Let be a variable of the Euclidean space , and real coefficients such that . A hyperplane H of is an -dimensional affine subspace . An arrangement of hyperplanes in is a finite set of hyperplanes. For example, the best known hyperplane arrangement is certainly
associated to the Coxeter group .
A chamber of a hyperplane arrangement is a connected component of the complement . Denote the set of all chambers of by .
Assign a variable to each hyperplane H of an arrangement . Let
be the ring of polynomials in variables . The module of -linear combinations of chambers of the hyperplane arrangement is
Let be the set of hyperplanes separating the chambers C and D in . The -bilinear symmetric form on the hyperplane arrangement defined by Varchenko  is
The matrix associated to the bilinear symmetric form is called the Varchenko matrix of the hyperplane arrangement . In terms of Markov chains, it is the matrix of random walks on whose walk probability from the chamber C to the chamber D is equal to . The Varchenko determinant of the hyperplane arrangement is the determinant
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 -algebra. It appeared more explicitly later, one year after that publication, albeit as a very special case, when Zagier studied a Hilbert space together with a nonzero distinguished vector , and a collection of operators satisfying the commutation relations
and the relation . To demonstrate the realizability of its model, he defined an inner product space with basis B consisting of n-particle states , and proved that [13, Theorems 1, 2]
It is the Varchenko determinant of with all hyperplanes weighted by q. Using the diagonal solutions of the Yang–Baxter equation, Duchamp et al. computed [6, Section 6.4.2]
each hyperplane having its own weight this time.
An edge of a hyperplane arrangement is a nonempty intersection of some of its hyperplanes. Denote the set of all edges of by . The weight of an edge E is
The multiplicity of an edge E is a positive integer computed as follows [5, Section 2]: First choose a hyperplane H of containing E. Then is half the number of chambers C of which have the property that E is the minimal intersection containing .
Due to Varchenko [12, Theorem 1.1], the formula for the determinant of a hyperplane arrangement is
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 ’s and the ’s. In this article, we prove that this is the case for the arrangements associated to finite Coxeter groups. Recall that a reflection in 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 . Coxeter groups find applications in practically all areas of mathematics. They are particularly studied in great depth in algebra , combinatorics , and geometry . 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
We begin with some definitions relating to the finite Coxeter group W.
The elements of the set are called the simple reflections of W. The set of all reflections of W is denoted by .
As usually, for a subset J of S, is the parabolic subgroup of W, the set of reflections of , the set of coset representatives of minimal length of , and 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 is irreducible.
For a subset K of J, we write for the normalizer of in , and for the set of double coset representatives
Let be a reduced expression of an element x of W. The support of x is
which is independent of the choice of the reduced expression [1, Proposition 2.16]. One says that x has full support if .
We finish with the set
Now, we come to the hyperplane arrangement associated to a finite Coxeter group called Coxeter arrangement. Let be the hyperplane of whose points are fixed by each element t of T. The hyperplane arrangement associated to the finite Coxeter group W is . In this case, explicit formulas for and can be given, and it is the aim of this article. For a subset U of T, we write for the edge
The aim of this article is to prove the following result.
Let E be an edge of . Then if and only if there exist an irreducible subset J of S and an element w of W such that . In this case, we clearly have . Moreover, let be a reflection with support J, a simple reflection and v an element of W such that a reduced expression of is . Then
Let be the set of the cosets of , and the set of the Coxeter classes of W such that J is irreducible. We deduce that the Varchenko determinant of the arrangement associated to a finite Coxeter group W is
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 , where and are two irreducible Coxeter groups, then
Recall that there is a one-to-one correspondence between the elements of W and the chambers of such that: if the chamber C corresponds to the neutral element e and to another element x, then with , the ’s being the hyperplanes one goes through from C to , see [1, Theorem 1.69].
Let be the subspace generated by the closed face of the chamber . Determining the multiplicity of the edge E contained in the hyperplane consists of counting the half of the chambers which have the property that E is the minimal edge containing . For the proof of Theorem 1.1, we need to introduce the three propositions that we prove in the next three sections.
Let E be an edge of . Then if and only if there exist an irreducible subset J of S and an element w of W such that .
Let us introduce the set .
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 . For a conjugate K of J, let be an element of W such that , and for a conjugate u of t with support J, let be an element of W such that . Then,
Let and let E be an edge contained in both and . Then
2 The Coxeter complex
Not all edges are relevant, or in other words, there are some edges whose multiplicities are null. We develop the condition for an edge E to be relevant which means .
A finite Coxeter group W is irreducible if and only if W has a reflection of full support.
If W is irreducible, then W has a highest root [9, Sections 2.10, 2.13], and the reflection corresponding to the highest root has full support.
Suppose that W is the product of nontrivial Coxeter groups and , and let t be a reflection in W. Without loss of generality, we can suppose that t is a conjugate of a simple reflection s of , hence t lies in and can not have full support. ∎
We continue our investigation by using the Coxeter complex. Recall that the Coxeter complex of W is the set of faces. The Coxeter complex is a combinatorial setup which permits to study the geometrical structure of . Indeed, is partitioned by , see [9, Section 1.15].
