Let $x=({x}_{1},\mathrm{\dots},{x}_{n})$ be a variable of the Euclidean space ${\mathbb{R}}^{n}$, and ${a}_{1},\mathrm{\dots},{a}_{n},b$ real coefficients such that $({a}_{1},\mathrm{\dots},{a}_{n})\ne (0,\mathrm{\dots},0)$. A hyperplane *H* of ${\mathbb{R}}^{n}$ is an $(n-1)$-dimensional affine subspace $H:=\{x\in {\mathbb{R}}^{n}\mid {a}_{1}{x}_{1}+\mathrm{\dots}+{a}_{n}{x}_{n}=b\}$. An arrangement of hyperplanes in ${\mathbb{R}}^{n}$ is a finite set of hyperplanes. For example, the best known hyperplane arrangement is certainly

${\mathcal{\mathcal{A}}}_{{A}_{n-1}}={\left\{\{x\in {\mathbb{R}}^{n}\mid {x}_{i}-{x}_{j}=0\}\right\}}_{1\le i<j\le n}$

associated to the Coxeter group ${A}_{n-1}$.

A *chamber* of a hyperplane arrangement $\mathcal{\mathcal{A}}$ is a connected component of the complement ${\mathbb{R}}^{n}\setminus {\bigcup}_{H\in \mathcal{\mathcal{A}}}H$. Denote the set of all chambers of $\mathcal{\mathcal{A}}$ by $\u212d(\mathcal{\mathcal{A}})$.

Assign a variable ${a}_{H}$ to each hyperplane *H* of an arrangement $\mathcal{\mathcal{A}}$. Let

${R}_{\mathcal{\mathcal{A}}}=\mathbb{Z}[{a}_{H}\mid H\in \mathcal{\mathcal{A}}]$

be the ring of polynomials in variables ${a}_{H}$. The module of ${R}_{\mathcal{\mathcal{A}}}$-linear combinations of chambers of the hyperplane arrangement $\mathcal{\mathcal{A}}$ is

${M}_{\mathcal{\mathcal{A}}}:=\{\sum _{C\in \u212d(\mathcal{\mathcal{A}})}{x}_{C}C\mid {x}_{C}\in {R}_{\mathcal{\mathcal{A}}}\}.$

Let $\mathcal{\mathscr{H}}(C,D)$ be the set of hyperplanes separating the chambers *C* and *D* in $\u212d(\mathcal{\mathcal{A}})$. The ${R}_{\mathcal{\mathcal{A}}}$-bilinear symmetric form $\U0001d5a1:{M}_{\mathcal{\mathcal{A}}}\times {M}_{\mathcal{\mathcal{A}}}\to {R}_{\mathcal{\mathcal{A}}}$ on the hyperplane arrangement $\mathcal{\mathcal{A}}$ defined by Varchenko [12] is

$\U0001d5a1(C,C):=1,\text{and}\mathit{\hspace{1em}}\U0001d5a1(C,D):=\prod _{H\in \mathcal{\mathscr{H}}(C,D)}{a}_{H}\text{if}C\ne D.$

The matrix ${(\U0001d5a1(C,D))}_{C,D\in \u212d(\mathcal{\mathcal{A}})}$
associated to the bilinear symmetric form $\U0001d5a1$ is called the Varchenko matrix of the hyperplane
arrangement $\mathcal{\mathcal{A}}$. In terms of Markov chains, it is the matrix of random walks on $\u212d(\mathcal{\mathcal{A}})$ whose walk probability from the chamber *C* to the chamber *D* is equal to $\U0001d5a1(C,D)$. The *Varchenko determinant* of the hyperplane arrangement $\mathcal{\mathcal{A}}$ is the determinant

$det\mathcal{\mathcal{A}}:=det{(\U0001d5a1(C,D))}_{C,D\in \u212d(\mathcal{\mathcal{A}})}.$

One of the first appearances of this bilinear form was in the work of Schechtman and Varchenko
[11, Section 1], in the implicit form of a symmetric bilinear form on a Verma module over a $\u2102$-algebra. It appeared more
explicitly later, one year after that publication, albeit as a very special case, when Zagier studied a Hilbert space $\mathbb{H}$ together with a nonzero distinguished vector $|0\u3009$, and a collection of operators ${a}_{k}:\mathbb{H}\to \mathbb{H}$ satisfying the commutation relations

