On Schur multiplier and projective representations of Heisenberg groups

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Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

On Schur multiplier and projective representations of Heisenberg groups

Sumana Hatui^a,∗, Pooja Singla^b

a DepartmentofMathematics,IndianInstituteofScience,Bangalore560012,India

bDepartmentofMathematicsandStatistics,IndianInstituteofTechnologyKanpur,Kanpur208016, India

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received11November2019 Receivedinrevisedform12 November2020

Availableonline13March2021 CommunicatedbyA.Solotar

MSC:

20C25;20G05;20F18

Keywords:

Schurmultiplier

Projectiverepresentations Representationgroup Heisenberggroup

Inthisarticle,wedescribetheSchurmultiplierandrepresentationgroupofdiscrete Heisenberggroupsandtheirt-variants.Wegiveaconstructionofallcomplexfinite- dimensionalirreducibleprojectiverepresentationsofthesegroups.

©2021ElsevierB.V.Allrightsreserved.

1. Introduction

Thetheory ofprojective representationsinvolves understandinghomomorphismsfrom agroupinto the projective linear groups. Schur [21–23] extensively studied it. These representations appear naturally in the study of ordinary representations of groupsand are knownto havemany applications in other areas of Physics and Mathematics. We refer the reader to Section3 for precise definitions and related results regardingprojective representations of agroup.Bydefinition, everyordinaryrepresentation ofagroup is projective, buttheconverseisnottrue.Therefore, understandingtheprojective representationsisusually more intricate. Recall, the Schur multiplier of a group G is the second cohomology group H²(G,C^×), whereC^× isatrivialG-module.TheSchurmultiplierofagroupplaysanimportantroleinunderstanding its projective representations. By definition, every projective representation ρ of G is associated with a 2-cocycle α : G×G → C^× such that ρ(x)ρ(y) =α(x,y)ρ(xy) for all x,y ∈ G. Inthis case, we say, ρis

* Correspondingauthor.

E-mailaddresses:sumanahatui@iisc.ac.in(S. Hatui),psingla@iitk.ac.in(P. Singla).

https://doi.org/10.1016/j.jpaa.2021.106742 0022-4049/©2021ElsevierB.V.Allrightsreserved.

an α-representation. Conversely,forevery2-cocycle αofG,there exists anα-representationof G, namely C^α(G) thetwistedgroupalgebraofG.So,thefirststeptowardsunderstandingtheprojectiverepresentations is to describe the 2-cocyclesof G upto cohomologous, i.e., to understandthe Schurmultiplier ofG. The second step involves constructing α-representations of G for all [α] ∈ H²(G,C^×), where [α] denotes the cohomology classofα.

The complexordinaryrepresentations offinite abelian groupsareeasy tounderstand. Forexample,all irreducibles are one dimensional. But this is not true for their projective representations. This problem has been studiedbymanyauthors,mostnotablyby Morris,Saeed-ul-Islam,and Thomasin[14],[15].All irreducible α-representations of (Z/nZ)^k for some special α have been described in [14]. This work was generalized toall finiteabelian groupsforsomespecial classof cocyclesin[15]. Theirresultsare outlined in [10, Chapter 3] and [11, Chapter 8]. Later, Higgs [4] constructed an irreducible α-representation of elementaryabelianp-groups(Z/pZ)^k, foreveryα.Also,hecountedthenumberof[α]∈H²((Z/pZ)^k,C^×) suchthatirreducibleα-representationsof(Z/pZ)^kcontinuetobeirreduciblewhenrestrictedtoasubgroup of index ≤ p². The corresponding results for (Z/p^rZ)^k with r > 1 are not yet known. The projective representations of dihedral groupsare also well knownin theliterature; see [10, Theorem 7.3].Schur [23]

studiedtheprojectiverepresentationsofthesymmetricgroupsS_n.HeprovedthattheSchurmultiplierofS_n for,n≥4,isZ/2ZanddescribedtherepresentationgroupofSn,see [16,24] formoredetails.Nazarov [18,19]

explicitlyconstructedtheprojectiverepresentationsofSnbyprovidingsuitableorthogonalmatricesforeach generatorofthesymmetricgroup.

Inthisarticle,ourgoalistodescribetheSchurmultiplierandtheprojectiverepresentationsofthediscrete Heisenberg groupsandtheirt-variants.Thet-variantsoftheHeisenberggroups,denotedbyH_2n+1^t (R),are defined as follows. Let R be a commutativering withidentity and t∈ R. Define the groupH_2n+1^t (R) by theset Rⁿ⁺¹⊕Rⁿ withmultiplicationgivenby,

(a, b₁, . . . , b_n, c₁, . . . , c_n)(a, b₁, b₂, . . . , b_n, c₁, c₂, . . . , c_n)

= (a+a+t(n

i=1b_ici), b1+b₁, . . . , bn+b_n, c1+c₁, . . . , cn+c_n).