The chamber is identified with the singleton . The coset is a face of if and only if . Then the closure of is
More generally, is a face of if and only if and . Then the closure of a face of is
Hyperplanes, and more generally edges, can be described as collections of the faces they consist of. We extend the definition of to a subset X of W in the following way:
Let . Then .
As W acts by right multiplication, if and only if , i.e. , i.e. , i.e. , i.e. . ∎
Let and . Then, .
Let be a face of the Coxeter complex. Then, the subspace generated by is the edge .
It is clear that the subspace generated by is equal to the subspace generated by . From the proof of Lemma 2.2, we know that being a hyperplane containing is equivalent to which is equivalent to . ∎
Let us take a reflection t of U. Recall that is half the number of chambers which have the property that is the minimal intersection containing . Minimality implies equality, that is, for the chamber to be counted, we must have , i.e. (by Lemma 2.3)
i.e. (by Lemma 2.4)
Moreover, since is a full support reflection of the group , we know from Lemma 2.1 that is an irreducible subset of S. So is of the form , where J is irreducible, otherwise . That proves Proposition 1.2.
3 The chambers to consider
For a relevant edge E and a hyperplane containing E, we determine the chambers such that .
Let J be an irreducible subset of S, t a reflection with support J, and E the relevant edge . For each K in the Coxeter class of J, let be the coset of minimal length of such that . Then,
Let with . Note that if and only if if and only if . We thus obtain the result. ∎
Recall that . We introduce the set
Let J be an irreducible subset of S. Consider a reflection t of with support J. Then,
We have , see [8, Corollary 3]. So
Since we always have , the remaining condition is . ∎
We write for the centralizer of the element x of W.
Let J be an irreducible subset of S. Consider a reflection with support J where s is a simple reflection and v an element of . For another reflection u of with support J and conjugate to t, let be the coset of minimal length of such that . Then,
The equality means that or , where u is a conjugate of t with support J. Then,
Since and , we get the result. ∎
4 Invariance of the multiplicity
For the calculation of the multiplicity of an edge E, one has to choose a hyperplane containing E. We prove that the result of the calculation is independent of the choice of the hyperplane containing E.
Consider two hyperplanes and of associated with the orthogonal vectors and , respectively, of equal length and from the positive root system. Let r be the rotation of , not necessarily in W, which transforms to . Then, .
Let be the 2-dimensional subspace :
on , the map r is the rotation of angle ,
on , the map r is the identity map.
We have with
For all H in , we have .
Let p be the projection on the subspace . The arrangement is the arrangement of a dihedral group whose angle between two certain hyperplanes is θ. Then, . Hence, for any H in , we have
which still belongs to .∎
We prove Proposition 1.4 now. Consider two hyperplanes and containing the edge E. Let and be the unit vectors of the positive root system associated to W which are orthogonal to and , respectively. We use the rotation r of Lemma 4.1 transforming to , and leaving invariant. For a given w in , we have
Hence , and , which is Proposition 1.4.
5 The multiplicity of an edge
We establish a formula for the multiplicity of a relevant edge of in this section.
We begin with the proof of Theorem 1.1. From Proposition 1.2, we know that the relevant edges are the intersections of hyperplanes with the condition that J is irreducible. The weight of is obviously so that the real problem concerns . Fixing a hyperplane containing , we see that the set of chambers taken into account to determine is . But Proposition 1.4 allows us to choose any hyperplane containing . Hence the multiplicity of is
Let J be an irreducible subset of S, and w an element of W. Then,
Let . We have
With the same argument, we obtain also . Hence . ∎
which finishes the proof of Theorem 1.1.
6 Computing the determinants of finite Coxeter groups
Before computing the Varchenko determinants of the irreducible finite Coxeter groups, we first have to determine their numbers of full support reflections.
The graph associated to a finite Coxeter group W is defined as follows:
the vertex set S is the simple reflection set of W,
the edge set M is composed of the pairs of simple reflections such that , and these edges are labeled by .
We only consider the graphs associated to the irreducible finite Coxeter groups [7, Table 1.1]. The reflections of an irreducible finite Coxeter group W form a single conjugacy class if and only if the edge labels of its associated graph are all odd [3, n1.3, Proposition 3]. This is the case of the Coxeter groups , , , , , , , and (m odd).
For the case of the Coxeter groups , , and (m even), removing the even labeled edge from their associated graphs leaves two graphs of type A. So their reflections form two conjugacy classes.
We can see the numbers of reflections of the irreducible finite Coxeter groups in the book of Björner and Brenti [2, Appendix A1] for example. Using the Principle of Inclusion and Exclusion, applied to the irreducible maximal parabolic subgroups, and Pascal’s triangle in the form , we get the number of full support reflections of the irreducible finite Coxeter groups in Table 1.
We are now able to compute the Varchenko determinant of a finite Coxeter group by using Theorem 1.1. The ingredients necessary to calculate this determinant are given in Tables 2–3 for all irreducible finite Coxeter groups. They are obtained with tools of Table 1, those in [7, Propositions 2.3.8, 2.3.10, 2.3.13], and [7, Tables A.1, A.2].
Let . We write for the subset of having the following properties:
the elements of are the elements of such that if ; and
if , then .
computed by Randriamaro  with combinatorial methods.
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About the article
Published Online: 2018-04-10
Published in Print: 2018-07-01
This research was supported through the programme “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2017.