Dihedral blocks with two simple modules

0.19 MB·English

by  Frauke M. Bleher

DIHEDRAL BLOCKS WITH TWO SIMPLE MODULES
FRAUKEM.BLEHER
Abstract. Letk beanalgebraicallyclosedfieldofcharacteristic2,andletGbeafinitegroup.
0 Suppose B is a block of kG with dihedral defect groups such that there are precisely two iso-
1 morphismclasses ofsimpleB-modules. ThedescriptionbyErdmannofthe quiverandrelations
0 of the basic algebra of B is usually only given up to a certain parameter c which is either 0 or
2 1. In this article, we show that c = 0 if there exists a central extension Gˆ of G by a group of
n order2together withablock Bˆ of kGˆ withgeneralized quaternion defect groups suchthat B is
a containedintheimageofBˆ underthenaturalsurjectionfromkGˆ ontokG. Asaspecialcase,we
J obtain that c=0 ifG=PGL2(Fq) for someodd primepower q and B isthe principalblock of
6 kPGL2(Fq).

]
R
G
1. Introduction
.
h
Letk be analgebraicallyclosedfieldofarbitrarycharacteristic,andletΛbe afinite dimensional
t
a k-algebra. One of the fundamental problems in representationtheory is to determine the indecom-
m posable Λ-modules up to isomorphism. Depending on the complexity of the set of isomorphism
[ classes of indecomposable Λ-modules, one distinguishes between three representation types: finite,

tame and wild. If G is a finite group and B is a block of kG, then B has finite representationtype
2
if and only if the defect groups of B are cyclic, whereas B has tame representation type if and
v
7 only if the characteristic of k is 2 and the defect groups of B are either dihedral, semidihedral or
8 generalizedquaternion. Inaseriesofarticlesculminatinginthemonograph[8],Erdmanndescribed
9 the quiversandrelationsofthe basicalgebrasofallblocksofgroupalgebrasoftame representation
0
type, by introducing the larger classes of algebras of dihedral, semidihedral and quaternion type.
.
2 For example, if k has characteristic 2, then the class of all k-algebras Λ of dihedral type includes,
1
up to Morita equivalence, all blocks of kG with dihedral defect groups.
9
Suppose now that k has characteristic 2. Let B consist of all blocks B of group algebras of
0 ℓ,d
: finite groups over k with dihedral defect groups of order 2d and precisely ℓ isomorphism classes of
v
simpleB-modules. In[4,Thm. 2],Brauershowedthatℓ≤3. LetΛbeak-algebraofdihedraltype
i
X such that Λ is Morita equivalent to a block B ∈ B . In [8, Chap. VI], the possible quivers for Λ
ℓ,d
r have been determined; it follows from the tables in [8, p. 294–297] that they only depend on the
a
decompositionmatrix ofB. If ℓ=3 thenthe relationsofΛ givenin [8, p. 294–297]only dependon
d. If ℓ=2 then the relations in [8, p. 294–297]additionally depend on a parameter c∈{0,1},and
it is shown in [8, Sect. VI.8] that the algebras with the same quiver, the same d but different c are
notMoritaequivalent. IfB isablockinB ,itisusuallydifficulttodecidewhetherc=0orc=1.
2,d
To our knowledge there are only a few cases of blocks B ∈B where c has been determined, such
2,d
as the principal 2-modular block of the symmetric group S (see [8, Cor. V.2.5.1]) or the principal
4
2-modular blocks of certain quotients of the general unitary group GU (F ), where q ≡ 3 mod 4
2 q
(see [9, Sect.1.5]).
In this paper, we consider the following situation, where as before k has characteristic 2. Let
G be a finite group, and suppose B is a block of kG belonging to B . Suppose further that
2,d
Date:January5,2010.
2000 Mathematics Subject Classification. Primary20C05; Secondary16G20.
Key words and phrases. Dihedral defect groups, generalized quaternion defect groups, projective general linear
groups.
Theauthor wassupportedinpartbyNSFGrantDMS06-51332.
