Math.Z.(2010)265:161–172
DOI10.1007/s00209-009-0508-9Mathematische Zeitschrift Trivial source modules in blocks with cyclic defect groups Shigeo Koshitani·Naoko Kunugi
Received:11August2006/Accepted:19February2009/Published online:27March2009
©Springer-Verlag2009
Abstract We shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non-identity p-elements infinite groups instead of doing huge calculation.The method is only for a case where defect groups are cyclic.Nevertheless, it works well at least when we want to prove Broué’s abelian defect group conjecture for blocks which have elementary abelian defect groups of order p2.
Keywords Trivial source module·Block·Cyclic defect group·Brauer tree Mathematics Subject Classification(2000)Primary20C20
1Introduction and main theorems
In modular representation theory offinite groups,one of the most important and interesting questions is Broué’s abelian defect group conjecture(ADGC)[3,6.2.Question,10,Con-jecture in p.132].It is still open and there are only several cases where the conjecture is checked.In recent papers of the authors with Waki we prove that Broué’s ADGC is true for several cases for a specific defect group which is elementary abelian of order nine and which is almost the smallest one in wild-representation type[11–13].Actually,one of the most efficient and essential methods in these papers is looking at trivial source(p-permutation) modules,where p is a prime.This means that we could have a chance to prove Broué’s ADGC for some unknown cases provided we get many(enough)trivial source modules.We S.Koshitani(B)
Department of Mathematics,Faculty of Science,Chiba University,Yayoi-cho,
Inage-ku,Chiba263-8522,Japan
e-mail:koshitan@math.s.chiba-u.ac.jp
N.Kunugi
Department of Mathematics,Tokyo University of Science,Kagurazaka1-3,
Shinjuku-ku,Tokyo162-8601,Japan
e-mail:kunugi@rs.kagu.tus.ac.jp
162S.Koshitani,N.Kunugi usuallyfind out trivial source modules by direct calculation,sometimes relying on computers, say GAP and MAGMA for example if thefinite groups are large.Therefore,if we know a more general and theoretical method to obtain trivial source modules,it should be nice and meaningful.
In this note,we shall present a method to get trivial source modules easily just by look-ing at values of ordinary characters at p-elements infinite groups instead of doing huge calculation.The method is only for a case where defect groups are cyclic.Nevertheless, it works quite well at least when we want to prove Broué’s ADGC for blocks which have elementary abelian defect groups of order p2because we often can reduce the task of getting trivial source modules to the case where the defect groups have central subgroups of order p. The proof given here is standard and based on celebrated work on blocks with cyclic defect groups,due to Dade[5],and also work of Rouquier[29].The result is,however,useful and convenient.Actually,our result is used to prove Broué’s ADGC for the Janko simple group J4in[14].Furthermore,nobody has remarked it before.Therefore,we believe that it would be worthwhile to present it here.
In this note,G is always afinite group and p is a prime.Let(K,O,k)be a splitting p-mod-ular system for all subgroups of G,namely,O is a complete discrete valuation ring of rank one such that K is the quotientfield of O with characteristic zero,such that k is the residue field O/rad(O)of O with characteristic p,and such that K and k are both splittingfields for all subgroups of G.Here,modules arefinitely generated right modules unless otherwise stated,and O G and kG are the group algebras.
Throughout this paper we are usually under the following situation.Namely,
Hypothesis1.1Suppose that k is an algebraically closedfield of characteristic p>0,G is afinite group,A is a block algebra of O G or kG with a cyclic defect group P,N=N G(P), and B is the Brauer correspondent of A in N,namely,B is a block algebra of O N with A=B G(block induction).Moreover,when P=1,let P1be a unique subgroup of P of order p,set N1=N G(P1),and let B1be the Brauer correspondent of A and B in N1,namely, B1is a block algebra of k N1with B N1=B1and B1G=A.Note that for any non-exceptional characterχof G in A and for any element u∈P−{1},the valueχ(u)is an integer by a result of Dade[7,Theorem68.1(8)(a)].