$a(l){a}^{\u2020}(k)-q{a}^{\u2020}(k)a(l)={\delta}_{k,l},$

and the relation $a(k)|0\u3009=0$. To demonstrate the realizability of its model, he defined an inner product space $(\mathbb{H}(q),\u3008\cdot ,\cdot \u3009)$ with basis *B* consisting of *n*-particle states ${a}^{\u2020}({k}_{1})\mathrm{\dots}{a}^{\u2020}({k}_{n})|0\u3009$, and proved that [13, Theorems 1, 2]

$det{(\u3008u,v\u3009)}_{u,v\in B}=\prod _{k=1}^{n-1}{(1-{q}^{{k}^{2}+k})}^{\frac{n!(n-k)}{{k}^{2}+k}}.$

It is the Varchenko determinant of ${\mathcal{\mathcal{A}}}_{{A}_{n-1}}$ with all hyperplanes weighted by *q*. Using the diagonal solutions of the Yang–Baxter equation, Duchamp et al. computed [6, Section 6.4.2]

$det{\mathcal{\mathcal{A}}}_{{A}_{n-1}}=\prod _{\begin{array}{c}I\in {2}^{[n]}\\ |I|\ge 2\end{array}}{\left(1-\prod _{\{i,j\}\in \left(\genfrac{}{}{0pt}{}{I}{2}\right)}{a}_{{H}_{i,j}}^{2}\right)}^{(|I|-2)!(n-|I|+1)!},$

each hyperplane $\{x\in {\mathbb{R}}^{n}\mid {x}_{i}-{x}_{j}=0\}$ having its own weight ${a}_{{H}_{i,j}}$ this time.

An *edge* of a hyperplane arrangement $\mathcal{\mathcal{A}}$ is a nonempty intersection of some of its hyperplanes. Denote the set of all edges of $\mathcal{\mathcal{A}}$ by $L(\mathcal{\mathcal{A}})$. The *weight*
$\U0001d5ba(E)$ of an edge *E* is

$\U0001d5ba(E):=\prod _{\begin{array}{c}H\in \mathcal{\mathcal{A}}\\ E\subseteq H\end{array}}{a}_{H}.$

The *multiplicity*
$l(E)$ of an edge *E* is a positive integer computed as follows [5, Section 2]: First choose a hyperplane *H* of $\mathcal{\mathcal{A}}$ containing *E*. Then $l(E)$ is half the number of chambers *C* of $\u212d(\mathcal{\mathcal{A}})$ which have the property that *E* is the minimal intersection containing $\overline{C}\cap H$.

Due to Varchenko [12, Theorem 1.1], the formula for the determinant of a hyperplane arrangement $\mathcal{\mathcal{A}}$ is

$det\mathcal{\mathcal{A}}=\prod _{E\in L(\mathcal{\mathcal{A}})}{\left(1-\U0001d5ba{(E)}^{2}\right)}^{l(E)}.$

From this formula, we see that we can get a more explicit or computable value of the determinant of an arrangement if we have computable forms of the $\U0001d5ba(E)$’s and the $l(E)$’s. In this article, we prove that this is the case for the arrangements associated to finite Coxeter groups. Recall that a reflection in ${\mathbb{R}}^{n}$ is a linear map sending a nonzero vector α to its negative while fixing pointwise the hyperplane *H* orthogonal to α. The finite reflection groups are also called finite Coxeter groups, since they have been classified by Coxeter [4]. Coxeter groups find applications in practically all areas of mathematics. They are particularly studied in great depth in algebra [7], combinatorics [2], and geometry [1]. And they are the foundation ingredients of mathematical theories like the descent algebras, the Hecke algebras, or the Kazhdan–Lusztig polynomials. A finite Coxeter group *W* has the presentation

$W:=\u3008{s}_{1},{s}_{2},\mathrm{\dots},{s}_{n}\mid {({s}_{i}{s}_{j})}^{{m}_{ij}}=1\u3009\mathit{\hspace{1em}}\text{with}{m}_{ii}=1,{m}_{ij}\ge 2,{m}_{ij}={m}_{ji}.$

We begin with some definitions relating to the finite Coxeter group *W*.