1
2 FRAUKEM.BLEHER
there exists a central extension Gˆ of G by a group of order 2 together with a block Bˆ of kGˆ with
generalizedquaterniondefectgroupsoforder2d+1 suchthatB iscontainedintheimageofBˆ under
the natural projection π : kGˆ → kG. We will prove in Theorem 2 that in this case the parameter
c in the description of the relations in [8] must be c = 0. If Gˆ is a finite group with generalized
quaternion Sylow 2-subgroups such that Gˆ has no normal subgroups of odd order, it follows by a
result of Brauer and Suzuki [5] that the center of Gˆ has order 2. Therefore, if G = Gˆ/Z(Gˆ) and
if the principal block of kG belongs to B , then the corresponding parameter c must also be zero
2,d
(seeCorollary3). Asaspecialcase,weobtainthatifq isanoddprimepowerand2d isthemaximal
2-power dividing (q2−1), then the parameter of the principal block of kPGL (F ), which belongs
2 q
to B , must also be c=0 (see Corollary 4).
2,d
The main idea for the proof of Theorem 2 is as follows. Let Λˆ (resp. Λ) be the basic algebra of
Bˆ (resp. B). Since B belongs to B , it follows that there are precisely two isomorphism classes
2,d
of simple Bˆ-modules and their representatives are given by the inflations of simple B-modules. In
particular,ifB istheprincipalblockofkG,thenBˆ istheprincipalblockofkGˆ. By[8,p. 303–304],
weknowthe quiverandrelationsofΛˆ upto aparametercˆ∈k. Sincethereis asurjectivek-algebra
homomorphism π : Bˆ → B, there is a surjective k-algebra homomorphism π : Λˆ → Λ which is
B Λ
compatible with π when we view Λˆ (resp. Λ) as a (non-unitary) subalgebra of Bˆ (resp. B). We
B
then determine possible generatorsof the kernel Ker(π ). To find these generators,we use that all
Λ
isomorphism classes of indecomposable Λ-modules are known, since Λ modulo its socle is a special
biserialalgebraby[8,Thm. VI.10.1]. ThisknowledgegivesusenoughcontrolonKer(π )tobeable
Λ
todeducefromtherelationsforΛˆ andfromthegeneratorsofKer(π )thattheconstantcoccurring
Λ
inthe relationsforΛ∼=Λˆ/Ker(πΛ)in[8]mustbe zero. The surprisingresultisthatalthoughmuch
lessisknownabout2-modularblockswithgeneralizedquaterniondefectgroupsthanfor2-modular
blocks with dihedral defect groups, the relationship we establish between the basic algebras Λˆ and
Λ is enough to determine the parameter c occurring in the description of Λ.
We finish this introduction with one application of our results. Let B ∈ B be a block of kG
2,d
for some finite group G and let V be a finitely generated kG-module belonging to B with stable
endomorphism ring k. If there exists a central extension Gˆ of G by a group of order 2 and a block
Bˆ of kGˆ with generalized quaternion defect groups as above, then the universal deformation ring
R(G,V)isisomorphictoasubquotientringofthegroupringofadefectgroupD ofB overthering
W(k)ofinfiniteWittvectorsoverk (see[3]). Infact,theconclusionaboutthestructureofR(G,V)
for all V belonging to an arbitrary block B in B with stable endomorphism ring k is equivalent
2,d
to the statement that the parameter c associated to each block B in B is equal to 0. This was
2,d
our original motivation in studying these parameters. In [2, Question 1.1] we asked whether the
conclusion about the structure of R(G,V) above holds for all V having stable endomorphism ring
k, without any condition on the block to which V belongs. We believe that further study of this
question will lead either to a counterexample or to new results about the structure of blocks of
group rings of finite groups.
For background on group algebras and blocks, we refer the reader to [7, Chaps. 1, 2 and 7].
For background on finite dimensional algebras, and in particular on algebras given by quivers and
relations, we refer the reader to [1, Chaps. I-III].
2. Dihedral blocks with two simple modules
Throughout this section we make the following assumptions.
Hypothesis 1. Let k be an algebraically closed field of characteristic 2, and let d ≥ 3 be a fixed
integer. Suppose G is a finite groupand B is a block of kG with dihedral defect groups of order 2d
such that there are precisely two isomorphism classes of simple B-modules.
