Coxeter multiarrangements with quasiconstant multiplicities. arXiv0708.3228

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

Coxeter multiarrangements with

quasi-constant multiplicities

Takuro Abe?Masahiko Yoshinaga?

September5,2007

Abstract

We study structures of derivation modules of Coxeter multiar-rangements with quasi-constant multiplicities by using the primitive

derivation.As an application,we show that the characteristic polyno-

mial of a Coxeter multiarrangement with quasi-constant multiplicity

is combinatorially computable.

1Introduction

Let V be an -dimensional Euclidean space over R with inner product I: V×V→R.Fix a coordinate(x1,···,x )and put S=S(V?)?R C= C[x1,...,x ].Let W?O(V,I)be a?nite irreducible re?ection group with

the Coxeter number h.It is proved by Chevalley in[2]that the invariant ring S W is a polynomial ring S W=C[P1,...,P ]with P1,...,P are homo-geneous generators.Suppose that deg P1≤···≤deg P .Then it is known that deg P1=2<deg P2≤···≤deg P ?1<deg P =h.Let A be the corresponding Coxeter arrangement,i.e.,the collection of all re?ecting hy-perplanes of W.Fix a de?ning linear formαH∈V?for each hyperplane H∈A.Let m:A→Z≥0be a map,called a multiplicity on A.Then the pair(A,m)is called a Coxeter multiarrangement.Let Der(S)denote the module of C-linear derivations of S.De?ne a graded S-module D(A,m)by D(A,m)={δ∈Der(S)|δαH∈(αH)m(H)for all H∈A}.

We say a multiarrangement(A,m)is free if D(A,m)is a free S-module. When(A,m)is free,we can choose a homogeneous basis{θ1,...,θ }for ?email:abetaku@math.sci.hokudai.ac.jp

?email:myoshina@math.kobe-u.ac.jp

1

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

D(A,m)and call the multiset(deg(θ1),...,deg(θ ))the exponents of a free multiarrangement(A,m)and denoted by exp(A,m),where the degree is the polynomial degree.The module D(A,m)was?rst de?ned by Ziegler([18]) and deeply studied for Coxeter multiarrangements with constant multiplicity by[11,13].In particular,Terao proved that if m is constant,then(A,m) is free and the exponents are expressed by using exponents of the Coxeter group and the Coxeter number h([13]).These facts played a crucial role in the proof of Edelman-Reiner conjecture([4,17]).

Another aspect of the above module is a relation with the Hodge?ltration of Der(S W)introduced by K.Saito in[8,9].It is proved in[14]that if m is a constant multiplicity with m=2k+1,then the S W-module D(A,m)W of all W-invariant vector?elds is precisely equal to the k-th Hodge?ltration of Der(S W).Based on these results,a geometrically expressed S-basis of the module D(A,m)for special kind of(not necessarily constant)multiplicities was constructed in[16].The purpose of this paper is to strengthen and generalize results in[13,16]by developing the“dual”version of[16].Indeed, we handle the following“quasi-constant”multiplicities.

De?nition1.A multiplicity m:A→Z≥0is said to be quasi-constant if max{ m(H)|H∈A}?min{ m(H)|H∈A}≤1.

It is clear that for a given quasi-constant multiplicity m,there exist an integer k and a{0,1}-valued multiplicity m:A→{0,1}such that m is either2k+m or2k?m.The above k∈Z≥0and m are uniquely determined unless m is the constant multiplicity with odd value.Our main results are concerning structures of derivation modules for Coxeter arrangements with quasi-constant multiplicities.

Theorem2.Let A be a Coxeter arrangement with the Coxeter number h and m:A→{0,1}be a{0,1}-valued multiplicity.Then

(1)D(A,2k+m)～=D(A,m)(?kh),

(2)D(A,2k?m)～=?1(A,m)(?kh),and

(3)The modules D(A,2k+m)(kh)and D(A,2k?m)(kh)are dual S-

modules to each other,

where M(n)denotes the degree shift by n for a graded S-module M.

