A},q^{k}) = \chi(\mathcal{A}\otimes\mathbb{F}_{q}^{k},q^{k}) 
= \# \left((\mathbb{F}_{q}^{k})^{\ell} \setminus \bigcup_{H \in \mathcal{A}}H\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q}^{k}\right). 
\end{align*}
Thus, if $\chi(\mathcal{A},q^{k})=0$ and $0 \leq j \leq k$, then $\chi(\mathcal{A},q^{j})=0$. 
\end{proof}

We define the \textbf{falling factorial} $t^{\underline{i}}$ for each $i \in \mathbb{Z}_{>0}$ by $t^{\underline{i}} \coloneqq t(t-1) \cdots (t-i+1)$. 
\begin{proposition}[{\cite[Theorem 15]{read1968introduction-joct}}]\label{stable partition}
Suppose $\chi(G, t) = \sum_{i=1}^{\ell}c_{i}t^{\underline{i}}$. 
Then $c_{i}$ coincides with the number of stable partitions of $G$ into $i$ blocks, where a stable partition of $G$ is a set partition of the vertex set such that no edge connects vertices within the same block.
In other words, $c_{i}$ coincides with the number of $i$-dimensional subspaces in $L(\mathcal{B}_{\ell})$ that are not contained in any hyperplanes in $\mathcal{A}_{G}$.
\end{proposition}

Note that the falling factorial $t^{\underline{i}}$ coincides with the characteristic polynomial $\chi(\mathcal{B}_{\ell},t)$. 
Define the polynomial $t_{q}^{\underline{i}}$ by 
\begin{align*}
t_{q}^{\underline{i}} \coloneqq \chi(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{i}),t) = (t-1)(t-q) \cdots (t-q^{i-1}). 
\end{align*}
The following is a $q$-version of \Cref{stable partition}. 
\begin{proposition}\label{q coefficients}
Let $\mathcal{A}$ be an arrangement in $\mathbb{F}_{q}^{\ell}$ and suppose 
$\chi(\mathcal{A},t) = \sum_{i=0}^{\ell}c_{i}t_{q}^{\underline{i}}$. 
Then $c_{i}$ is the number of $i$-dimensional subspaces in $\mathbb{F}_{q}^{\ell}$ that are not contained in any hyperplanes in $\mathcal{A}$. 
\end{proposition}
\begin{proof}
We proceed by double induction on $\ell$ and $|\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})\setminus \mathcal{A}|$. 
When $\ell = 1$, 
\begin{align*}
t = (t-1) + 1 &= t^{\underline{1}}_{q} + t^{\underline{0}}_{q} & (\text{for the empty arrangement}),\\
t-1 &= t^{\underline{1}}_{q} & (\text{for the single-point arrangement}). 
\end{align*}
Therefore the claim is true. 

Suppose that $\ell \geq 2$. 
Since $\chi(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell}),t) = t^{\underline{\ell}}_{q}$, the assertion is true for $\mathcal{A} = \mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$. 
Assume that $\mathcal{A} \subseteq \mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$ and $H \in \mathcal{A}$. 
Then, by the deletion-restriction formula \cite[Corollary 2.57]{orlik1992arrangements}, we have  
\begin{align*}
\chi(\mathcal{A}^{\prime}, t) &= \chi(\mathcal{A},t) + \chi(\mathcal{A}^{H},t) 
= \sum_{i=0}^{\ell}c_{i}t^{\underline{i}}_{q} + \sum_{i=0}^{\ell-1}d_{i}t^{\underline{i}}_{q} 
= t^{\underline{\ell}}_{q} + \sum_{i=0}^{\ell-1} (c_{i}+d_{i})t^{\underline{i-1}}_{q}. 
\end{align*}
By the induction hypothesis 
\begin{align*}
c_{i} &= \# \Set{X \in L(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})) | \dim X =i, X \not\subseteq K \text{ for any } K \in \mathcal{A} },  \\
d_{i} &= \# \Set{X \in L(\mathcal{A}_{\mathrm{all}}(H)) | \dim X =i, X \not\subseteq H \cap K \text{ for any } K \in \mathcal{A}\setminus\{H\} }.  
\end{align*}
Since 
\begin{multline*}
\Set{X \in L(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})) | \dim X = i, X \not\subseteq K \text{ for any } K \in \mathcal{A}\setminus\{H\} } \\
= \Set{X \in L(\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})) | \dim X =i, X \not\subseteq K \text{ for any } K \in \mathcal{A} } \\
\sqcup \Set{X \in L(\mathcal{A}_{\mathrm{all}}(H)) | \dim X =i, X \not\subseteq H \cap K \text{ for any } K \in \mathcal{A}\setminus\{H\} }, 
\end{multline*}
the claim follows. 
\end{proof}