Under these hypotheses, it follows from [8, Chaps. VI, IX and p. 294–295] that there exists
c ∈ {0,1} and i ∈ {1,2} such that the basic algebra of B is isomorphic to the symmetric algebra
DIHEDRAL BLOCKS WITH TWO SIMPLE MODULES 3
Λ , as defined in Figure 1, and the decomposition matrix of B is as in Figure 2. In particular, if
i,c
q is an odd prime power, 2d is the maximal 2-power dividing (q2−1) and B is the principal block
of kPGL (F ), then there exists c ∈ {0,1} such that B is Morita equivalent to Λ (resp. Λ ) if
2 q 1,c 2,c
q ≡1 mod 4 (resp. q ≡3 mod 4).
Figure 1. The basic algebras Λ =kQ /I and Λ =kQ /I .
1,c 1 1,c 2,c 2 2,c
0 β 1
Q1 = α ::•oo //• and
γ
I = hβγ,α2−c(γβα)2d−2,(γβα)2d−2 −(αγβ)2d−2i,
1,c
0 β 1
Q2 = α ::•oo //•dd η and
γ
I = hηβ,γη,βγ,α2−cγβα,γβα−αγβ,η2d−2 −βαγi.
2,c
Figure2. ThedecompositionmatrixforablockBofkGthatisMoritaequivalent
toΛ (resp. Λ ),where(∗)meansthatthelastrowisrepeated(2d−2−1)times.
1,c 2,c
1 0 1 0
     
1 0 1 0
 1 1  resp.  1 1  
     
 1 1    1 1  
     
 2 1  (∗)   0 1  (∗) 
Let i ∈ {1,2}. The projective indecomposable Λ -module corresponding to the vertex 0 (resp.
i,c
1) ofthe quiver Q is generatedby all paths in kQ modulo I that startat the vertex 0 (resp. 1).
i i i,c
Since Λ is a symmetric algebra, it follows that every projective indecomposable Λ -module has
i,c i,c
a simple socle (see for example [7, Prop. 9.12]). An elementary combinatorial argument using the
generatorsofthe ideal I shows thatthe element(γβα)2d−2 modulo I is not zeroandlies in the
1,c 1,c
socle of Λ . Similarly, (βαγ)2d−2 modulo I is not zero and lies in the socle of Λ . It follows
1,c 1,c 1,c
thatthe socleofthe projectiveindecomposableΛ -module correspondingtothe vertex0(resp. 1)
1,c
ofthe quiverQ isgeneratedby(γβα)2d−2 =(αγβ)2d−2 modulo I (resp. (βαγ)2d−2 modulo I ).
1 1,c 1,c
Thus the radicalserieslengthofthe projectiveindecomposableΛ -modulesis atleast3·2d−2+1.
1,c
Using the generators of the ideal I , it follows that all paths of length at least 3·2d−2 +1 are
1,c
zero modulo I . Hence the radical series length of the projective indecomposable Λ -modules is
1,c 1,c
exactly 3·2d−2 +1. Analyzing the projective indecomposable Λ -module corresponding to the
2,c
vertex 0 of the quiver Q , one similarly sees that its socle is generated by γβα=αγβ modulo I
2 2,c
and its radical series length is 4. For the projective indecomposable Λ -module corresponding to
2,c
the vertex 1 of the quiver Q , one has that its socle is generated by η2d−2 = βαγ modulo I and
2 2,c
its radical series length is 4 (resp. 2d−2+1) if d=3 (resp. if d>3).
It follows from the definition of string algebras in [6, Sect. 3] and from the above description of
the socle of Λ for i ∈ {1,2} that Λ /soc(Λ ) is a string algebra. Therefore, one can see as in
i,c i,c i,c
[8, Sect. I.8.11]thatthe isomorphismclassesofallnon-projectiveindecomposable Λ -modules are
i,c
given by string and band modules as defined in [6, Sect. 3].
4 FRAUKEM.BLEHER
Ifi∈{1,2}andB isablockofkGthatisMoritaequivalenttoΛ ,wecanusethedecomposition
i,c
matrix of B in Figure 2 to compute the composition series length of the projective indecomposable
B-modules (see [7, Thm. 18.26]). It follows that if i = 1, then the projective indecomposable B-
modulecorrespondingtothevertex0(resp. 1)ofthequiverQ has6+6·(2d−2−1)=6·2d−2 (resp.
1
4+3·(2d−2−1)=1+3·2d−2)compositionfactors. Ifi=2,thentheprojectiveindecomposableB-
modulecorrespondingtothevertex0(resp. 1)ofthequiverQ has6 (resp. 4+(2d−2−1)=3+2d−2)
2
composition factors.
