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Journal of Group Theory

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Volume 21, Issue 4

Issues

The Varchenko determinant of a Coxeter arrangement

Götz Pfeiffer
  • School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Hery Randriamaro
Published Online: 2018-04-10 | DOI: https://doi.org/10.1515/jgth-2018-0009

Abstract

The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.

1 Introduction

Let x=(x1,,xn) be a variable of the Euclidean space n, and a1,,an,b real coefficients such that (a1,,an)(0,,0). A hyperplane H of n is an (n-1)-dimensional affine subspace H:={xna1x1++anxn=b}. An arrangement of hyperplanes in n is a finite set of hyperplanes. For example, the best known hyperplane arrangement is certainly

𝒜An-1={{xnxi-xj=0}}1i<jn

associated to the Coxeter group An-1.

A chamber of a hyperplane arrangement 𝒜 is a connected component of the complement nH𝒜H. Denote the set of all chambers of 𝒜 by (𝒜).

Assign a variable aH to each hyperplane H of an arrangement 𝒜. Let

R𝒜=[aHH𝒜]

be the ring of polynomials in variables aH. The module of R𝒜-linear combinations of chambers of the hyperplane arrangement 𝒜 is

M𝒜:={C(𝒜)xCCxCR𝒜}.

Let (C,D) be the set of hyperplanes separating the chambers C and D in (𝒜). The R𝒜-bilinear symmetric form 𝖡:M𝒜×M𝒜R𝒜 on the hyperplane arrangement 𝒜 defined by Varchenko [12] is

𝖡(C,C):=1,and𝖡(C,D):=H(C,D)aH if CD.

The matrix (𝖡(C,D))C,D(𝒜) associated to the bilinear symmetric form 𝖡 is called the Varchenko matrix of the hyperplane arrangement 𝒜. In terms of Markov chains, it is the matrix of random walks on (𝒜) whose walk probability from the chamber C to the chamber D is equal to 𝖡(C,D). The Varchenko determinant of the hyperplane arrangement 𝒜 is the determinant

det𝒜:=det(𝖡(C,D))C,D(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 -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 |0, and a collection of operators ak: satisfying the commutation relations

a(l)a(k)-qa(k)a(l)=δk,l,

and the relation a(k)|0=0. To demonstrate the realizability of its model, he defined an inner product space ((q),,) with basis B consisting of n-particle states a(k1)a(kn)|0, and proved that [13, Theorems 1, 2]

det(u,v)u,vB=k=1n-1(1-qk2+k)n!(n-k)k2+k.

It is the Varchenko determinant of 𝒜An-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]

det𝒜An-1=I2[n]|I|2(1-{i,j}(I2)aHi,j2)(|I|-2)!(n-|I|+1)!,

each hyperplane {xnxi-xj=0} having its own weight aHi,j this time.

An edge of a hyperplane arrangement 𝒜 is a nonempty intersection of some of its hyperplanes. Denote the set of all edges of 𝒜 by L(𝒜). The weight 𝖺(E) of an edge E is

𝖺(E):=H𝒜EHaH.

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

Due to Varchenko [12, Theorem 1.1], the formula for the determinant of a hyperplane arrangement 𝒜 is

det𝒜=EL(𝒜)(1-𝖺(E)2)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 𝖺(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 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:=s1,s2,,sn(sisj)mij=1with mii=1,mij2,mij=mji.

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

The elements of the set S:={s1,s2,,sn} are called the simple reflections of W. The set of all reflections of W is denoted by T:={sixsiS,xW}.

As usually, for a subset J of S, WJ is the parabolic subgroup J of W, TJ the set of reflections TWJ of WJ, XJ the set of coset representatives of minimal length of WJ, 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 WJ is irreducible.

For a subset K of J, we write NWJ(WK) for the normalizer of WK in WJ, and X(J,K) for the set of double coset representatives

X(J,K):={wWJXKXK-1Kw=K}.

The normalizers of the parabolic subgroups were determined by Howlett [8]. He also proved that NWJ(WK)=WKX(J,K), see [8, Corollary 3].

Let si1sil be a reduced expression of an element x of W. The support of x is

J(x):={si1,,sil}

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

x:={yWJ(y)=J(x) and y is conjugate to x}.