For t = 1, we recover the classicalHeisenberg groupand throughout we denote H_2n+1¹ (R) by H_2n+1(R).

ExceptTheorem1.3,whichis trueforgeneralcommutativeringsRwithidentity, theringR willbeZ/rZ forr∈N∪ {0}.Itfollowsfrom[10,Corollary5.1.3] thattheprojectiverepresentationsofH_2n+1^t (Z/rZ) are obtainedfromthoseofH_2n+1^t (Z/p^m_i ⁱZ),wherer=p^m₁¹p^m₂²· · ·p^m_k^k istheprimedecompositionofr.Hence, forr∈N,wecanfurtherassumethatt|r.

Ourfirst resultdescribestheSchurmultiplierofH_2n+1^t (R) forR=Z/rZ.ThedescriptionoftheSchur multiplierofH_2n+1^t (R) forn>1 differsfromthecasen= 1.Forn= 1,wefurtherassumethateitherr= 0 or risanoddnaturalnumber.

Theorem 1.1.

(i) For n>1,

H²(H_2n+1^t (Z/rZ),C^×) =

(Z/rZ)^2n2−n−1×(Z/tZ)²ⁿ⁺¹, if r∈N, (C^×)^2n2−n−1×(Z/tZ)²ⁿ, if r= 0.

(ii) For r∈(2N+ 1)∪ {0},

H²(H₃^t(Z/rZ),C^×) =

(Z/rZ)²×Z/tZ, if r∈(2N+ 1), (C^×)², if r= 0.

The Schur multiplier of H₃(Z/rZ) was obtained in [8, Theorem 1.1]. A proof of the above result is includedinSection2.

Ournextaimisto describetheprojectiverepresentationsofH_2n+1^t (Z/rZ).Throughout thisarticle,we consider these groups as discrete (abstract) groups and therefore the obtainedprojective representations maynotbeunitaryorevencontinuous.ItiswellknownthattheprojectiverepresentationsofagroupGare obtainedfrom theordinaryrepresentations of itsrepresentation group;see Corollary 3.3. Ournext result describesarepresentationgroupofH₃^t(Z/rZ).Forr∈N∪ {0},defineagroupH(r,t) by

H(r, t) =x, y, z|[x, y] =z^t,[x, z] =z₁,[y, z] =z₂, x^r=y^r=z^rt= 1 .

Throughout the article,[x,y]=xyx⁻¹y⁻¹ andthe relations ofthe form [x,y] = 1 forgenerators xand y areomittedinthepresentationofagroup.

Theorem1.2. Forr∈(2N+ 1)∪ {0}andt|r,thegroupH(r,t) isarepresentationgroup ofH₃^t(Z/rZ).

See Section 3for the proof of this result. A construction of all finite-dimensional irreducible ordinary representationsofH(r, t) isincludedinSection4.Ournextresultfocusesontheprojectiverepresentations ofH_2n+1^t (R) forn>1.Recall thatthegroupH_2n+1^t (R) projectsontotheabelian groupR²ⁿ⊕R/tR(see (2.0.2)).Thefollowing resultis trueforgeneralcommutativeringsR withidentity.

Theorem 1.3.For n>1, every irreducibleprojectiverepresentation of H_2n+1^t (R) isobtained from an irre- ducibleprojective representationof theabelian groupR²ⁿ⊕R/tRviainflation.

Weobtainitsproof fromageneralresultregardingthecentralproductofgroups;seeCorollary3.4and Section3.1.Fromtheaboveresult,thequestionofdeterminingtheprojectiverepresentationsofH_2n+1^t (R) for n > 1 boils down to understanding the projective representations of abelian groups R²ⁿ⊕R/tR. As mentionedearlier, this resultisnotyet well understood.Next,forR=Z/rZ andn∈N, wedescribe the representationgroupofRⁿ⊕R/tR.DefinethegroupFn(r,t) asfollows.

Fn(r, t) =x_k, z_ij |1≤k≤n+ 1,1≤i < j≤n+ 1,[x_i, x_j] =z_ij, x^t₁=x^r_j = 1 .

Theorem1.4. Forr∈N∪ {0}andt|r,thegroupFn(r,t)isarepresentation groupof (Z/rZ)ⁿ⊕Z/tZ.