Theorem 2. Assume Hypothesis 1. Suppose there exists a central extension Gˆ of G by a group
of order 2 together with a block Bˆ of kGˆ with generalized quaternion defect groups such that B is
containedintheimageofBˆ underthenaturalprojection π :kGˆ →kG. ThenB isMorita equivalent
to either Λ or Λ , i.e. the parameter c is c=0.
1,0 2,0
Proof. Letz ∈Gˆbeacentralelementoforder2suchthatGisisomorphictoGˆ/hzi. Inthefollowing,
we identify G = Gˆ/hzi. Let π : kGˆ → kG be the natural projection. Since by assumption, B is
containedinthe imageofBˆ underπ,there isasimplekG-moduleS belongingtoB whoseinflation
via π belongs to Bˆ. Let T be an arbitrary simple kGˆ-module. Since z has order 2, it acts trivially
on T. Thus T is inflated via π from a simple kG-module which we also denote by T. By [7, Thm.
56.12], T belongs to B (resp. Bˆ) if and only if there is a sequence of simple kG-modules (resp.
simple kGˆ-modules) E ,E ,...,E such that E = S, E = T, and for 1 ≤ j < n, E and E
1 2 n 1 n j j+1
are equal or there is a non-split G-extension (resp. a non-split Gˆ-extension) of one of them by the
other. If E andE′ are simple kG-modules whose inflations fromG to Gˆ arealso denotedby E and
′
E , then the Lyndon-Hochschild-Serre spectral sequence gives an exact sequence
0→Ext1 (E,E′)→Ext1 (E,E′)→Hom (E,E′).
kG kGˆ kG
This implies immediately that Ext1 (E,E′) andExt1 (E,E′) have the same k-dimension for each
kGˆ kG
′
choice of non-isomorphic E and E . Therefore, the above characterization of the simple modules
which belong to B (resp. Bˆ) implies that Bˆ has the same number of isomorphismclasses of simple
modules as B. We can thus identify the simple Bˆ-modules with the simple B-modules. Therefore,
the restriction of the natural projection π to Bˆ gives a surjective k-algebra homomorphism
(1) π :Bˆ →B.
B
Moreover,since Ker(π)=(1+z)kGˆ, it follows that Ker(π )=(1+z)Bˆ. Thus if Pˆ is a projective
B
indecomposable Bˆ-module and P = B ⊗ Pˆ is the corresponding projective indecomposable B-
Bˆ
module, then we have a short exact sequence of Bˆ-modules
(2) 0→(1+z)Pˆ →Pˆ →P →0.
Since the map f : (1+z)Pˆ → B ⊗ Pˆ = P, defined by f((1+z)x) = 1⊗x for all x ∈ Pˆ, is a
Bˆ
Bˆ-module isomorphism, Pˆ is a non-trivial Bˆ-extension of P by itself. In particular, Pˆ has twice as
many composition factors as P.
Considering the possible decomposition matrices for blocks with generalized quaternion defect
groups and two isomorphism classes of simple modules as given in [8, p. 303–304] and using that
the projective indecomposable Bˆ-modules must have twice as many composition factors as the
corresponding projective indecomposable B-modules, we conclude the following: If B is Morita
equivalent to Λ (resp. Λ ), then there exists a constant cˆ∈k such that the basic algebra of Bˆ
1,c 2,c
is isomorphic to Λˆ (resp. Λˆ ), as defined in Figure 3, and the decomposition matrix of Bˆ is as
1,cˆ 2,cˆ
in Figure 4.
AswedidforB,wecanusethedecompositionmatrixofBˆinFigure4tocomputethecomposition
series length of the projective indecomposable Bˆ-modules. If Bˆ is Morita equivalent to Λˆ , then
1,cˆ
the projective indecomposable Bˆ-module corresponding to the vertex 0 (resp. 1) of the quiver Q
1
has 6+6·(2d−1−1) = 6·2d−1 (resp. 5+3·(2d−1−1) = 2+3·2d−1) composition factors. If
DIHEDRAL BLOCKS WITH TWO SIMPLE MODULES 5
Figure 3. The basic algebras Λˆ = kQ /Iˆ and Λˆ = kQ /Iˆ , where Q
1,cˆ 1 1,cˆ 2,cˆ 2 2,cˆ 1
and Q are as in Figure 1.