Now, we come to the hyperplane arrangement associated to a finite Coxeter group called Coxeter arrangement. Let Ht be the hyperplane ker(t-1) of n whose points are fixed by each element t of T. The hyperplane arrangement associated to the finite Coxeter group W is 𝒜W:={Ht}tT. In this case, explicit formulas for 𝖺(E) and l(E) can be given, and it is the aim of this article. For a subset U of T, we write EU for the edge

EU:=uUHu.

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

Theorem 1.1.

Let E be an edge of AW. Then l(E)0 if and only if there exist an irreducible subset J of S and an element w of W such that E=ETJw. In this case, we clearly have a(ETJw)=uTJaHuw. Moreover, let tJ be a reflection with support J, sJ a simple reflection and v an element of W such that a reduced expression of tJ is sJv. Then

l(ETJw)=|tJ||[J]||X(S,J)||X(J,{sJ})|.

Let YJ be the set of the cosets of NW(WJ), and (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

[J](S)wYJ(1-uTJaHuw2)|tJ||[J]||X(S,J)||X(J,{sJ})|.

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=W1W2, where W1 and W2 are two irreducible Coxeter groups, then

det𝒜W=(det𝒜W1)|W2|(det𝒜W2)|W1|.

Recall that there is a one-to-one correspondence between the elements of W and the chambers of 𝒜W such that: if the chamber C corresponds to the neutral element e and Cx to another element x, then Cx=Cx with x=t1tr, the Hti’s being the hyperplanes one goes through from C to Cx, see [1, Theorem 1.69].

Let C¯xHt be the subspace generated by the closed face C¯xHt of the chamber Cx. Determining the multiplicity of the edge E contained in the hyperplane Ht consists of counting the half of the chambers Cx which have the property that E is the minimal edge containing C¯xHt. 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 A. Then l(E)0 if and only if there exist an irreducible subset J of S and an element w of W such that E=ETJw.

Let us introduce the set L(E,t):={xWC¯xHt=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 sv. For a conjugate K of J, let cK,J be an element of W such that KcK,J=J, and for a conjugate u of t with support J, let cu,t be an element of W such that ucu,t=t. Then,

L(ETJ,t)=K[J]cK,JX(S,J)(utcu,tNWJ(W{s})v).

Proposition 1.4.

Let u,vT and let E be an edge contained in both Hu and Hv. Then

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

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 l(E)0.

Lemma 2.1.

A finite Coxeter group W is irreducible if and only if W has a reflection of full support.

Proof.

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 W1 and W2, 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 W1, hence t lies in W1 and can not have full support. ∎

We continue our investigation by using the Coxeter complex. Recall that the Coxeter complex 𝒞 of W is the set {WJxJS,xW} of faces. The Coxeter complex is a combinatorial setup which permits to study the geometrical structure of 𝒜W. Indeed, n is partitioned by 𝒞, see [9, Section 1.15].

The chamber Cx is identified with the singleton {x}. The coset WJw is a face of {x} if and only if WJw=WJx. Then the closure of Cx is

C¯x:={WJxJS}.(2.1)

More generally, WKw is a face of WJx if and only if WJxWKw and WKw=WKx. Then the closure of a face WJx of Cx is

WJx¯:={WKxJKS}.(2.2)

Hyperplanes, and more generally edges, can be described as collections of the faces they consist of. We extend the definition of J(x) to a subset X of W in the following way:

J(X):=xXJ(x).

Lemma 2.2.

Let tT. Then Ht={WJwJS,wW,J(tw-1)J}.

Proof.

As W acts by right multiplication, WJwHt if and only if WJwt=WJw, i.e. WJwt=WJw, i.e. tWJw, i.e. tw-1WJ, i.e. J(tw-1)J. ∎

Lemma 2.3.

Let xW and tT. Then, C¯xHt=WJ(tx-1)x¯.

Proof.

We have

C¯xHt={WJxJS}{WJwJS,wW,J(tw-1)J}={WJxJ(tx-1)J}=WJ(tx-1)x¯,

the first equality being a consequence of (2.1) and Lemma 2.2, and the last equality of (2.2). ∎

Lemma 2.4.

Let WJx be a face of the Coxeter complex. Then, the subspace WJx generated by WJx is the edge ETJx.