Theorem2generalizes[13,16]in the following three parts.In[16],the isomorphism D(A,2k+m)～=D(A,m)(?kh)is proved for the case(A,m)is free.In Theorem2,the assumption on the freeness is removed.Furthermore,

2

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

considerations on?1(A,m)instead of D(A,m)enable us to treat multiar-rangements of the type(A,2k?m)as well(2).The structure of the module D(A,m)is not so much known when it is not bining Theorem2

(1)and(2),we have an interesting relation Theorem2(3),i.e.,there exists

a natural pairing between the modules D(A,2k+m)and D(A,2k?m).It may be simply said that a relation between multiplicities gives an algebraic relation between derivation modules.

The organization of this paper is as follows.In§2we review Terao’s result about the derivation modules of Coxeter arrangements with constant multiplicity in[13]from the viewpoint of the di?erential modules.In§3 we prove Theorem2(2)and the rest in§4.In§5we apply these results to compute characteristic polynomials for Coxeter multiarrangements with quasi-constant multiplicities.

2An interpretation of Terao’s basis

In this section,we recall the main result of[13]and give an interpretation through the dual basis for?1(A,m).Let us?rst recall the de?nition of ?1(A,m).

De?nition3.Put Q(A,m)=

H∈A

αm(H)

H

and denote by?1

V

=S?C V?=

i=1

S·dx i the module of di?erentials.De?ne

?1(A,m)=

ω∈

1

Q(A,m)

?1

V

dαH∧ωdoes not have poles

along H,for any H∈A

.

It is known that?1(A,m)is the dual S-module of D(A,m)and vice versa ([7],[18]).Next we de?ne the a?ne connection?.

De?nition4.For a given rational vector?eldδ=

i=1

f i?

?x i

and a rational

di?erential k-formω=

i1,...,i k

g i

1,...,i k

dx i

1,...,i k

,de?ne?δωby

?δω=

i1,...,i k δ(g i

1,...,i k

)dx i

1,...,i k

.

The above?de?nes a connection.We collect some elementary properties of?which will be used later.

Proposition5.For a rational vector?eldδ,rational di?erential formωand f∈S,?has the following properties.

??δf=δ(f).

3

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

??fδω=f?δω.

?Leibniz rule:?δ(fω)=f?δω+(δf)ω.

?For any linear formα∈V?,?δ(dα∧ω)=dα∧?δω.

Now we?x a generating system P1,...,P of the invariant ring S W= C[P1,...,P ]as in§1.Note that we may choose P1(x)=I(x,x).Then?

?P i (i=1,..., )can be considered as a rational vector?eld on V with order one

poles along H∈A.Especially,we denote D=?

?P and call it the primitive

derivation.Since deg P i<deg P for i≤ ?1,the primitive derivation D is uniquely determined up to nonzero constant multiple independent of the choice of the generators P1,...,P ([8,9]).

For any constant multiplicity m∈Z≥0,Terao showed the freeness of ?1(A,m)by constructing a basis.

Theorem6.[13,Theorem1.1]

(1)If m=2k,then

??

?x1?k D dP1,??

?x2

?k D dP1,...,??

?x

?k D dP1

forms a basis for?1(A,2k).

(2)If m=2k+1,then

??

?P1?k D dP1,??

?P2

?k D dP1,...,??

?P

?k D dP1

forms a basis for?1(A,2k+1).

Originally in[13]a basis for D(A,m)is constructed.The above expression is obtained just by switching to?1(A,m)through?.

3Main results

Lemma7.Letδ1,...,δ be rational vector?elds.Suppose that they are linearly independent over S.Then

?δ1?k D dP1,?δ2?k D dP1,...,?δ

?k D dP1

are linearly independent over S.

4

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

Proof.Putδi=

j=1

a ij?j,where?j=?

?x j

.Then linearly independence of

{δ1,...,δ }is equivalent to det(a ij)=0.Now the assertion is clear from Theorem6(1)and

?δi?k D dP1=

j=1

a ij??j?k D dP1.

Lemma8.The pole order of?k

D dP1is exactly equal to2k?1.More pre-

cisely,?k

D dP1∈1

Q(A,2k?1)

?1

V

andα2k?2

H

?k D dP1has a pole along H for any

H∈A.

Proof.First note that since

?k D dP1=??