A proofofthis resultisincluded inSection3.SeeSection4, foraconstructionof allfinite-dimensional ordinaryirreduciblerepresentationsof Fn(r,t).Wealsoobtainresultsregardingtheprojectiverepresenta- tions of extra-special groups. Recall that a p-group G is called an extra-special group if its center Z(G) is cyclicof order pand thequotient G/Z(G) is anon-trivialelementary abelianp-group.It iswell known thatforeachn≥1,therearetwoextra-specialpgroupsoforder p²ⁿ⁺¹uptoisomorphismwithexponents eitherpor p². Wedenote the isomorphismclassesof extraspecial groupsoforder p²ⁿ⁺¹ with exponentp and p² byES_2n+1(p) and ES_2n+1(p²) respectively.From definition,thegroupsES_2n+1(p) areisomorphic to H2n+1(Z/pZ). Above, we have already stated the results regarding the projective representations for H2n+1(Z/pZ).Combiningthiswithournextresult,wecompletethepictureforextra-specialp-groups.

Corollary1.5.

(i) Everyprojectiverepresentation of ES3(p²)isequivalent toanordinary representation.

(ii) For n > 1, every irreducible projective representation of ES_2n+1(p²) is obtained from an irreducible projective representationof (Z/pZ)²ⁿ viainflation.

Above,(i)followsbecausetheSchurmultiplierofES₃(p²) istrivial;see[9,Theorem3.3.6].Fortheproof of (ii),seeSection3.1.

2. SchurmultiplierofH_2n+1^t (Z/rZ),r∈N∪ {0}

In this section, we prove Theorem 1.1. Throughout this article, we use x^y to denote the conjugation yxy⁻¹.ThecommutatorsubgroupandcenterofagroupGaredenotedbyG andZ(G),respectively.

Recall, foragroupGandi∈N, Hⁱ(G,C^×)=Zⁱ(G,C^×)/Bⁱ(G,C^×),whereZⁱ(G,C^×) andBⁱ(G,C^×) consists ofcocyclesandcoboundariesof Gⁱ respectively.Weshallcall elementsofZ²(G,C^×) as 2-cocycles (or sometimesjustcocycleswhenitis clearfromthecontext) andelements ofH²(G,C^×) thecohomology classes.Foranelementα∈Z²(G,C^×),thecorrespondingelementofH²(G,C^×) willbedenotedby[α].For 2-cocycles α,β∈Z²(G,C^×) wesayαis cohomologousto β,whenever[α]= [β].

A centralextension,

1→A→G→G/A→1 (2.0.1)

is called a stem extension, if A ⊆ Z(G)∩G. For a given stem extension (2.0.1), the Hochschild-Serre spectral sequence[5,Theorem2,p. 129] forcohomologyofgroupsyieldsthefollowingexactsequence.

1→Hom(A,C^×)−−→^tra H²(G/A,C^×)−−→^inf H²(G,C^×), where tra : Hom(A,C^×)→H²(G/A,C^×) givenbyf →[tra(f)],where

tra(f)(x, y) =f(μ(x)μ(¯y)μ( ¯xy)⁻¹), x, y∈G/A,

for a section μ : G/A → G, denotes the transgression homomorphism and the inflation homomorphism, inf : H²(G/A,C^×) → H²(G,C^×) is given by [α] → [inf(α)], where inf(α)(x,y)= α(xA,yA). For groups H_2n+1^t (R),we havethefollowingstemextension,

1→tR−→^f H_2n+1^t (R)−→^g R/tR⊕R²ⁿ →1, (2.0.2) given by

f(tr)→(tr,0,0,· · ·,0

2n-times

)

g(a, b1, . . . , bn, c1, . . . , cn) = (amod (tR), b1, . . . , bn, c1, . . . , cn).

Letα∈Z²(G1×G2,C^×).Recallthat

H²(G1×G2,C^×)∼=^θH²(G1,C^×)×H²(G2,C^×)×Hom(G1/G₁⊗G2/G₂,C^×) (2.0.3) is anisomorphismdefinedby

θ([α]) = (res^G_G¹₁^×G2([α]),res^G_G¹₂^×G2([α]), ν),

whereν : H²(G,C^×)→Hom(H⊗K,C^×) isahomomorphismgivenbyν([α])(˜g1⊗g˜2)=α(g1,g2)α(g2,g1)⁻¹, forg˜₁=g₁G₁ and˜g₂=g₂G₂.Wewillusethis resultwithoutexplicitlyreferringtoit.

Now, we recallthedefinitionof thecentralproductof groups.A groupGis calledacentralproduct of itstwo normalsubgroupsH and KamalgamatingAifG=HK withA=H∩K and[H,K]= 1.

Theorem2.1.([3,TheoremAandTheorem3.6])LetGbeacentralproductoftwonormalsubgroupsH and K amalgamatingA=H∩K.Set Z=H∩K.

(i) Thentheinflation mapinf : H²(G/Z,C^×)→H²(G,C^×)issurjectiveand H²(G,C^×)∼= H²(G/Z,C^×)/N, where N∼= Hom(Z,C^×).