Proof.

It is clear that the subspace generated by WJx¯ is equal to the subspace generated by WJx. From the proof of Lemma 2.2, we know that Ht being a hyperplane containing WJx is equivalent to J(tx-1)J which is equivalent to tTJx. ∎

Let us take a reflection t of U. Recall that l(EU) is half the number of chambers Cx which have the property that EU is the minimal intersection containing C¯xHt. Minimality implies equality, that is, for the chamber Cx to be counted, we must have C¯xHt=EU, i.e. (by Lemma 2.3)

WJ(tx-1)x¯=EU,

i.e. (by Lemma 2.4)

ETJ(tx-1)x=EU.

Moreover, since tx-1 is a full support reflection of the group WJ(tx-1), we know from Lemma 2.1 that J(tx-1) is an irreducible subset of S. So EU is of the form EJx, where J is irreducible, otherwise l(EU)=0. That proves Proposition 1.2.

3 The chambers to consider

For a relevant edge E and a hyperplane Ht containing E, we determine the chambers Cx such that C¯xHt=E.

Lemma 3.1.

Let J be an irreducible subset of S, t a reflection with support J, and E the relevant edge ETJ. For each K in the Coxeter class of J, let cK,J be the coset of minimal length of NW(WJ) such that KcK,J=J. Then,

L(E,t)=K[J]cK,J{xNW(WJ)J(tx-1)=J}.

Proof.

We have

L(E,t)={xWC¯xHt=E}={xWWJ(tx-1)x¯=E}={xWETJ(tx-1)x=ETJ},

the second and third equalities being consequences of Lemma 2.3 and Lemma 2.4, respectively. Denoting J(tx-1) by K, we see that the equality ETJ(tx-1)x=ETJ means that

  • |J(tx-1)|=|J|,

  • K[J], and

  • xcK,JNW(WJ).

Hence

L(E,t)=K[J]{xcK,JNW(WJ)|J(tx-1)|=|J|}.

Let x=cK,Jy with yNW(WJ). Note that |J(tx-1)|=|J| if and only if |J(ty-1)|=|J| if and only if J(ty-1)=J. We thus obtain the result. ∎

Recall that X(S,J):={xXJJx=J}. We introduce the set

WJ(t):={xWJJ(tx-1)=J}.

Lemma 3.2.

Let J be an irreducible subset of S. Consider a reflection t of WJ with support J. Then,

{xNW(WJ)J(tx-1)=J}=X(S,J)WJ(t).

Proof.

We have NW(WJ)=WJX(S,J), see [8, Corollary 3]. So

{xNW(WJ)J(tx-1)=J}  ={yzyWJ,zX(S,J),J(tz-1y-1)=J}.

Since we always have tz-1=t, the remaining condition is J(ty-1)=J. ∎

We write C(x) for the centralizer of the element x of W.

Lemma 3.3.

Let J be an irreducible subset of S. Consider a reflection t=sv with support J where s is a simple reflection and v an element of WJ. For another reflection u of WJ with support J and conjugate to t, let cu,t be the coset of minimal length of C(t) such that ucu,t=t. Then,

WJ(t)=utcu,tNWJ(W{s})v.

Proof.

The equality J(tx-1)=J means that x-1C(t)ct,u or xcu,tC(t), where u is a conjugate of t with support J. Then,

WJ(t)=utcu,tC(t).

Since C(t)=C(s)v and C(s)=NWJ(W{s}), we get the result. ∎

Proposition 1.3 is a combination of Lemma 3.1, Lemma 3.2, and Lemma 3.3.

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.

Lemma 4.1.

Consider two hyperplanes Hu and Hv of AW associated with the orthogonal vectors αu and αv, respectively, of equal length and from the positive root system. Let r be the rotation of Rn, not necessarily in W, which transforms αu to αv. Then, r(AW)=AW.

Proof.

Let Eu,v be the 2-dimensional subspace αu,αv:

  • on Eu,v, the map r is the rotation of angle θ=arccosαu.αvαuαv,

  • on Eu,v, the map r is the identity map.

We have 𝒜W=𝒜1𝒜2 with

𝒜1:={H𝒜Eu,vH}and𝒜2:={H𝒜dimEu,vH=1}.