?P

?k?1D dP1,

Theorem6implies that?k

D dP1∈?1(A,2k?1).Hence?k D dP1∈1

Q(A,2k?1)

?1

V

.

Suppose that there exists H∈A such thatα2k?2

H ?k D dP1does not have

poles along H.Let us de?ne the characteristic multiplicity m H by

m H(H )=

1if H =H, 0if H =H.

Then it is easily seen that?k

D dP1∈?1(A,2k?1?m H).Since??

?P j

increases

the pole order at most two,we have??

?P j

?k D dP1∈?1(A,2k+1?m H). However,this contradicts to Theorem6(2),for?(A,2k+1) ?1(A,2k+ 1?m

H

).

Remark9.Lemma8is a dual counterpart to[16,Lemma4].This property is related to the regularity of eigenvectors of the Coxeter element,which is of crucial importance in[8,9].

Let m:A→{0,1}be a{0,1}-valued multiplicity.The primitive deriva-tion and?enable us to compare D(A,m)and?1(A,2k?m).

Theorem10.Forδ∈D(A,m),Φk(δ):=?δ?k D dP1is contained in?1(A,2k?m).Furthermore,the map

Φk:D(A,m)(kh)?→?1(A,2k?m)

δ?→?δ?k D dP1

gives an S-isomorphism.

5

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

Proof.Since ?δincreases pole order at most one,from Lemma 8,?δ?k D dP 1∈1Q (A ,2k )?1V .Let H ∈A .Then α2k ?1H ·?k D dP 1has no poles along H .Thus ?δ α2k ?1H ·?k D dP 1 also has no poles along H .Suppose m (H )=1,and put δ(αH )=αH g .Then we have

?δ α2k ?1H ·?k D dP 1 =(2k ?1)α2k ?2H δ(αH )?k D dP 1+α2k ?1H ?δ?k D dP 1

=(2k ?1)α2k ?1H g ?k D dP 1+α2k ?1H ?δ?k D dP 1.

Hence α2k ?1H ?δ?k D dP 1has no pole along H .This shows that ?δ?k D dP 1∈1Q (A ,2k ?m )?1V

.Since dαH ∧?k D dP 1has no poles along H ,using Proposition 5,?δ(dαH ∧?k D dP 1)=dαH ∧?δ?k D dP 1also does not have poles along H .

This means Φk (δ)=?δ?k D dP 1∈?1(A ,2k ?m ).

Next we prove the injectivity.Let K be the ?eld of all rational functions.Since Φk is S -homomorphic,it can be extended to a K -linear map

Φk :D (A ,m )?S K ?

→?1(A ,2k ?m )?S K.Then Φk is isomorphic due to Lemma 7.Hence the induced map Φk is obviously injective.

Finally we prove the surjectivity.Let ω∈?1(A ,2k ?m ).Then clearly ω∈?1(A ,2k ).Hence from Theorem 6,there exists δ∈D (A ,0)= i S?i such that ω=?δ?k D dP 1.If m ≡0,there is nothing to prove.Otherwise,choose a hyperplane H ∈A such that m (H )=1.Then ?δ α2k ?1H ·?k D dP 1 =

(2k ?1)α2k ?2H δ(αH )?k D dP 1+α2k ?1H ωdoes not have poles along H .Hence α2k ?2H δ(αH )?k D dP 1does not have poles along H .From Lemma 8,δ(αH )has to be divisible by α

H .This shows that δ∈D (A ,m ).

4Conclusions

By using parallel arguments to §3in the context of [16],we can prove the following result.The notation is the same as above.

Theorem 11.Let m :A →{0,1}be a {0,1}-valued multiplicity and E = x i ?i be the Euler vector ?eld.Then for δ∈D (A ,m ),Ψk (δ):=?δ??k D E is contained in D (A ,2k +m ).Furthermore,the map

Ψk :D (A ,m )(?kh )?→D (A ,2k +m )

δ

?→?δ??k D E gives an S -isomorphism.

6

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

The action of?D shifts degree by?h.This proves the following results. Corollary12.For a{0,1}-valued multiplicity m:A→{0,1}and an inte-ger k>0,the following conditions are equivalent.

?(A,m)is free with exponents(e1,...,e ).