\begin{example}
Consider the empty arrangement in $\mathbb{F}_{q}^{\ell}$. 
Then we have 
\begin{align*}
t^{\ell} = \sum_{i=0}^{\ell} \qbinom{\ell}{i} t^{\underline{i}}_{q}, 
\end{align*}
where 
\begin{align*}
\qbinom{\ell}{i} = \dfrac{[\ell]_{q} [\ell-1]_{q} \cdots [\ell-i+1]_{q}}{[i]_{q} [i-1]_{q} \cdots [1]_{q}}
\end{align*}
is the $q$-binomial coefficient and $[k]_{q} = \frac{q^{k}-1}{q-1}$ denotes the $q$-integer.
%\begin{align*}
%t &= t^{\underline{1}}_{q} + 1 \\
%t^{2} &= t^{\underline{2}}_{q} + (q+1)t^{\underline{1}}_{q} + 1 \\
%t^{3} &= t^{\underline{3}}_{q} +(q^{2}+q+1)t^{\underline{2}}_{q} + (q^{2}+q+1)t^{\underline{1}}_{q} + 1. 
%\end{align*}
\end{example}

\begin{example}
Consider the Boolean arrangement in $\mathbb{F}_{q}^{\ell}$, which consists of all coordinate hyperplanes. 
Then we have 
\begin{align*}
(t-1)^{\ell} = \sum_{i=0}^{\ell} (q-1)^{\ell-i}S_{q}(\ell, i)t^{\underline{i}}_{q}, 
\end{align*}
where $S_{q}(\ell,i)$ denotes the $q$-Stirling number of the second kind defined by the following recurrence formula. 
\begin{align*}
S_{q}(\ell,i) = S_{q}(\ell-1, i-1) + [i]_{q} S_{q}(\ell-1, i), \qquad
S_{q}(0,i) = \delta_{0,i}. 
\end{align*}

This is a $q$-version of the following well-known formula for the Stirling number of the second kind. 
\begin{align*}
t^{\ell} = \sum_{i=0}^{\ell}S(\ell,i)t^{\underline{i}}, 
\qquad S(\ell,i) = S(\ell-1, i-1) + i S(\ell-1, i). 
\end{align*}
\end{example}

\subsection{\texorpdfstring{$q$}{q}-deformation of graphic arrangements}
We will focus on specific subarrangements of $\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$ that arise from simple graphs.
Note that the braid arrangement $\mathcal{B}_{\ell}$ corresponds to the graphic arrangement associated with the complete graph $K_{\ell}$. 
Thus, it seems natural to assign $K_{\ell}$ to $\mathcal{A}_{\mathrm{all}}(\mathbb{F}_{q}^{\ell})$. 
Furthermore, it seems natural to assign every clique of $G$ to to the set of all hyperplanes using the coordinates corresponding to the vertices in the clique. 
\begin{definition}
We define a \textbf{$q$-deformation of graphic arrangement} $\mathcal{A}_{G}^{q}$ in $\mathbb{F}_{q}^{\ell}$ as follows. 
\begin{align*}
\mathcal{A}_{G}^{q} \coloneqq \bigcup_{\{i_{1}, \dots, i_{r}\}} \Set{\{a_{i_{1}}x_{i_{1}} + \dots + a_{i_{r}}x_{i_{r}} = 0\} | (a_{i_{1}}, \dots, a_{i_{r}}) \in \mathbb{F}_{q}^{r}\setminus\{0\}}, 
\end{align*}
where $\{i_{1}, \dots, i_{r}\}$ runs over all cliques of $G$. 
\end{definition}

Besides this definition, Athanasiadis \cite{ATHANASIADIS1996193}, and later Postnikov and Stanley \cite{POSTNIKOV2000544}, also defined deformations of Coxeter arrangements as a generalization of generic arrangements, Linial arrangements, Shi arrangements, Catalan arrangements, etc. Their definition involves copies of translated hyperplanes, whereas the $q$-deformation focuses on the coefficients of variables and the choice of linear polynomials. \par
It is expected that $\mathcal{A}_{G}^{q}$ has a lot of properties similar to the graphic arrangement $\mathcal{A}_{G}$. 
The organization of this paper is as follows. 
In \Cref{section:characteristic polynomial}, we will show that the characteristic polynomial of $\mathcal{A}_{G}^{q}$ determines the chromatic polynomial $G$ when $q$ is large enough (\Cref{main theorem} and \Cref{cor}) and we will describe a relationship between the characteristic polynomial and  the numbers of stable partitions of $G$ (\Cref{q stable parition}). 
In \Cref{section:freeness}, we will show that $\mathcal{A}_{G}^{q}$ is free if and only if $G$ is chordal as in the case of graphic arrangements (\Cref{freeness for q-graphic arr}) and in \Cref{section:basis}, we construct an explicit basis for $D(\mathcal{A}_{G}^{q})$ for a chordal graph $G$ (\Cref{basis for q-graphic arr}).  