2
Iˆ = hγβγ−αγ(βαγ)2d−1−1, βγβ−βα(γβα)2d−1−1,
1,cˆ
α2−γβ(αγβ)2d−1−1−cˆ(αγβ)2d−1, βα2i,
Iˆ = hηβ−βα(γβα), γη−αγ(βαγ), βγ−η2d−1−1,
2,cˆ
α2−γβ(αγβ)−cˆ(αγβ)2, βα2i.
Figure4. ThedecompositionmatrixforablockBˆ ofkGˆthatisMoritaequivalent
to Λˆ (resp. Λˆ ), where (∗∗) means that the last row is repeated (2d−1 −1)
1,cˆ 2,cˆ
times.
1 0 1 0
     
1 0 1 0
 1 1    1 1  
  resp.   
 1 1    1 1  
     
 0 1    2 1  
     
 2 1  (∗∗)   0 1  (∗∗) 
Bˆ is Morita equivalent to Λˆ , then the projective indecomposable Bˆ-module correspondingto the
2,cˆ
vertex 0 (resp. 1) of the quiver Q has 12 (resp. 7+(2d−1−1)=6+2d−1) composition factors.
2
Let E and F be non-isomorphic simple B-modules. Then the inflations of E and F to Bˆ are
representatives of the isomorphism classes of simple Bˆ-modules, which we again denote by E and
F. Let eˆ and fˆ be two orthogonal primitive idempotents in Bˆ such that Bˆeˆ (resp. Bˆfˆ) is a
projectiveBˆ-modulecoverofE (resp. F). Thene=π (eˆ)andf =π (fˆ)areorthogonalprimitive
B B
idempotents in B, and Be (resp. Bf) is a projectiveB-module coverof E (resp. F). Let εˆ=eˆ+fˆ
and ε=e+f. Then π from (1) restricts to a surjective k-algebra homomorphism
B
(3) π :εˆBˆεˆ→εBε.
ε
Let i∈{1,2}, cˆ∈k and c∈{0,1} be such that there are k-algebra isomorphisms fˆ : εˆBˆεˆ→Λˆ
Λ i,cˆ
and f :εBε→Λ . Then there exists a surjective k-algebra homomorphism
Λ i,c
(4) π :Λˆ →Λ
Λ i,cˆ i,c
such that the diagram in Figure 5 commutes. Here nat.proj. stands for the natural projection
of Λˆ onto Λˆ /Ker(π ) and π is the k-algebra isomorphism induced by π . As in Figure 5,
i,cˆ i,cˆ Λ Λ Λ
define Λ′ =Λˆ /Ker(π ). We introduce this extra notation, since in the following we will need to
i,c i,cˆ Λ
′
distinguish between the isomorphic k-algebras Λ and Λ .
i,c i,c
We next show that the kernel of π is contained in the square of the radical of Λˆ . This
Λ i,cˆ
basically follows since Λˆ and Λ have the same quiver Q . However, for the convenience of
i,cˆ i,c i
the reader, we now give a more detailed explanation. Since π is surjective, π (radn(Λˆ )) ⊆
Λ Λ i,cˆ
radn(Λ ) for all n ≥ 1 by [7, Prop. 5.6]. Thus π induces a surjective k-algebra homomorphism
i,c Λ
π : Λˆ /radn(Λˆ ) → Λ /radn(Λ ) for all n ≥ 1. Let kQ+ be the ideal of kQ generated
Λ,n i,cˆ i,cˆ i,c i,c i i
by all arrows. By [1, Prop. III.1.6], rad(Λˆ ) (resp. rad(Λ )) is the image of kQ+ in Λˆ (resp.
i,cˆ i,c i i,cˆ
6 FRAUKEM.BLEHER
Figure 5. The commutative diagram relating Bˆ, Λˆ , B and Λ .
i,cˆ i,c
Bˆ πB // B
OO OO
εˆ(cid:31)B?ˆεˆ πε //ε(cid:31)B? ε
OO OO
fˆΛ fΛ
Λˆi,cˆnaKtK.KpKrKoKj.KKKK%% πΛ sssssπsΛssss99//Λi,c
Λˆ /Ker(π )
i,cˆ Λ
′
Λ
i,c
Λ ). Because Iˆ (resp. I ) is contained in (kQ+)2, it follows that for n ∈ {1,2} there is an
i,c i,cˆ i,c i
isomorphism between kQ /(kQ+)n and Λˆ /radn(Λˆ ) (resp. Λ /radn(Λ )). Hence π is a
i i i,cˆ i,cˆ i,c i,c Λ,n
k-algebra isomorphism for n ∈{1,2}, which implies Ker(π )⊂ rad2(Λˆ ). In particular, [1, Prop.