In principle,the Brauer tree B(A)of the block algebra A is taken arbitrarily(though as a matter of fact it is restricted for block algebras offinite groups,see[8]).On the other hand, the Brauer tree B(B1)of the bloc
k algebra B1is always a star with exceptional vertex in the center,see Dade[5],[7,Theorem68.1]and[1,Theorem17.2].Then,what about the block algebra B?As we can easily imagine,even stronger(better)situation happens for B. That is to say,the Brauer tree B(B)of the block algebra B is,of course,always a star with exceptional vertex in the center,and furthermore,all simple k N-modules in B are trivial source modules,see[29,Theorem10.1]and[15,III Lemma10.3].Namely,B1and B are always Morita equivalent but not necessarily Puig equivalent,see1.11for the definition of a Puig equivalence.By keeping these general relations between the structures of A,B1and B in mind,ourfirst result Theorem1.2gives characterizations of A so that A is Puig equivalent to B,namely,roughly speaking A essentially is the same as B from group representation theoretical point of view.On the other hand,our second result Theorem1.6might be more interesting.In other words,we look at a situation where B1and B are Puig equivalent.
Now we can state our main results.
Theorem1.2Assume1.1.Then,the following(1)∼(4)are equivalent:
Trivial source modules163 (1)For any non-exceptional characterχin A,it holds thatχ(u)>0for any element
u∈P−{1}.
(2)The Brauer tree of A is a star with exceptional vertex in the center(we consider that
this condition is satisfied if P=1),and there exists a non-exceptional characterχin
A such thatχ(u)>0for any element u∈P−{1}.
(3)The block algebras A and B are Puig equivalent,that is,there exists an(A,B)-bimod-
ule M such that M is a P-projective trivial source k[G×N]-module and M induces
a Morita equivalence between A and B.
(4)All simple kG-modules in A are trivial source modules.
Corollary1.3Assume1.1,and,in addition,suppose that A is nilpotent.Letχ1be a unique non-exceptional character in A.Then,the following three conditions are equivalent: (1)χ1(u)>0for any element u∈P−{1}.
(2)The block algebras A and B are Puig equivalent.
(3)The group algebra k P is a source algebra of A.
Corollary1.4Assume1.1,and that the Brauer tree of A is a star with exceptional vertex in the center.Then,we get the following:
(i)If there is a simple kG-module in A with a trivial source,then all simple kG-modules
in A are trivial source modules.
(ii)In particular,if A is the principal block algebra,then all simple kG-modules in A are trivial source modules.
Remark1.5The two statements in1.4,especially(ii),definitely have been known.They of course can be proved directly,while here we prove them as corollaries to one of our main results Theorem1.2.
Theorem1.6Assume1.1and P=1.Then,the following two conditions(1)and(2)are equivalent:
(1)The block algebras B1and B are Puig equivalent.
(2)There is a non-exceptional characterχin A such that one of the following two conditions
holds:
(i)χ(u)>0for all elements u∈P−{1}.
(ii)χ(u)<0for all elements u∈P−{1}.
Remark1.7It had been known that the situation(1)of Theorem1.6always happens when B1and B are the principal block algebras,see[4,6.4.]and[28,Remark9].
As a corollary to Theorem1.6we get a result Corollary1.8,where we obtain all trivial source modules in A with vertex P by looking at the character values of any irreducible ordinary characterχ1in A such that the edge which corresponds toχ1in the Brauer tree B(A)of A is at the end.
Corollary1.8Assume1.1.Letχ1be a non-exceptional character in A which is at an end in the Brauer tree of A,and let S1be a simple kG-module in A whose lift to O affordsχ1. Furthermore,let T=T(A)be the set of all trivial source kG-modules in A with vertex P, and let be the Heller operator for kG-modules.Then,we have the following:
164S.Koshitani,N.Kunugi
(i)Ifχ1(u)>0for all elements u∈P−{1},then T=
2n(S1)|n∈Z
.
