Chen Yang and Nan Ji-zhu

(School of Mathematical Sciences,Dalian University of Technology,Dalian,Liaoning,116024)

Communicated by Du Xian-kun

Abstract:In this paper,we describe the variety defined by the twisted transfer ideal.It turns out that this variety is nothing but the union of reflecting hyperplanes and the fixed subspaces of the elements of orderpinG.

Key words:invariant theory,twisted transfer,twisted transfer variety

1 Introduction

Let V be a vector space of dimension n over a field k,G a finite linear group of V.The induced action on the dual V?extends to the polynomial ring k[V]which is given by

The ring of invariants of G is the subring of k[V]given by

The transfer defined by

is a k[V]G-module homomorphism.It is surjective if and only if the characteristic of k does not divide the order of G,i.e.,in the nonmodular case.It provides a tool for constructing the ring of invariants.When the characteristic of k is a prime p and divides the order of G,i.e.,in the modular case,the image of the transfer,Im(TrG),is a proper non-zero ideal in k[V]G(see Theorem 2.2 of[1]).We may extend Im(TrG)to an ideal(Im(TrG))ein k[V]in the usual way.This extended ideal is also called the trace ideal and defines an algebraic set.

Definition 1.1[2]Let ρ:G,→ GL(n,k)be a representation of a finite group over the field k.The transfer variety,denoted by V((Im(TrG))e)?V,is defined by

where Tot(k[V])=⊕k[V]idenotes the totalization of k[V]throwing away the grading.

Remark 1.1Since V((Im(TrG))e)is an affine variety,we must use all polynomial functions to define it,and not just homogeneous ones.

The trace ideal defines exactly the wild ramification locus as shown first by Auslander and Rim[3]and first noted in the context of invariant theory by Feshbach[4].Kuhnigk and Smith[5]reproved Feshbach’s result and in addition described the transfer variety in the unpublished manuscript.It turns out that this variety has a particularly elegant description.Namely,it is the union of the fixed point sets of the elements of order p in G,where p is the characteristic of k.We refer the reader to[1]and[2]for the transfer variety.

Broer[6]defined a twisted transfer map and gave a characterisation of the direct summand property in terms of the image of twisted transfer maps.The direct summand property holds if and only if the image of the twisted transfer is a cyclic graded k[V]G-module.He conjectured that the invariant ring k[V]Gis a polynomial ring if G is generated by pseudoreflections,and k[V]Gis a direct summand of k[V].It seems worthwhile therefore to study the twisted transfer in modular invariant theory.This is what we do in this article.

We begin this paper by introducing the twisted transfer ideal.We then go on to describe the twisted transfer variety and obtain a decomposition of the radical of the twisted transfer ideal.

2 The Twisted Transfer Ideal

Let R?S be an integral extension of integral domains such that the extension of their quotient fields K?L is finite and separable.Multiplication by an element y∈L is a linear map of the finite dimensional K-vector space L.We write its trace as TrL/K(y).Then the symmetric K-bilinear form

is nondegenerate(see Lemma 3.7.2 of[7]).Classically,the(Dedekind)inverse different or complementary module is defined by

The(Dedekind)different is then

Broer[8]defined the twisted trace ideal as

In invariant theory,stronger hypotheses are satisfied for our extensions,so we are able to say more.In particular,we restrict ourselves to the following situation.Let K?L be a Galois extension with Galois group G.Then

Furthermore,we demand that R is a graded normal Noetherian domain and S is the integral closure of R in L and factorial.Then the different is a graded principal ideal,sayand the inverse different is the graded principal fractional ideal

where θ is a homogeneous generator ofThere is a group homomorphism χ:G → k such that

or in other words θ is a χ-semi-invariant.We call χ the differential character.The degree δ of the homogeneous generator θ is called the differential degree.When=Sθ,the twisted trace can be defined as the R-homomorphism

The twisted trace ideal is generated by the image of the twisted trace,

We can calculate the different locally as follows.

Proposition 2.1([7],Proposition 3.10.3) Suppose that K?L is a Galois extension with Galois group G.If P is a height one prime in S with inertia group GP,then

where vPis the discrete valuation associated with P.

From now on we assume that S:=k[V]and R:=k[V]G.In particular,the trace is just the ordinary transfer of invariant theory.As the twisted trace we define the twisted transfer TrGχ:k[V]→ k[V]Gθ by the formula

It is a homomorphism of graded k[V]G-modules.The twisted transfer ideal is

In the nonmodular case,Reynolds operator is a projection operator of k[V]G-modules.In other words,k[V]Gis a direct summand of k[V]as a k[V]G-module.Broer[6]studied the direct summand property in modular invariant theory.