\section{Characteristic polynomial}\label{section:characteristic polynomial}


\begin{theorem}\label{main theorem}
For any $k \in \mathbb{Z}_{\geq 0}$,
\begin{align*}
\dfrac{\chi(\mathcal{A}_{G}^{q},q^{k})}{(q-1)^{\ell}} \equiv \chi(G,k) \pmod{q-1}. 
\end{align*}
\end{theorem}
\begin{proof}

Let $V = \mathbb{F}_{q}^{k}$. 
Define 
\begin{align*}
\underline{\chi}_{G}^{q}(V) \coloneqq \Set{(v_{1}, \dots, v_{\ell}) \in V^{\ell} |\begin{array}{l}
\text{ $\{v_{i_{1}}, \dots, v_{i_{p}}\}$ is linearly independent} \\
\text{ over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$.} \\
\end{array}  }.
\end{align*}
Note that $\chi(\mathcal{A}_{G}^{q},q^{k}) = \#\underline{\chi}_{G}^{q}(V)$. 
The group $(\mathbb{F}_{q}^{\times})^{\ell}$ acts on $\underline{\chi}_{G}^{q}(V)$ by 
\begin{align*}
(a_{1}, \dots, a_{\ell}) \cdot (v_{1}, \dots, v_{\ell}) \coloneqq (a_{1}v_{1} \dots, a_{\ell}v_{\ell}). 
\end{align*}
Therefore $\# \underline{\chi}_{G}^{q}(\mathbb{P}(V)) = \frac{\chi(\mathcal{A}_{G}^{q}, q^{k})}{(q-1)^{\ell}}$, where 
\begin{align*}
\underline{\chi}_{G}^{q}(\mathbb{P}(V)) \coloneqq \Set{(\overline{v}_{1}, \dots, \overline{v}_{\ell}) \in \mathbb{P}(V)^{\ell} | \begin{array}{l}
\text{ $\{v_{i_{1}}, \dots, v_{i_{p}}\}$ is linearly independent} \\
\text{ over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$.} 
\end{array} }. 
\end{align*}


Let $T \coloneqq (\mathbb{F}_{q}^{\times})^{k}/\mathbb{F}_{q}^{\times}(1,1, \dots, 1)$. 
Let $x = \begin{pmatrix}
x_{1} \\ \vdots \\ x_{k}
\end{pmatrix} \neq 0$. 
Then $T$ acts on $\mathbb{P}(V)$ by 
\begin{align*}
\overline{(a_{1}, \dots, a_{k})} \cdot \overline{x} = \overline{\begin{pmatrix}
a_{1}x_{1} \\ \vdots \\ a_{k}x_{k}
\end{pmatrix}}. 
\end{align*}
Let $\nu(x) \coloneqq \#\Set{i \in [k] | x_{i} \neq 0}$. 
Then it is easily seen that 
\begin{align*}
\# (T\cdot \overline{x}) = (q-1)^{\nu(x)-1}.
\end{align*}
In particular, $\overline{x} \in \mathbb{P}(V)$ is a fixed point if and only if $\overline{x} = \overline{e}_{i}$ for some $i$. 
Consider the diagonal $T$-action on $\mathbb{P}(V)^{\ell}$. 
This induces a $T$-action on $\underline{\chi}_{G}^{q}(\mathbb{P}(V))$. 
The fixed point set is 
\begin{align*}
\underline{\chi}_{G}^{q}(\mathbb{P}(V))^{T} = \Set{(\overline{e}_{s_{1}}, \dots, \overline{e}_{s_{\ell}}) | \begin{array}{l}
\text{ $\{e_{s_{i_{1}}}, \dots, e_{s_{i_{p}}}\}$ is linearly independent} \\
\text{ over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$.}. 
\end{array}}
\end{align*}
Note that $\{e_{s_{i_{1}}}, \dots, e_{s_{i_{p}}}\}$ is linearly independent if and only if they are mutually different. 
Therefore we have 
\begin{align*}
\# \underline{\chi}_{G}^{q}(\mathbb{P}(V))^{T} = \chi(G,k). 
\end{align*}
Any other orbits have cardinalities divisible by $q-1$. 
This completes the proof. 
\end{proof}