(ii) The subgroup Hom(Z,C^×) embeds in H²(H/A,C^×)/L⊕H²(K/A,C^×)/M via tra : Hom(Z,C^×) → H²(G/Z,C^×),whereL∼= Hom (A∩H)/Z,C^×

,M∼= Hom (A∩K)/Z,C^× . Lemma2.2. Letr∈N∪ {0}andt divides r.

(i) H²(Z/tZ⊕(Z/rZ)^k,C^×)∼= (Z/tZ)^k⊕(Z/rZ)^k(k−1)2 .Further,any α∈Z²(Z/tZ⊕(Z/rZ)^k,C^×)with k≥2satisfies [α]= [ν] forν∈Z²(Z/tZ⊕(Z/rZ)^k,C^×)suchthat

ν (m1, m2, . . . , mk, mk+1),(n1, n2, . . . , nk, nk+1)

=

1≤i<j≤k+1

μⁿ_i,j^imj,

forsomeμi,j∈C^× satisfying μ^r_i,j= 1 for2≤i< j≤k+ 1andμ^t_1,l = 1 for2≤l≤k+ 1.

(ii) Anyα∈Z²(H₃^t(Z/rZ),C^×)satisfies[α]= [σ]forσ∈Z²(H₃^t(Z/rZ),C^×)suchthatforx= (m1,n1,p1) andy= (m₂,n₂,p₂)we have,

σ(x, y) =

λ^{(m2p1+tn2p1 (p21−1))}μ^{(n1m2+tp1n2 (n22−1)+tp1n1n2)}, r= 0, λ^{(m2p1+tn2p1 (p21−1))}μ^{(n1m2+tp1n2 (n22−1)+tp1n1n2)}δ^(p1n2), r∈N,

forsomeλ,μ,δ∈C^× suchthatλ^r=μ^r=δ^t= 1.

Proof. (i) Schurmultiplier offinitely generated abelian groupsfollows from (2.0.3).We use[13, Theorem 9.4] forthecocycledescription.WeobtainthateverycocycleofZ/tZ⊕Z/rZiscohomologoustoacocycle oftheform

α((m1, m2),((n1, n2)) =σ1(m1, n1)σ2(m2, n2)g(n1, m2),

whereσ1∈H²(Z/tZ,C^×),σ2∈H²(Z/rZ,C^×) andg:Z/tZ⊕Z/rZ→C^× isamapsuchthatg(n1,m2)= g(1,1)^n1m2 =μⁿ_1,2^1m2.Thegeneralresultfollowsusinginductionargumentonk.

(ii)TheproofofthisresultgoesalongthesamelinesasPacker [20,Proposition 1.1].Followingthecited proof,we obtainthateveryα∈Z²(H₃^t(Z/rZ),C^×) iscohomologousto acocycleoftheform

β((m1, n1, p1),(m2, n2, p2)) =

λ^{(m2p1+tn2p1 (p21−1))}μ^{(n1m2+tp1n2 (n22−1)+tp1n1n2)}δ^(p1n2),

forsomeλ,μ,δ∈C^× suchthatλ^r=μ^r=δ^r= 1 Firstassumethatr= 0.Choosesomeδ1∈C^× suchthat δ^t₁=δ.Now,define afunctionb:H₃^t(Z)→C^× byb(m,n,p)=δ^m₁ .Then

b(m1, n1, p1)⁻¹b(m2, n2, p2)⁻¹b(m1+m2+tp1n2, n1+n2, p1+p2) =δ^p1n2

isacoboundary.Hence,everycocycleα∈Z²(H₃^t(Z),C^×) iscohomologoustoacocycleoftheform

σ((m₁, n₁, p₁),(m₂, n₂, p₂)) =λ^{m2p1+tn2p1 (p21−1)}μ^{n1m2+tp1n2 (n22−1)+tp1n1n2}, forsomeλ,μ∈C^×.

Now, assumer∈N.Ifwedefine amap b:H₃^t(Z/rZ)→C^× byb(m1,n1,p1)=δ^m1,thenwe have b(m₁, n₁, p₁)⁻¹b(m₂, n₂, p₂)⁻¹b(m₁+m₂+tp₁n₂, n₁+n₂, p₁+p₂) =δ^tp1n2,

which saysthatδ^tp1n2 is cohomologousto atrivialcocycle.Then everycocycleα∈Z²(H₃^t(Z/rZ),C^×) is cohomologousto acocycleoftheform

σ((m₁, n₁, p₁),(m₂, n₂, p₂)) =λ^{m2p1+tn2p1 (p21−1)}μ^{n1m2+tp1n2 (n22−1)+tp1n1n2}δ^p1n2, forsomeλ,μ,δ∈C^× suchthatλ^r=μ^r=δ^t= 1.