The elements of the set $S:=\{{s}_{1},{s}_{2},\mathrm{\dots},{s}_{n}\}$ are called the simple reflections of *W*. The set of all reflections of *W* is denoted by $T:=\{{s}_{i}^{x}\mid {s}_{i}\in S,x\in W\}$.

As usually, for a subset *J* of *S*, ${W}_{J}$ is the parabolic subgroup $\u3008J\u3009$ of *W*, ${T}_{J}$ the set of reflections $T\cap {W}_{J}$ of ${W}_{J}$, ${X}_{J}$ the set of coset representatives of minimal length of ${W}_{J}$, and $[J]$ its Coxeter class that is the set of *W*-conjugates of *J* which happen to be subsets of *S*.

One says that *J* is irreducible if the relating parabolic subgroup ${W}_{J}$ is irreducible.

For a subset *K* of *J*, we write ${N}_{{W}_{J}}({W}_{K})$ for the normalizer of ${W}_{K}$ in ${W}_{J}$, and $X(J,K)$ for the set of double coset representatives

$X(J,K):=\{w\in {W}_{J}\cap {X}_{K}\cap {X}_{K}^{-1}\mid {K}^{w}=K\}.$

The normalizers of the parabolic subgroups were determined by Howlett [8]. He also proved that ${N}_{{W}_{J}}({W}_{K})={W}_{K}\cdot X(J,K)$, see [8, Corollary 3].

Let ${s}_{{i}_{1}}\mathrm{\dots}{s}_{{i}_{l}}$ be a reduced expression of an element *x* of *W*. The support of *x* is

$J(x):=\{{s}_{{i}_{1}},\mathrm{\dots},{s}_{{i}_{l}}\}$

which is independent of the choice of the reduced expression [1, Proposition 2.16]. One says that *x* has full support if $J(x)=S$.

We finish with the set

$\lfloor x\rceil :=\{y\in W\mid J(y)=J(x)\text{and}y\text{is conjugate to}x\}.$

Now, we come to the hyperplane arrangement associated to a finite Coxeter group called Coxeter arrangement. Let ${H}_{t}$ be the hyperplane $\mathrm{ker}(t-1)$ of ${\mathbb{R}}^{n}$ whose points are fixed by each element *t* of *T*. The hyperplane arrangement associated to the finite Coxeter group *W* is ${\mathcal{\mathcal{A}}}_{W}:={\{{H}_{t}\}}_{t\in T}$. In this case, explicit formulas for $\U0001d5ba(E)$ and $l(E)$ can be given, and it is the aim of this article. For a subset *U* of *T*, we write ${E}_{U}$ for the edge

${E}_{U}:=\bigcap _{u\in U}{H}_{u}.$

The aim of this article is to prove the following result.

#### Theorem 1.1.

*Let **E* be an edge of ${\mathcal{A}}_{W}$. Then $l\mathit{}\mathrm{(}E\mathrm{)}\mathrm{\ne}\mathrm{0}$ if and only if there exist an irreducible subset *J* of *S* and an element *w* of *W* such that $E\mathrm{=}{E}_{{T}_{J}^{w}}$. In this case, we clearly have $\mathrm{a}\mathit{}\mathrm{(}{E}_{{T}_{J}^{w}}\mathrm{)}\mathrm{=}{\mathrm{\prod}}_{u\mathrm{\in}{T}_{J}}{a}_{{H}_{{u}^{w}}}$. Moreover, let ${t}_{J}$ be a reflection with support *J*, ${s}_{J}$ a simple reflection and *v* an element of *W* such that a reduced expression of ${t}_{J}$ is ${s}_{J}^{v}$. Then

$l({E}_{{T}_{J}^{w}})=|\lfloor {t}_{J}\rceil |\cdot |[J]|\cdot |X(S,J)|\cdot |X(J,\{{s}_{J}\})|.$