\begin{corollary}\label{cor}
If $q > \ell^{\ell}$, then $\chi(\mathcal{A}_{G}^{q},t)$ determines $\chi(G,t)$. 
\end{corollary}
\begin{proof}
Note that $\chi(G,t)$ is determined by the values $\chi(G,1), \chi(G,2), \dots, \chi(G,\ell)$. 
For $k \in [\ell]$, we have $\chi(G,k) \leq k^{\ell} < q-1$. 
Hence $\chi(G,k)$ is the remainder of $\dfrac{\chi(\mathcal{A}_{G}^{q},q^{k})}{(q-1)^{\ell}}$ divided by $q-1$. 
\end{proof}

\begin{remark}
There exists a pair of graphs $(G,H)$ depicted in the following picture such that $\chi(\mathcal{A}_{G}^{2},t) = \chi(\mathcal{A}_{H}^{2},t)$ but $\chi(G,t) \neq \chi(H,t)$. 

\begin{center}
\begin{tikzpicture}
\draw (0,0) node[v](6){};
\draw (-0.5,0.5) node[v](1){};
\draw (-1,0) node[v](5){};
\draw (-0.5,-0.5) node[v](3){};
\draw (0.5,0.5) node[v](4){};
\draw (0.5,-0.5) node[v](0){};
\draw (1,0) node[v](2){};
\draw (0)--(2);
\draw (0)--(4);
\draw (0)--(6);
\draw (1)--(3);
\draw (1)--(4);
\draw (1)--(5);
\draw (1)--(6);
\draw (2)--(4);
\draw (2)--(6);
\draw (3)--(5);
\draw (3)--(6);
\draw (4)--(6);
\draw (2) to [distance=50] (5);

\draw (0)--(3);
\draw (0,-1.25) node{$G$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\draw (0,0) node[v](6){};
\draw (-0.5,0.5) node[v](1){};
\draw (-1,0) node[v](5){};
\draw (-0.5,-0.5) node[v](3){};
\draw (0.5,0.5) node[v](4){};
\draw (0.5,-0.5) node[v](0){};
\draw (1,0) node[v](2){};
\draw (0)--(2);
\draw (0)--(4);
\draw (0)--(6);
\draw (1)--(3);
\draw (1)--(4);
\draw (1)--(5);
\draw (1)--(6);
\draw (2)--(4);
\draw (2)--(6);
\draw (3)--(5);
\draw (3)--(6);
\draw (4)--(6);
\draw (2) to [distance=50] (5);

\draw (5) to [distance=-40] (0);
\draw (0,-1.25) node{$H$};
\end{tikzpicture}
\end{center}
The characteristic polynomials are as follows. 
\begin{align*}
\chi(\mathcal{A}_{G}^{2},t) = \chi(\mathcal{A}_{H}^{2},t) &= t^{7} - 30 t^{6} + 376 t^{5} - 2545 t^{4} + 9934 t^{3} - 21880 t^{2} + 24384 t - 10240, \\
\chi(G,t) &= t^{7} - 14t^{6} + 83t^{5} - 265t^{4} + 474t^{3} - 441t^{2} + 162t, \\
\chi(H,t) &= t^{7} - 14t^{6} + 83t^{5} - 264t^{4} + 468t^{3} - 430t^{2} + 156t. 
\end{align*}
\end{remark}

The characteristic polynomial $\chi(\mathcal{A}_{G}^{q},t)$ can also determine the numbers of stable partitions of $G$ by the following theorem. 

\begin{theorem}\label{q stable parition}
Let $\chi(\mathcal{A}_{G}^{q},t) = \sum_{i=0}^{\ell}c_{i}t_{q}^{i}$. 
Then $c_{i}/(q-1)^{\ell-i}$ is a non-negative integer and congruent to the number of stable partitions of $G$ into $i$ blocks modulo $q-1$. 
\end{theorem}
\begin{proof}
By \Cref{q coefficients}, the coefficient $c_{i}$ is equal to the number of subspaces that are not contained in any hyperplanes in $\mathcal{A}_{G}^{q}$. 
Since every $i$-dimensional subspace in $\mathbb{F}_{q}^{\ell}$ corresponds to a matrix consisting of $\ell$ columns with rank $i$ in reduced row echelon form, $c_{i}$ coincides with the number of reduced row echelon form $(v_{1}, \dots, v_{\ell}) \in V^{\ell}$ such that $\{v_{i_{1}}, \dots, v_{i_{p}}\}$ is linearly independent over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$, where $V = \mathbb{F}_{q}^{i}$. 
Therefore $c_{i}/(q-1)^{\ell-i}$ equals the number of $(\overline{v}_{1}, \dots, \overline{v}_{\ell}) \in \mathbb{P}(V)^{\ell}$ such that $\{v_{i_{1}}, \dots, v_{i_{p}}\}$ is linearly independent over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$. 