Corollary 2.3.Letr >1andμ isaprimitiver-th rootof unity.Thenα∈Z²(Z/rZ⊕Z/rZ)definedby α((m, n),(m, n)) =μ^nm,

correspondstoanon-trivial elementof H²(Z/rZ⊕Z/rZ,C^×).

2.1. Proofof Theorem1.1

Proof. (i)SchurmultiplierofH_2n+1^t (Z/rZ)forn>1:LetG=H_2n+1^t (Z/rZ),r∈N∪{0}andn>1.Then thegroupGisacentralproduct ofK₁=H_2n^t ₋₁(Z/rZ) andK₂=H₃^t(Z/rZ) amalgamatingatA=Z(G).

ConsiderZ=K₁∩K₂ whichisisomorphictotZ/rZ.HereG/Z∼=A/Z⊕(K₁/A⊕K₂/A)∼=Z/tZ⊕(Z/rZ)²ⁿ. ByTheorem2.1,itfollowsthatthehomomorphisminf of thefollowing exactsequence issurjective.

1→Hom(Z,C^×)−−→^tra H²(G/Z,C^×)−−→^inf H²(G,C^×).

Also,Hom(tZ/rZ,C^×) embeds inH²(K₁/A,C^×)⊕H²(K₂/A,C^×) via tra homomorphism.Hence, H²(G,C^×)∼= ^H

2(K1/A,C^×)×H²(K2/A,C^×)

Hom(Z,C^×) ×Hom((Z/rZ)⁴ⁿ⁻⁴,C^×)×(Z/tZ)²ⁿ

∼= ^Hom((Z/rZ)2n

2−5n+4,C^×)

Hom(tZ/rZ,C^×) ×Hom((Z/rZ)⁴ⁿ⁻⁴,C^×)×(Z/tZ)²ⁿ

∼= ^Hom((Z/rZ)

2n2−n,C^×)

Hom(tZ/rZ,C^×) ×(Z/tZ)²ⁿ. (2.1.1)

Here the map inf : H²(G/Z,C^×)→H²(G,C^×) is surjective, soevery cocycleofZ²(H_2n+1^t (Z/rZ),C^×) is cohomologoustoacocycleoftheform

β((l1, m1, . . . m2n),(l₁, m₁, . . . m_2n)) =

1≤i<j≤2n

μi,jm_imj

2n

k=1

μkl₁mk,

forsomeμi,j,μk∈C^× andμ^t_k= 1 for 1≤k≤2n,followsfrom Lemma2.2(i).

If r = 0, then μ_i,j ∈ C^× and μ^t_k = 1 for 1 ≤ i < j ≤ 2n, 1 ≤ k ≤ 2n. Let δ ∈ C^× and define a map b : H_2n+1^t (Z) → C^× such that b(l₁,m₁,. . . m_2n) = (δ^1/t)^l1. By using the map b, we ob- tain that δ^{(1≤i≤nmimn+i)} is cohomologous to a trivial cocycle. Therefore, up to cohomologous we can choose (μ_i,n+i)_1≤i≤n ∈ (C^×)ⁿ/(δ,δ,δ,· · ·,δ) | δ ∈ C^× which is isomorphic to (C^×)ⁿ⁻¹. As by (2.1.1), (C^×)^2n2−n−1×(Z/tZ)²ⁿ embeds inH²(H_2n+1^t (Z),C^×),hence

H²(H_2n+1^t (Z),C^×)∼= (C^×)^2n2−n−1×(Z/tZ)²ⁿ.

If r ∈ N, then μ^r_i,j = 1 for 1 ≤ i < j ≤ 2n and μ^t_k = 1 for 1 ≤ k ≤ 2n. We observe that x^{(t1≤i≤nmimn+i)} is cohomologous to a trivial cocycle, by using the map b : H_2n+1^t (Z/rZ) → C^× such thatb(l₁,m₁,. . . m_2n)=x^l1, for x∈ C^×,x^r = 1.So, upto cohomologous, we canchoose(μ_i,n+i)_1≤i≤n ∈ (Z/rZ)ⁿ/(x^t,x^t,x^t,· · ·,x^t)|x∈Z/rZ ∼= (Z/rZ)ⁿ⁻¹×Z/tZ.Therefore,by(2.1.1),

H²(H_2n+1^t (Z/rZ),C^×)∼= (Z/rZ)^2n2−n−1×(Z/tZ)²ⁿ⁺¹.