Let ${Y}_{J}$ be the set of the cosets of ${N}_{W}({W}_{J})$, and $\mathcal{\mathcal{I}}(S)$ the set of the Coxeter classes $[J]$ of *W* such that *J* is irreducible. We deduce that the Varchenko determinant of the arrangement associated to a finite Coxeter group *W* is

$\prod _{[J]\in \mathcal{\mathcal{I}}(S)}\prod _{w\in {Y}_{J}}{\left(1-\prod _{u\in {T}_{J}}{a}_{{H}_{{u}^{w}}}^{2}\right)}^{|\lfloor {t}_{J}\rceil |\cdot |[J]|\cdot |X(S,J)|\cdot |X(J,\{{s}_{J}\})|}.$

We compute the Varchenko determinants associated to the irreducible Coxeter groups in Section 6. We deduce the Varchenko determinant associated to a finite Coxeter group from the following remark: If $W={W}_{1}{W}_{2}$, where ${W}_{1}$ and ${W}_{2}$ are two irreducible Coxeter groups, then

$det{\mathcal{\mathcal{A}}}_{W}={(det{\mathcal{\mathcal{A}}}_{{W}_{1}})}^{|{W}_{2}|}{(det{\mathcal{\mathcal{A}}}_{{W}_{2}})}^{|{W}_{1}|}.$

Recall that there is a one-to-one correspondence between the elements of *W* and the chambers of ${\mathcal{\mathcal{A}}}_{W}$ such that: if the chamber *C* corresponds to the neutral element *e* and ${C}_{x}$ to another element *x*, then ${C}_{x}=Cx$ with $x={t}_{1}\mathrm{\dots}{t}_{r}$, the ${H}_{{t}_{i}}$’s being the hyperplanes one goes through from *C* to ${C}_{x}$, see [1, Theorem 1.69].

Let $\u3008{\overline{C}}_{x}\cap {H}_{t}\u3009$ be the subspace generated by the closed face ${\overline{C}}_{x}\cap {H}_{t}$ of the chamber ${C}_{x}$. Determining the multiplicity of the edge *E* contained in the hyperplane ${H}_{t}$ consists of counting the half of the chambers ${C}_{x}$ which have the property that *E* is the minimal edge containing $\u3008{\overline{C}}_{x}\cap {H}_{t}\u3009$. For the proof of Theorem 1.1, we need to introduce the three propositions that we prove in the next three sections.

#### Proposition 1.2.

*Let **E* be an edge of $\mathcal{A}$. Then $l\mathit{}\mathrm{(}E\mathrm{)}\mathrm{\ne}\mathrm{0}$ if and only if there exist an irreducible subset *J* of *S* and an element *w* of *W* such that $E\mathrm{=}{E}_{{T}_{J}^{w}}$.

Let us introduce the set $L(E,t):=\{x\in W\mid \u3008{\overline{C}}_{x}\cap {H}_{t}\u3009=E\}$.

#### Proposition 1.3.

*Let **J* be an irreducible subset of *S*, *t* a reflection with support *J*, *s* a simple reflection, and *v* an element of *W* such that a reduced expression of *t* is ${s}^{v}$. For a conjugate *K* of *J*, let ${c}_{K\mathrm{,}J}$ be an element of *W* such that ${K}^{{c}_{K\mathrm{,}J}}\mathrm{=}J$, and for a conjugate *u* of *t* with support *J*, let ${c}_{u\mathrm{,}t}$ be an element of *W* such that ${u}^{{c}_{u\mathrm{,}t}}\mathrm{=}t$. Then,

$L({E}_{{T}_{J}},t)=\bigsqcup _{K\in [J]}{c}_{K,J}X(S,J)\left(\bigsqcup _{u\in \lfloor t\rceil}{c}_{u,t}{N}_{{W}_{J}}{({W}_{\{s\}})}^{v}\right).$

#### Proposition 1.4.

*Let $u\mathrm{,}v\mathrm{\in}T$ and let **E* be an edge contained in both ${H}_{u}$ and ${H}_{v}$. Then

$|L(E,u)|=|L(E,v)|.$

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