Λ i,cˆ
III.1.6] implies that the radical of Λ′ is the image of kQ+ in Λ′ .
i,c i i,c
Eventhoughπ in(1)isexplicitandnaturalandthedomainandrangeofπ in(4)areexplicitly
B Λ
given,wedonothaveanexplicitdescriptionofπ ,owingtothefactthatMoritaequivalences,given
Λ
by f and fˆ , were used to define π in Figure 5. Therefore, we need to show that π matches
Λ Λ Λ Λ
′
up the simple modules corresponding to the vertices of Q . Let S (resp. S ) be the simple Λ -
i 0 1 i,c
module corresponding to the vertex 0 (resp. 1) in Q . Then the inflations of S and S to Λˆ are
i 0 1 i,cˆ
the simple Λˆ -modules, which also correspond to the vertices 0 and 1 of Q , respectively. Let T
i,cˆ i 0
(resp. T ) be the simple Λ -modules corresponding to the vertex 0 (resp. 1) in Q . We computed
1 i,c i
the composition series lengths of the projective indecomposable Λ -modules P(T ) and P(T ) in
i,c 0 1
the paragraphbefore the statementofTheorem2, andwe computed the compositionserieslengths
of the projective indecomposable Λˆ -modules Pˆ(S ) and Pˆ(S ) in the paragraph after defining
i,cˆ 0 1
Λˆ and Λˆ . Comparing these composition series lengths, we see that Pˆ(S ) has twice as many
1,cˆ 2,cˆ u
′
composition factors as P(T ) if and only if u = v. Thus the Morita equivalence between Λ and
v i,c
Λ induced by the isomorphism π must send S to T for u∈{0,1}.
i,c Λ u u
To complete the proof of Theorem 2, we will make use of Λˆ -modules corresponding to certain
i,cˆ
paths in kQ . Let w = ζ ···ζ ζ be a path of length n ≥ 1 in kQ whose image modulo Iˆ does
i n 2 1 i i,cˆ
not lie in soc (Λˆ ). For 1≤j ≤n, let v be the end vertex of ζ , and let v be the starting vertex
2 i,cˆ j j 0
of ζ . Define a kQ -module M of k-dimension n+1 with respect to a given k-basis {b ,...,b }
1 i w 0 n
as follows. Let 0 ≤ j ≤ n. If v is a vertex in Q , define vb = b if v = v , and vb = 0 otherwise.
i j j j j
If ζ is an arrow in Q , define ζb = b if ζ = ζ and j ≤ n−1, otherwise define ζb = 0. By
i j j+1 j+1 j
our assumption on w, the ideal Iˆ of kQ acts as zero on M . Hence M defines a Λˆ -module,
i,cˆ i w w i,cˆ
whichwealsodenotebyM . Moreover,M isauniserialΛˆ -modulewithdescendingcomposition
w w i,cˆ
factors S ,S ,...,S .