?(A,2k+m)is free with exponents(kh+e1,...,kh+e ).

?(A,2k?m)is free with exponents(kh?e1,...,kh?e ).

Remark13.The?rst condition in Corollary12is equivalent to say the subarrangement m?1(1)?A is free.For the Coxeter arrangement of type A,free subarrangements(A,m)are completely classi?ed in[12].See also[3].

Another conclusion is the following.

Theorem14.Let(A,m)be a Coxeter arrangement with a{0,1}-valued multiplicity m and k>0.Then D(A,2k+m)(kh)and D(A,2k?m)(kh) are dual S-module to each other.

bining Theorem10and11,we have the following isomorphisms of graded S-modules.

D(A,2k+m)(kh)～=D(A,m)～=?1(A,2k?m)(?kh).

Since?1(A,2k?m)～=D(A,2k?m)?,we have D(A,2k+m)(kh)～=?1(A,2k?m)(?kh)～=D(A,2k?m)(kh)?.

5Characteristic polynomials

In a recent paper[1],the characteristic polynomialχ((A,m),t)∈Z[t]for a multiarrangement(A,m)is de?ned.In this section,we apply results in the previous sections to compute the characteristic polynomials.Let us?rst recall the de?nition of the characteristic polynomial brie?y.

Let(A,m)be a multiarrangement of rank .Then the module D p(A,m) and?p(A,m)are de?ned for0≤p≤ (see Introduction of[1]and[18]), and de?ne functions

ψ(A,m;t,q)=

p=0

H(D p(A,m),q)(t(q?1)?1)p,

φ(A,m;t,q)=

p=0

H(?p(A,m),q)(t(1?q)?1)p,

7

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

in t and q,where H(M,q)is the Hilbert series of a graded S-module M.In [1],ψandφare proved to be polynomials in t and q and(?1) ψ(A,m;t,1)=φ(A,m;t,1).The characteristic polynomial of(A,m)is by de?nition

χ((A,m),t)=(?1) ψ(A,m;t,1)=φ(A,m;t,1).

Note that the above de?nition is a generalization of so called Solomon-Terao’s formula([10]),that is,χ((A,1),t)is equal to the combinatorially de?ned characteristic polynomialχ(A,t)of A([6]).

In general the computation of the characteristic polynomialχ((A,m),t), especially the constant term,is di?cult.One of the reasons is thatχ((A,m),t) is not a combinatorial invariant.However,we can compute it combinatorially for Coxeter multiarrangements with quasi-constant multiplicities.

Theorem15.Let A be a Coxeter arrangement with the Coxeter number h, and m:A→{0,1}be a{0,1}-valued multiplicity as in the previous sections. Let k∈Z>0.Then

(1)χ((A,2k+m),t)=χ((A,m),t?kh),and

(2)χ((A,2k?m),t)=(?1) χ((A,m),kh?t).

For the proof,we need the following lemmas.

Lemma16.Let m=(x1,...,x )?S be the graded maximal ideal of S. Let(A,m)be any multiarrangement.Then?p(A,m)is saturated in the

following sense,that is,ifω∈1

Q(A,m)?p

V

satis?es m·ω??p(A,m),then

ω∈?p(A,m).Similarly,ifδ∈Der p(S)satis?es m·δ?D p(A,m),then δ∈D p(A,m).

Proof.We may assume the coordinate(x1,...,x )is generic so that no coordinate hyperplane{x i=0}is contained in A.From the assump-tion,dαH∧x iωhas no poles along H,obviously,so does dαH∧ω.Hence ω∈?p(A,m).For D p(A,m)the proof is similar.

Lemma17.Let(A,m)be as in Theorem15.

D p(A,2k+2±m)～=D p(A,2k±m)(?ph),and

?p(A,2k+2±m)～=?p(A,2k±m)(ph).

Proof.We only give a proof for?p.The other case is immediate from the fact that D p and?p are dual S-modules to each other.