Using the same group action in the proof of \Cref{main theorem}, we can show that $c_{i}$ is congruent to the number of reduced row echelon form $(e_{s_{1}}, \dots, e_{s_{\ell}})$ such that $\{e_{s_{i_{1}}}, \dots, e_{s_{i_{p}}}\}$ is linearly independent over $\mathbb{F}_{q}$ if $\{i_{1}, \dots, i_{p}\}$ is a clique of $G$, modulo $q-1$. 
From such a matrix $(e_{s_{i_{1}}}, \dots, e_{s_{i_{\ell}}})$, we can construct a stable partition of $G$ consisting of $i$ blocks $\Set{j \in [\ell] | s_{j}=k} \ (1 \leq k \leq i)$.  
\end{proof}

\begin{remark}
Some of properties of the chromatic polynomials for finite graphs have $q$-analogues. For example, let us 
denote by $G+K_m$ the join of a graph $G$ and the complete graph $K_m$. Then 
it is easily seen that $\chi(G+K_m, t)=t(t-1)\cdots(t-m+1)\chi(G, t-m)$. 
As a $q$-analogue of this formula, we can prove 
\begin{align*}
\chi(\mathcal{A}_{G+K_m}^{q}, t)=(t-1)(t-q)\cdots (t-q^{m-1})q^{m\ell}\chi(\mathcal{A}_G^{q}, q^{-m}t). 
\end{align*}
\end{remark}

\section{Freeness}\label{section:freeness}

An arrangement $\mathcal{A}$ is \textbf{supersolvable} if and only if there exists a filtration 
\begin{align*}
\varnothing=\mathcal{A}_{0} \subseteq \mathcal{A}_{1} \subseteq \mathcal{A}_{2} \subseteq \dots \subseteq \mathcal{A}_{\ell} = \mathcal{A}
\end{align*}
such that, for each $i \in [\ell]$, $\rank \mathcal{A}_{i} = i$  and for any distinct hyperplanes $H,H^{\prime} \in \mathcal{A}_{i}\setminus\mathcal{A}_{i-1}$ there exists $H^{\prime\prime}$ such that $H \cap H^{\prime} \subseteq H^{\prime\prime}$ \cite[Theorem 4.3]{bjorner1990hyperplane-dcg}. 
Every supersolvable arrangement is inductively free by \cite[Theorem 4.2]{jambu1984free-aim}. 
When $\mathcal{A}$ is supersolvable with the filtration above, the characteristic polynomial decomposes as 
\begin{align*}
\chi(\mathcal{A},t) = \prod_{i=1}^{\ell}(t-|\mathcal{A}_{i}\setminus\mathcal{A}_{i-1}|). 
\end{align*}

A vertex $v$ of a simple graph $G$ is called \textbf{simplicial} if the neighborhood $N_{G}(v) \coloneqq \Set{u \in V_{G} | \{u,v\} \in E_{G}}$ is a clique. 
An ordering $(v_{1}, \dots, v_{\ell})$ of the vertices of $G$ is a \textbf{perfect elimination ordering} if $v_{i}$ is simplicial in the subgraph of $G$ induced by $\{v_{1}, \dots, v_{i}\}$ for each $i \in [\ell]$. 