  • For all H in 𝒜1, we have r(H)=H.

  • Let p be the projection on the subspace Eu,v. The arrangement p(𝒜2) is the arrangement of a dihedral group whose angle between two certain hyperplanes is θ. Then, r(p(𝒜2))=p(𝒜2). Hence, for any H in 𝒜2, we have

    r(H)=r(p(H)Eu,v)=r(p(H))Eu,v

    which still belongs to 𝒜2.∎

We prove Proposition 1.4 now. Consider two hyperplanes Hu and Hv containing the edge E. Let αu and αv be the unit vectors of the positive root system associated to W which are orthogonal to Hu and Hv, respectively. We use the rotation r of Lemma 4.1 transforming αu to αv, and leaving 𝒜 invariant. For a given w in L(E,u), we have

C¯wHu=C¯wE(C¯wHu)r=(C¯wE)r(C¯w)r(Hu)r=(C¯w)r(E)r(C¯w)rHv=(C¯w)rE.

Hence L(E,u)r=L(E,v), and |L(E,u)|=|L(E,v)|, which is Proposition 1.4.

5 The multiplicity of an edge

We establish a formula for the multiplicity of a relevant edge of 𝒜W 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 ETJw with the condition that J is irreducible. The weight of ETJw is obviously uTJquw so that the real problem concerns l(ETJw). Fixing a hyperplane Ht containing ETJw, we see that the set of chambers taken into account to determine l(ETJw) is L(ETJw,t). But Proposition 1.4 allows us to choose any hyperplane containing ETJw. Hence the multiplicity of ETJw is

l(ETJw)=12|L(ETJw,t)|.

Lemma 5.1.

Let J be an irreducible subset of S, and w an element of W. Then,

L(ETJw,tw)=L(ETJ,t)w.

Proof.

Let xL(ETJ,t). We have

C¯xHt=C¯xETJ(C¯xHt)w=(C¯xETJ)w(C¯x)w(Ht)w=(C¯x)w(ETJ)wC¯xwHtw=C¯xwETJw.

Then

L(ETJ,t)wL(ETJw,tw).

With the same argument, we obtain also L(ETJw,tw)w-1L(ETJ,t). Hence L(ETJw,tw)=L(ETJ,t)w. ∎

We see in Lemma 5.1 that we just need to investigate L(ETJ,t) for each Coxeter class [J]. From Proposition 1.3, we get

l(ETJ)=12|L(ETJ,t)|=12|K[J]cK,JX(S,J)(utcu,tNWJ(W{s})v)|=|t||[J]||X(S,J)||X(J,{s})|

since

12|NWJ(W{s})|=|X(J,{s})|,

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 Γ=(S,M) 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 {si,sj} such that mij3, and these edges {si,sj} are labeled by mij.

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 An-1, Dn, E6, E7, E8, H3, H4, and I2(m) (m odd).

For the case of the Coxeter groups Bn, F4, and I2(m) (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 (n2)-(n-12)=n-1, we get the number of full support reflections of the irreducible finite Coxeter groups in Table 1.

Table 1

Number of full support reflections of the irreducible finite Coxeter groups.

Table 2

Multiplicities of the Coxeter classes – part 1.

Table 3

Multiplicities of the Coxeter classes – part 2.

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 23 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 [±n]:={-n,,-2,-1,1,2,,n}. We write 2[±n]¯ for the subset of 2[±n] having the following properties:

  • the elements of 2[±n]¯ are the elements {i1,,it} of 2[±n] such that |ir||is| if rs; and

  • if {i1,,it}2[±n]¯, then {-i1,,-it}2[±n]¯.

Using Theorem 1.1 and Tables 23, we refind, for example, the determinants

det𝒜Bn=J2[±n]¯|J|2(1-{i,j}(J2)aHi,j2)2n-|J|+1(|J|-2)!(n-|J|+1)!  ×I2[n]|I|1(1-iIaHi2{i,j}(I2)aHi,j2aH-i,j2)2n-1(|I|-1)!(n-|I|)!,

computed by Randriamaro [10] with combinatorial methods.

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About the article


Received: 2017-11-13

Revised: 2018-03-07

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.


Citation Information: Journal of Group Theory, Volume 21, Issue 4, Pages 651–665, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgth-2018-0009.

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