The case p=1is obvious from Theorem10and11.Put m =2k±m. Consider the coherent sheaf E p(A,m ):=

?p(A,m )on P ?1=Proj S cor-responding to the graded S-module?p(A,m )([5]).Recall that E p(A,m )

8

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

is known to be a re?exive O -module,and from Lemma 16?p (A ,m )can

be recovered from E p (A ,m )by taking the global section Γ?(E p (A ,m )):= d ∈Z Γ(P

?1,E p (A ,m )(d ))=?p (A ,m ).Let L (A )be the intersection lat-tice,and denote by L k (A )the set of intersections of codimension k .For X ∈L 2(A ),denote by X ?P ?1the corresponding ?at.Consider the open subset U =P ?1\ X ∈L 2(A )

X

with the inclusion i :U →P ?1.Since E p (A ,m )is re?exive,hence normal,we have i ?E p (A ,m )U ～=E p (A ,m ).Furthermore,since E p (A ,m )U is locally free on U ,we have

E p (A ,m )U ～=∧p E 1(A ,m )U .

Combining these facts,we have

E p (A ,m +2)=

i ?E p (A ,m +2)U =

i ? ∧p E 1(A ,m +2)U =

i ? ∧p E 1(A ,m )U ?O (h )U =

i ?(E p (A ,m )U ?O (ph )U )

=E p (A ,m )?O (ph ).

By taking the global section,we have ?p (A ,2k +2±m )～=?p (A ,2k ±m )(ph ).

Proof of Theorem 15.Let us prove (2).From Theorem 10and Lemma 17,we obtain the isomorphism ?p (A ,2k ?m )～=D p (A ,m )(pkh )of graded S -modules.Hence their Hilbert series are related by the relation

H (?p (A ,2k ?m ),q )=H (D p (A ,m ),q )q ?pkh .

From the de?nitions of φand ψ,

φ(A ,2k ?m ;t,q )=

p =0H (?p (A ,2k ?m ),q )(t (1?q )?1)p =

p =0H (D p (A ,m ),q )q ?pkh (t (1?q )?1)p =

p =0H (D p (A ,m ),q ){q ?kh (t (1?q )?1)}p ,

=ψ(A ,m ;q ?kh ?11?q

?q ?kh t,q ).9

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

Now we haveφ(A,2k?m;t,1)=ψ(A,m;kh?t,1)as q→1and obtain(2). The proof of(1)is similar.

Example18.Suppose A is de?ned by xyz(x+y)(y+z)(x+y+z),which is linearly isomorphic to the Coxeter arrangement of type A3and h=4.Let m:A→{0,1}be de?ned by m?1(1)=xyz(x+y+z).Thenχ((A,m),t)= t3?4t2+6t?3.Thus we have from Theorem15that

χ((A,2k+m),t)=(t?4k)3?4(t?4k)2+6(t?4k)?3

χ((A,2k?m),t)=(t?4k)3+4(t?4k)2+6(t?4k)+3.

Theorem15says that for any quasi-constant multiplicity m on a Coxeter arrangement A with the Coxeter number h,the formula

χ((A,m +2k+2),t)=χ((A,m +2k),t?h)

holds.Some computational examples show that similar formula holds for any multiplicity m :A→Z≥0,namely,supporting the following conjecture. Conjecture19.Let A be a Coxeter arrangement with the Coxeter number h.Let m:A→Z≥0be a multiplicity.Then there exists a constant N= N(A,m)such that

χ((A,m+2k+2),t)=χ((A,m+2k),t?h)

is satis?ed for any integer k>N.

Acknowledgment T.A.is supported by21st Century COE program “Mathematics of Nonlinear Structures via Singularities”Hokkaido Univer-sity.M.Y.is supported by JSPS Postdoctoral Fellowship for Research Abroad.This paper was begun when T.A.was staying at the Adbus Salam ICTP as a short time visitor.Both authors are grateful to the Abdus Salam ICTP for the hospitality.

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We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant mul

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Math.Soc.,Providence,RI,1989.

Takuro Abe

Department of Mathematics,

Hokkaido University,

Sapporo060–0810,Japan.

abetaku@math.sci.hokudai.ac.jp

Masahiko Yoshinaga

Department of Mathematics

Graduate School of Science

Kobe University

1-1,Rokkodai,Nada-ku,

Kobe657-8501,Japan

myoshina@math.kobe-u.ac.jp

12

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