\begin{theorem}[Stanley (See {\cite[Theorem 3.3]{edelman1994free-mz}}), Dirac \cite{dirac1961rigid-aadmsduh}, Fulkerson--Gross \cite{fulkerson1965incidence-pjom}]\label{graphic arrangement freeness}
The following are equivalent. 
\begin{enumerate}[label=(\arabic*)]
\item\label{PEO} $G$ has a perfect elimination ordering. 
\item $\mathcal{A}_{G}$ is supersolvable. 
\item $\mathcal{A}_{G}$ is free. 
\item\label{chordal} $G$ is chordal. 
\end{enumerate}
Moreover, when $(v_{1}, \dots, v_{\ell})$ is a perfect elimination ordering of $G$
\begin{align*}
\chi(\mathcal{A}_{G}, t) = \prod_{i=1}^{\ell}(t-|N_{G_{i}}(v_{i})|), 
\end{align*}
where $G_{i} \coloneqq G[\{v_{1}, \dots, v_{i}\}]$ and $N_{G_{i}}(v_{i})$ is the set of adjacent vertices of $v_{i}$ in $G_{i}$. 
\end{theorem}
%\begin{proof}
%$(1) \Rightarrow (2)$
%Let $(v_{1}, \dots, v_{\ell})$ be a perfect elimination ordering and $(x_{1}, \dots, x_{\ell})$ the corresponding coordinates. 
%Let $\mathcal{A}_{i}$ be the subarrangement of $\mathcal{A}_{G}$ consisting hyperplanes whose defining linear form contains $x_{1}, \dots x_{i}$. 
%Then the filtration 
%\begin{align*}
%\mathcal{A}_{1} \subseteq \mathcal{A}_{2} \subseteq \dots \subseteq \mathcal{A}_{\ell} = \mathcal{A}_{G}
%\end{align*}
%guarantees that $\mathcal{A}_{G}$ is supersolvable and $|\mathcal{A}_{i}\setminus \mathcal{A}_{i-1}| = |N_{G_{i}}(v_{i})|$. 
%
%$(2)\Rightarrow(3)$ 
%In general, every supersolvable arrangement is inductively free and hence free by Jambu--Terao. 
%
%$(3)\Rightarrow(4)$ 
%It suffices to show that $\chi(\mathcal{A}_{C_{\ell}},t)$ does not factor into the product of linear forms over $\mathbb{Z}$ when $\ell \geq 4$. 
%Since $\chi(\mathcal{A}_{C_{\ell}},t) = (t-1)^{\ell}+(-1)^{\ell}(t-1)$, we have 
%\begin{align*}
%\chi(\mathcal{A}_{C_{\ell}},x+1) = x^{\ell} + (-1)^{\ell}x = x(x^{\ell-1}+(-1)^{\ell}). 
%\end{align*}
%Since $x^{\ell}+(-1)^{\ell}$ has an imaginary root, $\chi(\mathcal{A}_{C_{\ell}},t)$ does not factor over $\mathbb{Z}$. 
%
%$(4)\Rightarrow (1)$
%This is proven by Dirac and Fulkerson--Gross. 
%\end{proof}

\begin{theorem}\label{freeness for q-graphic arr}
The following conditions are equivalent to the conditions in \Cref{graphic arrangement freeness}. 
\begin{enumerate}[label=(\arabic*)]
\setcounter{enumi}{4}
\item\label{q SS} $\mathcal{A}_{G}^{q}$ is supersolvable. 
\item\label{q free} $\mathcal{A}_{G}^{q}$ is free. 
\end{enumerate}
Moreover, when $(v_{1}, \dots, v_{\ell})$ is a perfect elimination ordering of $G$
\begin{align*}
\chi(\mathcal{A}_{G}^{q}, t) = \prod_{i=1}^{\ell}\left(t-q^{|N_{G_{i}}(v_{i})|}\right), 
\end{align*}
where $G_{i} \coloneqq G[\{v_{1}, \dots, v_{i}\}]$ and $N_{G_{i}}(v_{i})$ is the set of adjacent vertices of $v_{i}$ in $G_{i}$. 
\end{theorem}
\begin{proof}
The proof is very similar to the proof of \Cref{graphic arrangement freeness}. 

To show $\ref{PEO}\Rightarrow\ref{q SS}$, 
let $(v_{1}, \dots, v_{\ell})$ be a perfect elimination ordering and $(x_{1}, \dots, x_{\ell})$ the corresponding coordinates. 
Let $\mathcal{A}_{i}$ be the subarrangement of $\mathcal{A}_{G}^{q}$ consisting hyperplanes whose defining linear form contains $x_{1}, \dots, x_{i}$. 
Then the filtration 
\begin{align*}
\mathcal{A}_{1} \subseteq \mathcal{A}_{2} \subseteq \dots \subseteq \mathcal{A}_{\ell} = \mathcal{A}_{G}^{q}
\end{align*}
guarantees that $\mathcal{A}_{G}^{q}$ is supersolvable and $|\mathcal{A}_{i}\setminus \mathcal{A}_{i-1}| = q^{|N_{G_{i}}(v_{i})|}$. 

The implication $\ref{q SS} \Rightarrow \ref{q free}$ follows since every supersolvable arrangement is (inductively) free (Jambu--Terao).
To prove $\ref{q free}\Rightarrow\ref{chordal}$ 
It suffices to show that $\chi(\mathcal{A}_{C_{\ell}}^{q},t)$ does not factor into the product of linear forms over $\mathbb{Z}$ when $\ell \geq 4$ by Terao's factorization theorem. 
Since $\chi(\mathcal{A}_{C_{\ell}}^{q},t) = (t-q)^{\ell}+(-1)^{\ell}(q-1)^{\ell-1}(t-q)$ by the following \Cref{characteristic poynomial cycle graph}, we have 
\begin{align*}
\chi(\mathcal{A}_{C_{\ell}},t) = (q-1)^{\ell}(x^{\ell}+(-1)^{\ell}) = (q-1)^{\ell}x(x^{\ell-1}+(-1)^{\ell}), 
\end{align*}
where $x = \frac{t-q}{q-1}$. 
Since $x^{\ell}+(-1)^{\ell}$ has an imaginary root, $\chi(\mathcal{A}_{C_{\ell}}^{q},t)$ does not factor over $\mathbb{Z}$. 
\end{proof}

