Xiang TANG Hsian-Hua TSENG

1 Introduction

Extensions of finite groups play an important role in the theory of finite groups.For example,the composition serious of a finite groupHconsists of a sequence of subgroupsHi

such thatHiis a strict normal subgroup ofHi+1with a simple quotient groupHi+1/Hi,fori=0,···,n−1.Therefore,with the classification theorem of finite simple groups,the study of extensions of finite groups would describe and classify all finite groups.

The structure of extensions of finite groups has been studied for a long time(see[7]).In this paper,we look at extensions of finite groups from a geometric point of view.A finite groupGis a groupoid with one unit.In the language of stacks(see[1]),such a group(oid)corresponds to the classifying stackBGof principalG-bundles.An extension of a finite groupQby a finite groupG

is equivalent to aG-gerbe

a bundle ofBGoverBQ(see[5]).

Our study of extensions of finite groups is motivated by a conjecture in mathematical physics(see[4]).Letbe the finite set of isomorphism classes of irreducible unitary representationsofG.The above extensionHofQbyGgives a natural action ofQon.Consider the transformation groupoidThere is a canonical classcinassociated to the extensionH.The decomposition conjecture in[4]suggests that the geometry of aG-gerbe associated to the extensionHis equivalent to the geometry of the orbifold associated to the groupoidtwisted byc.We studied this conjecture in[8]from the view point of noncommutative geometry.In particular,we proved that the group algebra ofHis Morita equivalent to thec-twisted groupoid algebra ofThe details of this are reviewed in Section 2.

In this short note,we present two results from our analysis of the structure of CH.One result concerns the relations between conjugacy classes ofHandQ(see Section 3).The other result concerns a generalized orthogonality relation between characters ofG(see Section 4).

2 Group Algebras of Finite Group Extensions

Consider an extension of finite groups as in

As part of our study of gerbe duality,the structure of the group algebra CHis analyzed in[8].We briefly recall the results.

Choose a sections:Q→Hofj:H→Qabove such thatj◦s=id,ands(1)=1.SinceGandQare finite groups,such a sectionsalways exists.Forq1,q2∈Q,defineτ(q1,q2):=s(q1)s(q2)s(q1q2)−1.It is easy to see thatτ(q1,q2)∈ker(j)=G,so we obtain

Clearlyτis trivial(i.e.,τ(−,−)=1)if and only ifs:Q→His a group homomorphism,which in turn is equivalent to the extension(2.1)being a split extension.

The definition ofτmay be written as

By associativity,we haveIt follows that

Given the sections,we can define a set-theoretic bijection betweenHandG×Q:

The inverse ofαis

The group structure onHinduces a new group structure·onG×Qviaα.This group structure is given by

where Adh(·)denotes the conjugation action of an elementh∈HonG,which is an automorphism ofGbecauseGis normal inH.Denote by

the setG×Qwith the group structure given by(2.4).The definition implies thatαis a group isomorphism:

It is easy to check that dif f erent choices of the sectionsyield isomorphic groups

The group isomorphismαnaturally induces an isomorphism of group algebras

Givensandτ,we let an elementq∈Qact on CGby conjugation bys(q).This does not give an action ofQon CG,and the failure of this to be an action is governed byτ.In other words,this de fines aτ-twisted action ofQon CG.Hence the group algebracan be written as a twisted crossed product algebra

Letbe the set of isomorphism classes of irreducible complex linear representations ofG.Furthermore,for every element[ρ]in,we choose an irreducible representation in the class[ρ]denoted by

whereVρis a certain finite dimensional C-vector space.The group algebra CGis isomorphic to a direct sum of matrix algebrasEnd(Vρ):

This is well-known(see e.g.[3,Proposition 3.29]).

Next we define an action ofQon.Letρ:G→End(Vρ)be a C-linear representation ofG.Givenq∈Q,we obtain anotherGrepresentationdefined by

It is easy to see thatis irreducible if and only ifρis.Ifis another section ofj,then we haveρ◦Sinceis an inner automorphism ofG.Henceρ◦Ads(q)andρ◦are isomorphicG-representations.Therefore the assignmentyields a rightQ-action on;namely,q∈Qsends the classto the classFor notational convenience,we write this right action as a left action.We denote the image of the isomorphism classunder the action byqby By abuse of notation,we denote the chosen irreducibleG-representation that represents the classalso byEndLet

be the groupoid associated to thisQ-action on.

By construction,the representationis equivalent to the representationdefined byTherefore there exists a C-linear isomorphism

that intertwines the two representations,namely

We may chooseto be the identity map onVρ.It can be shown that there are constantssuch thatistimes the identity map.In other words,

Since the collection{ρ}consists of unitary representations,the isomorphismsT[ρ]qcan also be chosen to be unitary.Therefore,c[ρ](q1,q2)actually takes value inU(1).By[8,Proposition 3.1],the function

is a 2-cocycle on the groupoidsuch thatfor anyThe cohomology class defined bycis independent of the choices of the sectionsand the operator

LetCbe the twisted groupoid algebra associated to the cocycleconWe explain the definition ofand refer the readers to[9]for more details.By definition,is the set of-valued functions onQ,i.e.,C-valued functions onBy abuse of notation,forwe also denote bythe function onwhich takes value 1 at([ρ],q)and 0 elsewhere.The collectionof functions onforms an additive basis ofThe setis endowed with a product structure defined by

The cocycle condition ofcimplies that this product is associative.