(ii)SchurmultiplierofH₃^t(Z/rZ):ThegroupG=H₃^t(Z/rZ) isthesemidirectproductofnormalsubgroup N =(m,n) ∼=Z/rZ⊕Z/rZandasubgroupT =p ∼=Z/rZ, wheretheactionof T onN is definedby p.(m,n)= (m+tpn,n).HereT actonHom(N,C^×) by(x.f)(n)=f(x.n) forf ∈Hom(N,C^×),n∈N,x∈ T.Then

H¹(T,Hom(N,C^×)) = Z¹(T,Hom(N,C^×)) B¹(T,Hom(N,C^×)), where

Z¹(T,Hom(N,C^×)) ={f :T →Hom(N,C^×)|f(xy) = (x.f(y))f(x)∀x, y∈T}

andB¹(T,Hom(N,C^×)) consistsoff ∈Z¹(T,Hom(N,C^×)) suchthatthere existsg∈Hom(N,C^×) satis- fyingf(x)= (x.g)g⁻¹ forallx∈T.

Givenα∈Z²(N,C^×),letα^x∈Z²(N,C^×) bedefinedbyα^x(n,n)=α(x.n,x.n) forx∈T andn,n∈N.

LetH²(N,C^×)^T denotetheT-stablesubgroupofH²(N,C^×),i.e.,

H²(N,C^×)^T ={[α]∈H²(N,C^×)|[α^x] = [α]∀x∈T}. Wehavethefollowing exactsequence.

1→H¹(T,Hom(N,C^×))−→^ψ H²(G,C^×)−−→^res H²(N,C^×)^T,

whichfollowsfrom[9,Theorem2.2.5] and[12,Corollary2.5] forthefiniteandinfinitediscretecasesrespec- tively.Here themapψisdefinedby

ψ([χ])((m1, n1, p1),(m2, n2, p2)) =χ(p1)(m2, n2),

for χ ∈ H¹(T,Hom(N,C^×)). Since, by Corollary 2.3, every cocycleα ∈ Z²(N,C^×) is cohomologous to a cocycleoftheformα((m1,n1),(m2,n2))=μ^n1m2,soforp∈T,wehave

α^p((m1, n1),(m2, n2)) =α((m1+tpn1, n1),(m2+tpn2, n2)) =μ^n1m2+tpn1n2. Then[α^p]= [α] as

αα^p−1((m₁, n₁),(m₂, n₂)) =b(m₁, n₁)b(m₂, n₂)b(m₁+m₂, n₁+n₂)⁻¹, whereb:N →C^× definedbyb(m,n)=μ^tpn2/2 (asrisodd).Hence,

H²(N,C^×)^T = H²(N,C^×).

Now,wedefine amap φ: H²(N,C^×)→H²(G,C^×) givenby[α]→[φ[α]], where

φ([α])((m₁, n₁, p₁),(m₂, n₂, p₂)) =μ^{n1m2+tp1n2 (n22−1)+tp1n1n2}.

Thenthecompositionmapres◦φ: H²(N,C^×)→H²(N,C^×) becomestheidentityhomomorphism.Hence, φisinjectiveandres issurjectivemap.

Thuswe have

H²(H₃^t(Z/rZ),C^×)∼= H¹(T,Hom(N,C^×))×H²(N,C^×). (2.1.2) Now onwards,weconsider thecasesr= 0 andr∈N separately.

Case 1:r= 0.Wefollowtheproofof[12,Theorem2.11].Weshowthat H¹(T,Hom(N,C^×))∼=C^×.

Define a map τ : Z¹(T,Hom(N,C^×)) → (C^×)² by τ(χ) = (χ(1)(1,0),χ(1)(0,1)). Then τ is injec- tive. For c₁,c₂ ∈ C^×, define χ(p)(m,n) = c^(mp+tn₁ ^{p(p−1)2 )}c^pn₂ . By [12, Lemma 2.7], it follows that χ∈Z¹(T,Hom(N,C^×)) andτ(χ)= (c₁,c₂).So,τ issurjective.Hence,viatheisomorphismτ,wehave

Z¹(T,Hom(N,C^×))∼= (C^×)².

Here B¹(T,Hom(N,C^×)) isthesetofallf :T →Hom(N,C^×) satisfyingthefollowing, f(p)(m, n) =g(m+tpn, n)g(m, n)⁻¹ forg∈Hom(N,C^×), m, n∈N, p∈T.

Observethatτ(f)= (1,g((1,0)^t)) andhence,τ(B¹(T,Hom(N,C^×)))∼=C^×.Thusitfollows that H¹(T,Hom(N,C^×))∼=C^×.

Hence,by(2.1.2),

H²(H₃^t(Z),C^×)∼= (C^×)². Case 2:r∈N.Forthiscase,ourclaimis

H¹(T,Hom(N,C^×))∼=Z/rZ⊕Z/tZ.