There is a well-known explicit formula for the chromatic polynomial of a cycle graph for $\ell \geq 3$.
\begin{align*}
\chi(\mathcal{A}_{C_{\ell}},t) = (t-1)^{\ell}+(-1)^{\ell}(t-1). 
\end{align*}

The following lemma is a $q$-version of this formula. 
\begin{lemma}\label{characteristic poynomial cycle graph}
Let $\ell \geq 4$. 
Then 
\begin{align*}
\chi(\mathcal{A}_{C_{\ell}}^{q},t) = (t-q)^{\ell}+(-1)^{\ell}(q-1)^{\ell-1}(t-q).
\end{align*}
\end{lemma}
\begin{proof}
First consider the arrangement $\mathcal{B}$ in $\mathbb{F}_{q}^{3}$ consisting of the following hyperplanes. 
\begin{align*}
x_{1}=ax_{2},\ x_{2}=ax_{3},\ x_{3}=ax_{1} \quad (a \in \mathbb{F}_{q}).
\end{align*}
Then $\mathcal{B}$ is supersolvable with exponents $(1,q,2q-1)$. 
Using the deletion-restriction theorem \cite[Corollary 2.57]{orlik1992arrangements} $q-1$ times, we have 
\begin{align*}
\chi(\mathcal{A}_{C_{4}}^{q}, t) &= \chi(\mathcal{A}_{P_{4}}^{q}, t) - (q-1)\chi(\mathcal{B},t) \\
&= (t-1)(t-q)^{3} - (q-1)(t-1)(t-q)(t-2q+1) \\
&= (t-1)(t-q)\left((t-q)^{2}-(q-1)(t-2q+1)\right) \\
&= (t-1)(t-q)\left((t-q)^{2}-(q-1)(t-q) +(q-1)^{2}\right) \\
&= (t-1)(t-q)\dfrac{(t-q)^{3}+(q-1)^{3}}{(t-q)+(q-1)} \\
&= (t-q)^{4}+(-1)^{4}(q-1)^{3}(t-1). 
\end{align*}

Now suppose that $\ell \geq 5$. 
Using the deletion-restriction theorem $q-1$ times, we have 
\begin{align*}
\chi(\mathcal{A}_{C_{\ell}}^{q},t) &= \chi(\mathcal{A}_{P_{\ell}}^{q})-(q-1)\chi(\mathcal{A}_{C_{\ell-1}}^{q},t) \\
&= (t-1)(t-q)^{\ell-1}-(q-1)(t-q)^{\ell-1}-(-1)^{\ell-1}(q-1)^{\ell-1}(t-q) \\
&= (t-q)^{\ell}+(-1)^{\ell}(q-1)^{\ell-1}(t-q). 
\end{align*}
\end{proof}

\section{Basis construction}\label{section:basis}

Let $G$ be a chordal graph with a perfect elimination ordering $(v_{1}, \dots, v_{\ell})$ and $(x_{1}, \dots, x_{\ell})$ the corresponding coordinates. 
Define the sets $C_{\geq k}$ and $E_{<k}$ by 
\begin{align*}
C_{\geq k} &\coloneqq \{k\} \cup \left \{~ i \in [\ell] | \text{there exists a path } v_{k}v_{j_{1}}\cdots v_{j_{n}}v_{i} \text{ such that } k < j_{1} < \dots < j_{n} < i ~\right \}, \\
E_{<k} &\coloneqq \Set{j \in [\ell] | j < k \text{ and } \{v_{j},v_{k}\} \in E_{G}}. 
\end{align*}

Let $\Delta(x_{1}, \dots, x_{k})$ denote the Vandermonde determinant: 
\begin{align*}
\Delta(x_{1}, \dots, x_{k}) \coloneqq \begin{vmatrix}
1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{k-1} \\
1 & x_{2} & x_{2}^{2} &  \dots & x_{k}^{k-1} \\
\vdots & \vdots & \vdots &  & \vdots \\
1 & x_{k} & x_{k}^{2} & \dots & x_{k}^{k-1}
\end{vmatrix} 
= \prod_{1 \leq i < j \leq k}(x_{j}-x_{i}). 
\end{align*}