Theorem 2.1([6],Theorem 1) Let R?S be an integral extension of integral domains such that the extension of their quotient fields K?L is finite and separable.Then R is a direct summand of S as an R-module if and only if

3 The Twisted Transfer Variety

By a pseudoreflection we mean a nontrivial linear transformation leaving all points of a hyperplane fixed,a transvection means a nondiagonalisable pseudoreflection.These can only exist in positive characteristic.A linear subspace is a reflecting hyperplane if and only if it has codimension 1 in V and its isotropy subgroup is nontrivial.

The same as the transfer variety,we define the twisted transfer variety in the following.

Definition 3.1Let ρ:G,→ GL(n,k)be a representation of a finite group over the field k.The twisted transfer variety,denoted byis defined by

Lemma 3.1Let ρ:G,→ GL(n,k)be a representation of a finite group over the field k and x∈V.Then for any f∈k[V]of positive degree we have

where Gx≤G is the isotropy group of x.

Proof.Choose elements g1,···,gm∈ G such that are simultaneously a left and right transversal for Gxin G.Then···,is also a left transversal for Gxin G and

by the definition of the relative twisted transfer.

Lemma 3.2Let G be a finite group and k a field.Let χ:G → k×be a linear character,where k×denotes the multiplicative group of k.Then

where 0kis the zero element in k,and|G|is the order of the group G.

Proof. First assume that χ is trivial,i.e.,for all g ∈ G we have χ(g)=1 ∈ k.Then the result is clear.Thus,let us assume that there exists g′in G such that χ(g′)=1 ∈ k.Notice that for any element g′∈ G the map

is clearly a bijection.Then

Since χ(g′)=1 and k is a fi eld,it follows that

as desired.

Proposition 3.1Let ρ :G,→ GL(n,k)be a representation of a finite group over the field k of characteristic p.Then a point x∈V belongs to the twisted transfer variety if and only if Gxcontains a pseudoreflection or p divides|Gx|.

Proof. If x is fixed by a pseudoreflection,then θ(x)=0 by Proposition 2.1.It follows thatIf p divides|Gx|,then x is fixed by a element g of order p.Since χ(g)=1,we have

On the other hand,suppose that x is not fixed by any pseudoreflection in G.Then θ(x)=0.For any g ∈ Gx,

Then χ(g)=1.However,since the orbit Gxis finite,there exists an f ∈ k[V]such that f(x)is nonzero and

Suppose that k[V]=k[X1,···,Xn].Then one such function is given by

where i(w)is the least i such that Xi(x)=Xi(w).Note that f is not homogeneous and

Therefore,if p does not divide|Gx|,we conclude that

Corollary 3.1Let ρ:G,→ GL(n,k)be a representation of a finite group over the field k of characteristic p.Then

in other words,the twisted transfer variety is the union of the fixed point sets of the pseudoreflections and the elements of order p in G.

Let g∈GL(n,k)and define the element?g=1?g∈Matn,n(k),the algebra of n×n matrices over k.The element?gacts on the algebra k[V]as a linear twisted differential,i.e.,

Let Ig? k[V]denote the ideal generated by?g(V?),where as usual V?,the space of linear,forms on V=kn.

Lemma 3.3([2],Lemma 2.4.1) Let g ∈ GL(n,k)and Im(?g)the image of the action of?gon k[V].Then Im(?g)? Ig.

Lemma 3.4([2],Lemma 6.4.1) Let g∈GL(n,k)and V(Ig)?V the affine variety defined by the ideal Ig?Tot(k[V]).Then V(Ig)=Vg,where Vg?V is the fixed point set of the element g acting on V.

Theorem 3.1Let ρ:G,→ GL(n,k)be a representation of a finite group over the field k of characteristic p.Then

and the ideals Ig,where g is a pseudoreflection or|g|=p,are prime.Hence

Proof. Since the hypotheses and conclusion are not altered by passage to a field extension we may suppose that k is algebraically closed,where Hilbert’s Nullstellensatz holds.By Corollary 3.1,passing back to ideals leads to the equality

Remark 3.1For the trivial different character,the twisted transfer is just the transfer and θ is a invariant.For example,G is generated by the p-order elements over the field k of characteristic p.

Remark 3.2When k[V]Gis polynomial,we haveHence,the twisted transfer variety is union of reflecting hyperplanes.

Example 3.1We refer to Nakajima Notation 4.2 of[9]for the representation of symmetric group.Let G=S5be the symmetric group in its 3-dimensional irreducible 5-modular representation obtained as the nontrivial constituent of the permutation module.Suppose that k is the prime field of characteristic 5 and n=3.Letbe vector spaces with natural kS5-module structure,where S5is the symmetric group of degree 5.Let:S5→ GL5(k)(resp.F′:S5→ GL4(k))be the matrix representation of S5on the basis{e1,e2,e3,e4,e5}(resp.{e2?e1,e3?e1,e4?e1,e5?e1})and putIm(resp.G′=Im(F′)).Let

We denote by G the subgroup of GL3(k),

Furthermore,