Let ζ be a primitive r-th root of unity and Hom(N,C^×) ∼= φ1,φ2 where φ1 : N → C^× is defined by φ1(1,0) = ζ,φ1(0,1) = 1 and φ2(1,0) = 1,φ2(0,1) = ζ. Now, T acting on Hom(N,C^×) by ^pφ1(1,0) = φ1(1,0) and ^pφ1(0,1) =φ1(tp,1) =ζ^pt. So, ^pφ1 = φ1φ^pt₂ . Similarlyit is easy to see that^pφ2 =φ2. Now, define amap N orm: Hom(N,C^×)→Hom(N,C^×) by

N orm(φ) =

p∈T pφ.

Consider anothermap h: Hom(N,C^×)→Hom(N,C^×) defined byh(φ)=^pφφ⁻¹, where pis agenerator ofT.ItisawellknownresultthatH¹(T,Hom(N,C^×)∼=^{ker(N orm)}_image(h) (seestep3intheproof ofTheorem5.4 of [6]).Since risodd,itiseasytocheck thatN orm(φ1)= 1 andN orm(φ2)= 1.Therefore,ker(N orm)= φ₁,φ₂ andimage ofhis< φ^t₂>.Therefore,H¹(T,Hom(N,C^×)∼=Z/rZ⊕Z/tZ.Thusby(2.1.2),

H²(H₃^t(Z/rZ),C^×)∼= (Z/rZ)²×Z/tZ.

3. Projective representationsofH_2n+1^t (R)

Inthissection,wefirstrecallsomebasicdefinitionsandresultsregardingprojectiverepresentationsofa groupandthenproveTheorems 1.2,1.3,and1.4.

LetV beacomplexvectorspace.AprojectiverepresentationofagroupGisahomomorphismofGinto theprojectivegenerallineargroup,PGL(V)= GL(V)/Z(V).Equivalently,aprojective representationisa mapρ:G→GL(V) suchthat

ρ(x)ρ(y) =α(x, y)ρ(xy), ∀x, y∈G,

forsuitablescalarsα(x,y)∈C^×.BytheassociativityofGL(V),themap(x,y)→α(x,y) givesa2-cocycle of G, i.e., an element of Z²(G,C^×). Wedenote this cocycle byα itself and say ρ is an α-representation.

Twoprojectiverepresentations ρ1:G→GL(V) and ρ2 :G→GL(W) are called projectivelyequivalent if thereis aninvertible T ∈Hom(V,W) andamapb:G→C^× suchthat

b(g)T ρ₁(g)T⁻¹=ρ₂(g)∀ g∈G.

Equivalentprojective representationsaresaidto haveequivalent2-cocycles. Thustwo cocyclesα,α :G× G→C^×areequivalentifthereexistsamapb:G→C^× suchthatα(x,y)= ^b(x)b(y)_b(xy) α(x,y) forallx,y∈G.

IntermsofSchurmultiplier,thismeansthattherepresentationsρandρ areequivalentimpliesthattheir cocyclesα and α are cohomologous, i.e., [α]= [α] in H²(G,C^×).It is to be noted thatto determine all projective representations of G up to equivalence, it is enough to determine projectively inequivalent α- representationsofGforasetofall2-cocyclerepresentativesofelementsofH²(G,C^×).Wefurthernotethat twoprojectivelyequivalentα-representations(ρ1,V) and(ρ2,W) arecalledlinearlyinequivalentifb(g)= 1 forall g∈G.Any α-representation ρofGsuch thatαis cohomologousto trivial2-cocycle, willbe called equivalentto anordinaryrepresentationofG.

Theset of allinequivalentirreducible ordinaryrepresentationsof agroupGwillbe denotedby Irr(G).

LetIrr^α(G) be thesetofcomplexlinearlyinequivalentirreduciblerepresentationsofGcorresponding toa 2-cocycleα.Wecanfurtherassumethatαisnormalizedcocycle,i.e., α∈Z²(G,C^×) satisfies

α(g,1) =α(1, g) = 1, ∀g∈G. (3.0.1) Throughout this section, we assume that the cocycle representative of [α] with which we work, satisfies (3.0.1).Next,werecallthedefinitionofarepresentationgroup(alsocalled acoveringgroup) ofagroupG from [21,Page23].

Definition 3.1(Representation groupof G).A groupG^∗ is called arepresentation groupofG, ifthere isa centralextension

1→A→G^∗→G→1 suchthatcorresponding transgressionmap

tra : Hom(A,C^×)→H²(G,C^×) isanisomorphism.

In[21],Schurprovedthattherepresentationgroupofafinitegroupalwaysexists.Forinfinitegroups,the parallelresultisalsoknown;see[2] forrelatedresults.Thenextresultrelatestheprojectiverepresentations ofagroupGanditscertainquotientgroup.