When $E_{<k} = \{j_{1}, \dots, j_{m}\}$ with $j_{1} < \dots < j_{m}$, 
\begin{align*}
\Delta(E_{<k}) \coloneqq \Delta(x_{j_{1}}, \dots, x_{j_{m}}) 
\quad \text{ and } \quad
\Delta(E_{<k}, x_{i}) \coloneqq \Delta(x_{j_{1}}, \dots, x_{j_{m}}, x_{i}). 
\end{align*}

\begin{theorem}[{\cite[Theorem 4.1]{suyama2019vertex-weighted-dcg}}]
Let 
\begin{align*}
\theta_{k} \coloneqq \sum_{i \in C_{\geq k}}\dfrac{\Delta(E_{<k}, x_{i})}{\Delta(E_{<k})}\partial_{i} \qquad (1 \leq k \leq \ell). 
\end{align*}
Then $\{\theta_{1}, \dots, \theta_{\ell}\}$ forms a basis for $D(\mathcal{A}_{G})$. 
\end{theorem}
\noindent
Let $\Delta_{q}(x_{1}, \dots, x_{k}) \in \mathbb{F}_{q}[x_{1}, \dots, x_{k}]$ denote the determinant of the Moore matrix. 
Namely 
\begin{align*}
\Delta_{q}(x_{1}, \dots, x_{k}) = \begin{vmatrix}
x_{1} & x_{1}^{q} & x_{1}^{q^{2}} & \dots & x_{1}^{q^{k-1}} \\
x_{2} & x_{2}^{q} & x_{2}^{q^{2}} &  \dots & x_{k}^{q^{k-1}} \\
\vdots & \vdots & \vdots &  & \vdots \\
x_{k} & x_{k}^{q} & x_{k}^{q^{2}} & \dots & x_{k}^{q^{k-1}}
\end{vmatrix} 
= \prod_{i = 1}^{k}\prod_{c_{1}, \dots, c_{i-1} \in \mathbb{F}_{q}}(c_{1}x_{1} + \dots + c_{i-1}x_{i-1} + x_{i}). 
\end{align*}

\begin{theorem}\label{basis for q-graphic arr}
Let 
\begin{align*}
\theta_{k} \coloneqq \sum_{i \in C_{\geq k}}\dfrac{\Delta_{q}(E_{<k}, x_{i})}{\Delta_{q}(E_{<k})}\partial_{i} \qquad (1 \leq k \leq \ell). 
\end{align*}
Then $\{\theta_{1}, \dots, \theta_{\ell}\}$ forms a basis for $D(\mathcal{A}_{G}^{q})$.
\end{theorem}
\begin{proof}
Let $K = \{i_{1}, \dots, i_{m}\}$ be a clique of $G$ with $i_{1} < \dots < i_{m}$. 
Let $\alpha = c_{1}x_{i_{1}} + c_{2}x_{i_{2}} + \dots + c_{m}x_{i_{m}} $ be a nonzero linear form over $\mathbb{F}_{q}$. 
If $K \cap C_{\geq k} = \varnothing$, then $\theta_{k}(\alpha) = 0$. 

Suppose that $K \cap C_{\geq k} \neq \varnothing$ and take $i_{s} \in K \cap C_{\geq k}$ with minimal $s$. 
Then one can show that $\{i_{1}, \dots, i_{s-1}\} \subseteq E_{<k}$ and $\{i_{s}, i_{s+1}, \dots, i_{m}\} \subseteq C_{\geq k}$. 
Then 
\begin{align*}
\theta_{k}(\alpha) 
&= \sum_{u=s}^{k}\dfrac{\Delta_{q}(E_{<k},x_{i_{u}})}{\Delta_{q}(E_{<k})} \cdot c_{i_{u}} 
=\dfrac{\Delta_{q}(E_{<k}, \ c_{i_{s}}x_{i_{s}} + \dots + c_{i_{m}}x_{i_{m}})}{\Delta_{q}(E_{<k})} 
= \dfrac{\Delta_{q}(E_{<k}, \ \alpha)}{\Delta_{q}(E_{<k})} \in (\alpha). 
\end{align*}
Using Saito's criterion \cite[Theorem 4.19]{orlik1992arrangements}, we can prove $\{\theta_{1}, \dots, \theta_{\ell}\}$ is a basis. 
%Since the Saito's matrix $M(\theta_{1}, \dots, \theta_{\ell})$ is triangular, it is easy to show that $\det M(\theta_{1}, \dots, \theta_{\ell}) \doteq Q(\mathcal{A}_{G}^{q})$. 
\end{proof}



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