(ii)Ifχ1(u)<0for all elements u∈P−{1},then T=
2n+1(S1)|n∈Z
.
Remark1.9Actually,in1.2(1),1.3(1),1.6(2)(i)and1.8(i),the condition“χ(u)>0”is
replaced by“χ(u)is a positive integer”,see1.1.
Remark1.10The statement1.2(3)and hence1.3(2)–(3),1.4and1.6(1)hold not only over k
but also over O since a Puig equivalence lifts from k to O by a result of Puig[23,7.8.Lemma]
(see[30,(38.8)Proposition]).
Notation and terminology1.11For a ring R we denote by1R and rad(R)the unit ele-
ment and the Jacobson radical of R,respectively.We say that X is an O G-lattice if X is
an O G-module and is free offinite rank as an O-module.For such an X and an element
a∈O G,tr X(a)denotes the trace of the endomorphism of M induced by the action of a.We write O G and k G for the trivial O G-and kG-modules,respectively.Let M be a kG-mod-
ule.We denote by P(M)and I(M)the projective cover and the injective envelope(hull)of
M,respectively.We write for the Heller operator,namely, (M)=Ker(P(M) M),
and −1(M)=Coker(M I(M)).Similarly,we define (L)for an O G-lattice L,see [30,p.35].Let H be a subgroup of G,and let N be a k H-module.Then,M↓H and N↑G, respectively,denote the restriction of M from G to H and the induced module N⊗k H kG. Similarly,ifχand χrespectively are characters of G and H,we denote byχ↓H and χ↑G the restriction ofχfrom G to H and the induction(induced character)of χfrom H to G,respectively.If N is a normal subgroup of G and if L is an O[G/N]-lattice,we write Def G G/N(L)for the deflation of L from G/N to G,that is to say,it is the same as L as an O-lattice and it is considered as an O G-lattice by the new action ·g= ·gN for ∈L and g∈G.We write Irr(G)and IBr(G),respectively,for the sets of all irreducible ordinary and Brauer characters of G.We denote by1G∈Irr(G)the trivial character of G.For another kG-module M we write M |M if M is(isomorphic to)a direct summand of M as a kG-module.We say that M is a trivial source module if M is indecomposable and has a source k Q
where Q is a vertex of M.We write G for the diagonal copy of G in G×G,that is, G={(g,g)∈G×G|g∈G}.We denote by Z(G)the center of G,and also by Z(R) the center of a ring R.For g,x∈G,set x g=g−1xg.We write G=N H when G is a semi-direct product of N by H,namely,N is a normal subgroup of G and H is a subgroup of G.
Let A and B,respectively,be block algebras of kG and k H forfinite groups G and H
such that A and B have a common defect group P(and hence P⊆G∩H).Then,we
say that an(A,B)-bimodule M gives a Puig(or splendid Morita)equivalence between A
and B if a pair(M,M∨)gives a Morita equivalence between these blocks such that M is a trivial source module with vertex P as a right k[G×H]-module via m(g,h)=g−1mh for m∈M,g∈G and h∈H.It follows from a result of Puig and independently of Scott,see [25,Remark7.5]and[19,Theorem4.1],that this is equivalent to say that source algebras of A and B are isomorphic as interior P-algebras.The same is true even over O as well by a result of Puig,see[23,7.8.Lemma]and[30,(38.8)Proposition].We also say that there is a (two-sided)splendid Rickard equivalence between A and B as in[29,p.204, .−10],and there is a stable equivalence of Morita type between A and B as in[10,p.123, .−3].By saying that there is a derived Rickard equivalence between A and B we mean the statement in [29,10.2.3in pp.203∼204],
where the equivalence is called a Rickard equivalence,see also [27,9.2].We write A X,Y B and A Z B when X,Y and Z are a left A-module,a right B-module and an(A,B)-bimodule,respectively.We denote by1A the block idempotent(block)of the block algebra A.We write Irr(A)for the set of all irreducible ordinary charactersχ∈Irr(G)
Trivial source modules 165which belong to the block algebra A .Similarly,we write IBr (A )for the set of all irreducible Brauer characters φ∈IBr (G )which belong to A .We mean by IBr (A )also the set of all non-isomorphic simple kG -modules which belong to A .If H is a subgroup of G and B is a block algebra of k H ,we write B G for the block induction in the sense of Brauer if it is defined.A PIM means a projective indecomposable module.