Theorem 3.2. Let A be a subgroup of a finitely generated group G such that A ⊆ G∩Z(G) and, [α] ∈ H²(G,C^×)beintheimageofinf : H²(G/A,C^×)→H²(G,C^×).Then

{[β]∈H²(G/A,C^×)|inf([β])=[α]}Irr^β(G/A) and Irr^α(G)are inbijectivecorrespondencevia inflation.

Proof. Wehavethefollowingexactsequence

1→Hom(A,C^×)−→^tra H²(G/A,C^×)−→^inf H²(G,C^×).

Fix a [β] ∈ H²(G/A,C^×) such that inf([β]) = [α]. Due to the exactness of the above sequence, the set

χ∈Hom(A,C^×)[β]tra(χ) consists ofalldistinct elementsofH²(G/A,C^×) that mapto[α] viainf.

Let ρ: G→ GL(V) be anirreducible α-representation of G. Then there exists arepresentative of [β], denotedby β itself, suchthatα(g,h)=β(gA,hA) for allg,h∈G.Therefore,for alla∈Aand g∈G, we haveα(g,a)=α(a,g)= 1.Hence,

ρ(g)ρ(a) =ρ(a)ρ(g), ∀a∈A, g∈G.

Sinceeveryirreduciblerepresentationinourcaseiscountabledimensional,bySchur’slemma(duetoDixmier forcountabledimensionalcomplexrepresentations),foralla∈A,ρ(a) isascalarmultipleofidentity.Further α(a,a)= 1 for all a,a ∈A, so ρ|A is ahomomorphism on A. Let μ: G/A→G be asection of G/Ain G such thatgA =μ(gA)A for all g ∈G. Every element g ∈G canbe written uniquelyg = agμ(gA) for some ag ∈A. Note thattra(ρ|A)(gA,hA)=ρ(μ(gA)μ(hA)μ(ghA)⁻¹). Now, define ρ˜: G/A→GL(V) by

˜

ρ(gA)=ρ(μ(gA)).Then

˜

ρ(gA) ˜ρ(hA) ˜ρ(ghA)⁻¹=ρ(μ(gA))ρ(μ(hA))ρ(μ(ghA))⁻¹

=β(gA, hA)ρ(μ(gA)μ(hA))ρ(μ(ghA))⁻¹

=β(gA, hA)ρ(μ(gA)μ(hA)μ(ghA)⁻¹μ(ghA))ρ(μ(ghA))⁻¹

= (βtra(ρ|A))(gA, hA)α⁻¹(μ(gA)μ(hA)μ(ghA)⁻¹, μ(ghA))

= (βtra(ρ|A))(gA, hA),

(3.0.2)

where α⁻¹(μ(gA)μ(hA)μ(ghA)⁻¹,μ(ghA))= 1 as μ(gA)μ(hA)μ(ghA)⁻¹ ∈A. Thusρ˜isβ-representation of G/Asuchthat[β]= [β][tra(ρ|A)] andinf([β])= [α]. Sinceρis irreduciblerepresentationand ρ(a) isa scalarmultipleofidentityfora∈A,ρ˜isalsoanirreduciblerepresentation.

Define amap

φ: Irr^α(G)−→

{[β]∈H²(G/A,C^×)|inf([β])=[α]}

Irr^β(G/A)

by φ(ρ) = ˜ρ. It is easy to see that φ is awell defined map. Next, we prove that φ is injective. Suppose ρ,ρ∈Irr^α(G) andφ(ρ)= ˜ρ,φ(ρ)= ˜ρ suchthatρ˜andρ˜ arelinearlyequivalent,i.e.,ρ˜(gA)=Tρ(gA)T˜ ⁻¹ forallg∈GandforsomeT ∈GL(V).Sinceρ˜andρ˜ areβtra(ρ|A) andβtra(ρ|A)-representationsofG/A respectively, tra(ρ|A)= tra(ρ|A). Buttra is injective,so ρ|A=ρ|A. Nowit is easyto check thatρ(g)= T ρ(g)T⁻¹for g∈G.Hence,φisinjective.Itremainstoshow thatφissurjective. Letρ˜:G/A→PGL(V) be anirreducibleβ1-projective representationsuchthatinf(β1)=α.Define ρ:G→PGL(V) via inflation, i.e., ρ(g)= ˜ρ(gA).Then ρisanirreducible α-representationofGandφ(ρ)= ˜ρ.

Corollary 3.3.LetAbe acentralsubgroup ofafinitelygenerated groupG^∗suchthat G^∗ isarepresentation group ofG=G^∗/A.Thenthereisabijection betweenthesets ∪[α]∈H²(G,C^×)Irr^α(G)andIrr(G^∗).