Letbe the C-vector space spanned by elements of the form(xρ,q),wherexρis an element in End(Vρ)withWe equip this space with a product◦defined as follows:

Let

be the spaceEndwith the product◦defined above.We call this the twisted crossed product algebra.This algebra plays an important role in the following structure result on the group algebra CH.

Proposition 2.1(see[8,Proposition 3.2])The map

de fines an algebra isomorphism from the group algebrato the twisted crossed productalgebraQ.Hence,is an algebra isomorphism.

Proposition 2.1 is used in[8,Section 3.2]to prove the following structure result of CH.

Theorem 2.1(see[8,Theorem 3.1])The group algebraCH is Morita equivalent to thetwisted groupoid algebra

We remark that the proof of Theorem 2.1 is done by explicitly constructing Morita equivalence bimodules between the two algebras.

Sincej:H→Qis a surjective group homomorphism,jinduces a surjective homomorphism of algebras from CHto CQ.It is well-known that the center of CQhas a canonical additive basis indexed by the conjugacy classes ofQ.This decomposition of the centerZ(CQ)and the surjection CH→CQimply that the center of CH,as a vector space,decomposes into a direct sum of subspacesindexed by conjugacy classes

As shown in[8,Section 3.2],the centerdecomposes into a direct sum of subspacesindexed by conjugacy classes ofQ,

The explicit Morita equivalence bimodules in the proof of Theorem 2.1 yield an algebra isomorphism from the center of CHto the center of,which we denote byI.

Proposition 2.2(see[8,Proposition 3.4])The isomorphism

is compatible with the decompositions into subspaces indexed by conjugacy classes of Q,i.e.,Iis an isomorphism from

In the rest of this paper,we discuss some group-theoretic applications of our analysis of the group algebra CH.

3 Counting Conjugacy Classes in Group Extensions

Letj:H→Qbe a surjective homomorphism of finite groups.Letbe a conjugacy class ofQ.The pre-imageHmay be partitioned into a disjoint union of conjugacy classes ofH.It is natural to ask the following question.

Question 3.1How many conjugacy classes ofHare contained in

In this section,we discuss an answer to this question.

LetGbe the kernel ofThen we are in the situation of the exact sequence(2.1).The homomorphisminduces a surjective homomorphism j:between group algebras.This,in turn,induces a homomorphism j:between centers.The centersviewed as vector spaces,admit natural bases,andindexed by conjugacy classes.These bases satisfy the requirement thatifthe mapis surjective.Let

By construction,the dimension dimis the number of conjugacy classes ofHthat are contained inBy Proposition 2.2,the isomorphismrestricts to an additive isomorphism

Clearly,the answer to Question 3.1 is the dimension dimwhich we now compute.

Letbe the subset consisting of elements fixedbe the centralizer subgroup ofq.Then,by[6],we have thatis additively isomorphic to thec-twisted orbifold cohomologyDecomposeinto a disjoint union ofC(q)-orbits:

For eachC(q)-orbitOi,pick a representative[ρi]and denote byQi:=StabC(q)([ρi])⊂C(q)the stabilizer subgroup of[ρi].Consider the homomorphism

Here,c[ρ](−,−)is the cocycle defined in(2.5).It follows from(3.1)that

By[6,Example 6.4],we have thatif the following condition holds:

Moreover,if(3.2)does not hold,thenIt follows that dimis equal to

In summary,we have obtained the following theorem as an answer to Question 3.1.

Theorem 3.1Letbe an extension of Q by G.Consider the canonicalquotient map j:H→Q.For q∈Q,the number of conjugacy classes of H that are mapped tothe conjugacy classis equal to

In the following,we discuss a few special cases of Theorem 3.1.

Example 3.1If the groupGis abelian,then all irreducible representations ofGare 1-dimensional,and all intertwiners in(2.5)can be taken to be the identity.In this case,(3.3)can be simplified into

Example 3.2If the groupGis abelian andHis a semi-direct product ofGandQ,then the cocycleτ(−,−)can be taken to be trivial.In this case,(3.3)can be simplified into

Example 3.3If theQ-action onis trivial1Equivalently,this means that the band of the gerbe BH→BQ is trivial.,thenand all intertwiners in(2.5)can be taken to be the identity.In this case,(3.3)can be simplified into

4 An Orthogonality Relation of Characters

The material in this section is inspired by the proof of the orthogonality relation given in[2,Chapter 2,Section 12].Using Proposition 2.1,we prove a generalization of the orthogonality relation between characters ofG.Forh∈H,write the centralizer subgroup ofhbyCH(h),and the number of elements inCH(h)by|CH(h)|.

Theorem 4.1be an extension of Q by G.be thecharacter of the G-representation Vρ.For(g1,g2)∈G×G,

ProofConsider(2.1)again.The groupH×Hacts naturally on the group algebra CHviaIn this way,we may view CHas a representation ofH×H.Its charactercan be calculated as follows:

We now consider CHas a representation of the subgroupG×G.The above calculation gives the character of this representation:

We calculate the characterby another method.By Proposition 2.1,there is an isomorphism of algebras

Under this isomorphism,theG×Gaction on CHis identified with the followingG×Gaction on

where◦is the algebra structure on

For eachρ, fix an isomorphism of End(Vρ)with a matrix algebra,and letdenote the standard basis of this matrix algebra.We use the symbol(xρ)stto denote thes,t-entry ofxρ∈End(Vρ).Then we haveTherefore,

wheredenote the characters of theG-representationsρandq([ρ]).Summing overandwe find that

Combining the above with(4.2),we obtain the desired identity:

AcknowledgementsTo the best of our knowledge,the results in this paper are new.We would like to thank I.M.Isaacs for discussions related to Question 3.1.

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