Let A be an arbitrary G -algebra,see [30,Sect.10].Then,for a subgroup P of G we use the notation A P to mean the set of P -fixed elements of A ,LP  A P  the set of all local points of A P ,see [30,Sect.14],and Br A :A P →A (P )the Brauer homomorphism corresponding to P ,see [30,Sect.11].
For other notation and terminology,see the books of Nagao–Tsushima [20]and Thévenaz
[30].
This paper is organized as follows.In Sect.2we provide sevaral propositions and lemmas for block algebras with cyclic defect groups,which are quite useful to prove our main results.In Sect.3we give complete proofs of our main results and consider an example.
2Notation and lemmas
In this section we list a number of lemmas and propositions,which are useful to prove our main results on blocks with cyclic defect groups.However,note that the first two lemmas are for an arbitrary finite p -group.
Lemma 2.1Let P be an arbitrary finite p-group with P =1,and let E be an abelian p  -subgroup of Aut (P ).Set G =P  E ,and assume that C G (P )⊆P.Suppose that V is an indecomposable endo-permutation O P-lattice with p |rank O V.Set S =End O (V ),T =O G and R =S ⊗O T.Then ,the unit element 1R of R is primitive in R P ,namely 1R is a unique primitive idempotent of R P .
block truncatedProof Since it suffices to discuss over k instead of O ,we do so.Set Q =C G (P ).Since S (P )∼=k by [22,p.200]or [30,Proof of (28.11)Corollary],and since T (P )∼=kC G (P )=k Q by [30,Exercise (11.5)],we know from [30,(28.3)Proposition]that R (P )∼=S (P )⊗k T (P )∼=k Q .Namely,R (P )is a local ring.Then,there is a primitive idempotent e of R P with Br P (e )=0.There is a local point α∈LP  R P  su
ch that e ∈α.Then,the multiplicity of αis one and Br P (e )=1R (P )since R (P )is a local ring,see [30,Sect.4].
Now,take any simple right R -module L .Then,L =V ⊗k X for a simple right T -mod-ule X .Note that V is considered as a unique simple right S -module.Since E is abelian,dim X =1.Hence,L k P ∼=V k P ⊗k X ↓P =V k P ⊗k k ∼
=V k P .Set f =1R −e ∈R P .Then,L =Le ⊕L f ,and this is a direct sum of right k P -modules.Let f =f 1+···+f n be a decomposition of orthogonal primitive idempotents of R P for a positive integer n .Thus,Br P (f i )=0for any i since R (P )is a local ring and ef i =f i e =0.Then,for each i ,we get by Rosenberg’s lemma [30,(4.9)Proposition]that L f i is relatively Q i -projective for a proper subgroup Q i of P ,and hence p |dim (L f i )by [20,Chapter 4,Theorem 7.5].Thus,p |dim (L f ).Since dim L =dim V ≡0(mod p ),we have Le =0.Then,L f =0since L k P ∼=V k P and V k P is indecomposable.This yields that f ∈rad (R ),which means f =0.  Lemma 2.2Let P ,E ,G and T be the same as in 2.1.Assume that W is an indecom-posable endo-permutation O P-lattice with p |rank O W ,U is a projective O P-lattice and X =W ⊕U.Set S =End O (X ).Suppose ,in addition ,that e is a primitive idempotent of

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