v0 v1 vn
Suppose first that i = 1. As seen in the second paragraph after Hypothesis 1, the radical series
lengths of the projective Λ -module covers of T and T are both 3·2d−2+1. Using the Morita
1,c 0 1
′
equivalence between Λ and Λ induced by π , it follows that the radical series lengths of the
1,c 1,c Λ
DIHEDRAL BLOCKS WITH TWO SIMPLE MODULES 7
projectiveΛ′ -module coversofS and S arealso both 3·2d−2+1. Since both the radicalof Λˆ
1,c 0 1 1,cˆ
′
and the radicalof Λ are generated by arrows,all paths in kQ of length greater than or equal to
1,c 1
3·2d−2+1 modulo Iˆ must lie in Ker(π ). In particular,
1,cˆ Λ
γβ(αγβ)2d−1−1+cˆ(αγβ)2d−1 modulo Iˆ
1,cˆ
lies in Ker(π ). Thus it follows from the description of Iˆ in Figure 3 that α2 modulo Iˆ lies in
Λ 1,cˆ 1,cˆ
Ker(π ). Using similar arguments, we see that
Λ
(5) α2,βγβ,γβγ,(αγβ)2d−2α,(γβα)2d−2γ,(βαγ)2d−2β modulo Iˆ
1,cˆ
alllieinKer(π ). Takingthepathβγ inkQ ,wecandefinetheuniserialΛˆ -moduleM asabove
Λ 1 1,cˆ βγ
withdescendingcompositionfactorsS ,S ,S . UsingthatΛ /soc(Λ )isastringalgebra,wesee
1 0 1 1,c 1,c
that there is no uniserial Λ -module with descending composition factors T ,T ,T . Because of
1,c 1 0 1
′
theMoritaequivalencebetweenΛ andΛ inducedbyπ ,itfollowsthatthereisalsonouniserial
1,c 1,c Λ
′
Λ -modulewithdescendingcompositionfactorsS ,S ,S . Therefore,theremustexistanelement
1,c 1 0 1
x∈Ker(π )thatactsnon-triviallyonM . TheonlyelementsofΛˆ thatactnon-triviallyonM
Λ βγ 1,cˆ βγ
arek-linearcombinationsofthepaths1 ,1 , β, γ andβγ moduloIˆ . SinceKer(π )⊂rad2(Λˆ ),
0 1 1,cˆ Λ 1,cˆ
it follows that x=βγ+x′ modulo Iˆ where x′ is a (possibly zero) k-linear combination of paths
1,cˆ
z of length at least 2 that are different from βγ. Multiplying x on both sides with 1 modulo Iˆ ,
1 1,cˆ
′
we may assume that all the paths z coming up in the description of x both start and end at the
vertex1. Moreover,wemayomitallpathsz thatmoduloIˆ lieinKer(π )accordingto(5). Thus
1,cˆ Λ
we may assume that x′ is a k-linear combination of (βαγ)j for 1≤j ≤2d−2. It follows that there
exist constants a1,...,a2d−2 in k such that
2d−2
(6) x = βγ + a (βαγ)j modulo Iˆ
X j 1,cˆ
j=1
lies in Ker(π ). Suppose not all a are zero, and let 1 ≤ j ≤ 2d−2 be minimal with a 6= 0. If
Λ j 0 j0
j <2d−2, we canmultiply x onthe left withγ(βαγ)2d−2−j0−1 modulo Iˆ (resp. onthe rightwith
0 1,cˆ
(βαγ)2d−2−j0−1β modulo Iˆ ) and use (5) to see that
1,cˆ
γ(βαγ)2d−2−1 and (βαγ)2d−2−1β modulo Iˆ
1,cˆ
lie in Ker(π ). But this implies that all paths of length 3·2d−2 modulo Iˆ lie in Ker(π ), and
Λ 1,cˆ Λ
hence rad3·2d−2(Λ′ )=0, which is a contradiction. Thus a =0 for 1≤j <2d−2 and we obtain
1,c j
(7) π βγ−c ((βαγ)2d−2 modulo Iˆ =0
Λ(cid:16) 1 1,cˆ(cid:17)
for some c ∈ k. Using that Λ /soc(Λ ) is a string algebra, we see that there is no uniserial
1 1,c 1,c
Λ -module of length 3·2d−2+1 with descending composition factors
1,c
(T ,T ,T ,T ,T ,...,T ,T ,T ,T ,T )
0 0 1 0 0 1 0 0 1 0
(resp. (T ,T ,T ,T ,T ,...,T ,T ,T ,T ,T )).
0 1 0 0 1 0 0 1 0 0
′
Because of the Morita equivalence between Λ and Λ induced by π , we see that there is also
1,c 1,c Λ
no uniserial Λ′ -module of length 3·2d−2+1 with descending composition factors
1,c
(S ,S ,S ,S ,S ,...,S ,S ,S ,S ,S )
0 0 1 0 0 1 0 0 1 0
(resp. (S ,S ,S ,S ,S ,...,S ,S ,S ,S ,S )).
