The Poincaré Series of Relative Invariants of Finite Pseudo-reflection Groups in Finite Fields

QIN Xiaoer, YAN Li

(College of Mathematics and Computer Science, Yangtze Normal College, Chongqing 408100)

1 Introduction

Invariants theory is an important branch of algebra. In 1890, using the Hilbert basis theorem, Hilbert[1]proved the invariants ofGLn(C) is finitely generated. In 1955, for every finite reflection group, Chevalley[2]got that each invariant of finite reflection groups can be represented by the polynomial of elementary invariants, and this result was extended to finite pseudo-reflection groups. Poincaré series is an algebraic invariant. In 1897, Molien[3]gave a formula to compute the Poincaré series of the general linear groups. The invariants and relative invariants have relations with Poincaré series, such as the coefficients of Poincaré series are the dimensions of the invariants corresponding degrees. The relative invariants of finite pseudo-reflection groups are similar to the invariants of finite pseudo-reflection groups, and the relative invariants of finite pseudo-reflection groups have some relation with the 1-dimension representation of the groups. Wan[4]introduced the invariants and relative invariants theory of finite reflection groups. Smith[5]gave the relation between invariants and relative invariants of finite pseudo-reflection groups. Nan and Qin[6]did some researches on relative invariants of finite pseudo-reflection groups in the general fields, and computed the Poincaré series of relative invariants of finite pseudo-reflection groups. Recently, invariants of groups have been an interesting subject of study. Nan and his students[7-11]did many researches on this topic, and more and more scholars begin to study invariants, we refer the readers to [12-15].

LetFqbe a finite field withq=pm,m≥1, andVbe then-dimensional vector space overFq. The pseudo-reflection and the reflecting hyperplane are defined as follows

σ∈GL(V),H={ξ∈V|σξ=ξ}.

If dimH=n-1, thenσis called a pseudo-reflection, and subspaceHis called the reflecting hyperplane ofσ. A vectorv≠0 in Im(σ-1) is called a reflecting vector ofσ.

For convenience, we always supposeGis a finite pseudo-reflection group that is generated by the fundamental pseudo-reflectionss1,s2,…,sn,Fqdenotes a fixed finite field with characteristicp, unless the contrary is explicitly stated.σhas finite order,pdoes not divide the order ofσ(which we shall call the nonmodular case), thusσmust be diagonalizable.

2 The 1-dimensional representation of finite pseudo-reflection groups in finite field

We can now give the first main result of this paper.

Theorem2.1LetPbe aχ-relative invariant of the groupG, i.e. for eachσ∈G,σ·P=χ(σ)P,P≠0,Fqbe a finite field andGbe a finite pseudo-reflection group. Forσ∈G, let |σ|=r. Ifr|q-1, thenχ(σ)=1 orχ(σ)=(detσ)α, where 1≤α≤r-1.

ProofLetUbe a reflecting hyperplane of a pseudo-reflectionσ, letGU=〈σ〉,|σ|=r. Take a basisε1,ε2,…,εn, such that

σi(εj) =εj, 1≤j≤n-1,

σ·P=χ(σ)P,

Suppose that

then

which is equivalent to

Sincer|q-1, comparing coefficients of thexn, we get that

χ(σ)=1 orP0=0;

χ(σ)ξσ=1 orP1=0;

……

0≤m1,m2≤r-1,

0≤α≤r-1.

This completes the proof of Theorem 2.1.

In what follows we shall characterize the relation between theχ-relative invariants and invariants ofG. LetH(G)={Hs|s∈G} denote the set of reflecting hyperplanes of all pseudo-reflections inG,

Hs={λ∈V|ls(x1,x2,…,xn)(λ)=0}

is defined byls(x1,x2,…,xn)=0, wherels(x1,x2,…,xn)=0 is a homogeneous linear polynomial. IfU∈H(G) is a reflecting hyperplane ofG, denotesGUthe pointwise stabilizer ofUinG. This is the group generated by all the pseudo-reflections inGwithUa reflecting hyperplane together with 1. For everyU∈H(G), chooseaU∈Nminimal such that

χ(sU)=det(sU)aU

and introduce the form

In the following, we shall show that

divides everyχ-relative invariant ofG. We need the following lemmas.

If none ofl1,l2,…,lkis nonzero multiples ofls, thenα1α2…αk=1 andL=α1α2…αkis a invariant ofs.

Writing

Thus

i.e.

is aχ-relative invariant.

By Lemma 2.5, we can make the conclusion that the difference between relative invariants and invariants is only one divisor

3 The Poincaré series of relative invariants of finite pseudo-reflection groups in finite fields

Fq[V*] is a gradedFq-algebra, the Poincaré series ofFq[V*] is defined as follows

whereFq[V*]dis aFq-subspace consisting of all homogeneous polynomial functions of degreedinFq[V*]. For the finite subgroup of the general linear group, its Poincaré series of invariants can be characterized by Molien’s Theorem. In what follows, we give the second main result of this paper.

Theorem3.1LetVbe a finite dimensionFqvector space. LetG∈GL(V) be a finite nonmodular subgroup. Ifpdoes not divide |G|, then

we defineσ·fas

then

Supposeλ1(σ-1),λ2(σ-1),…,λn(σ-1) are the eigenvalue of the linear tranformationσ-1, then

Sinceλi(σ-1)=λi(σ)-1,i=1,2,…,n,

Thus

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