0 1 0 0 1 0 0 1 0 0
On the other hand, we can define a uniserial Λˆ -module M corresponding to the path w =
1,cˆ w
(γβα)2d−2 (resp. w = (αγβ)2d−2) in kQ with such descending composition factors. Thus there
1
must exist an element y ∈ Ker(π ) that acts non-trivially on M . The only elements of Λˆ that
Λ w 1,cˆ
act non-trivially on M are k-linear combinations of 1 , 1 , α, β, γ and subpaths of w of length
w 0 1
at least 2 modulo Iˆ . Since Ker(π ) ⊂ rad2(Λˆ ), there exists a subpath w′ of w of length at
1,cˆ Λ 1,cˆ
8 FRAUKEM.BLEHER
least 2 such that y = w′+y′ modulo Iˆ and y′ is a (possibly zero) k-linear combination of paths
1,cˆ
′ ′
z 6= w of length at least 2. Since w = w w w for certain subpaths w and w of w, we can
1 2 1 2
multiply y on the left with w modulo Iˆ and on the right with w modulo Iˆ to be able to
1 1,cˆ 2 1,cˆ
assume that w′ = w. Multiplying y on both sides with 1 modulo Iˆ , we may assume that all
0 1,cˆ
′
the paths z coming up in the description of y both start and end at the vertex 0. Moreover,
we may omit all paths z that modulo Iˆ lie in Ker(π ) according to (5). If w˜ is the path such
1,cˆ Λ
that {w,w˜}={(γβα)2d−2,(αγβ)2d−2}, we may thus assume that y′ is a k-linear combination of w˜,
(γβα)j and (αγβ)j for 1 ≤ j ≤ 2d−2−1. It follows that there exist constants a,a ,a in k for
j,1 j,2
1≤j ≤2d−2−1 such that
2d−2−1 2d−2−1
(8) y = w + aw˜ + a (γβα)j + a (αγβ)j modulo Iˆ
X j,1 X j,2 1,cˆ
j=1 j=1
liesinKer(π ). Arguinginasimilarwayto(7),weseethatalla =0=a for1≤j ≤2d−2−1.
Λ j,1 j,2
We obtain
(9) π (γβα)2d−2)−c (αγβ)2d−2 modulo Iˆ = 0,
Λ(cid:16) 2 1,cˆ(cid:17)
(10) π (αγβ)2d−2)−c (γβα)2d−2 modulo Iˆ = 0
Λ(cid:16) 3 1,cˆ(cid:17)
for certain c ,c ∈ k. If one of c ,c were zero, we could also choose the other one to be zero.
2 3 2 3
′
This wouldimply that the radicalserieslength of the projective Λ -module coverof S is atmost
1,c 0
3·2d−2, which is a contradiction. Thus c ,c have to be non-zero. Therefore, there exist certain
2 3
c ∈k, c ∈k∗ such that Ker(π ) contains the image Jˆ in Λˆ of the ideal
1 2 Λ c1,c2 1,cˆ
J =hα2,βγ−c (βαγ)2d−2,(γβα)2d−2 −c (αγβ)2d−2i ⊂ kQ .
c1,c2 1 2 1
Wecancomputethek-dimensionofkQ1/I1,c =Λ1,c (resp. ofkQ1/Jc1,c2 ∼=Λˆ1,cˆ/Jˆc1,c2)bycounting
k-linearly independent paths in kQ modulo I (resp. modulo J ). Since these k-dimensions
1 1,c c1,c2
are equal, it follows that Ker(πΛ) = Jˆc1,c2 and thus Λ′1,c ∼= kQ1/Jc1,c2. Because Λ′1,c ∼= kQ1/Jc1,c2
is symmetric, there exists a k-linear map ϕ : kQ /J → k such that ϕ(ab) = ϕ(ba) for all
1 c1,c2
a,b∈kQ /J andsuchthatKer(ϕ)doesnotcontainanynon-zeroleftidealofkQ /J . Asin
1 c1,c2 1 c1,c2
the second paragraph after Hypothesis 1, one sees that the socle of the projective indecomposable
kQ /J -modulecorrespondingtothevertex0isgeneratedby(γβα)2d−2 moduloJ . Itfollows
1 c1,c2 c1,c2
that
ϕ (γβα)2d−2 modulo J =ϕ (αγβ)2d−2 modulo J
(cid:16) c1,c2(cid:17) (cid:16) c1,c2(cid:17)
is non-zero, which implies c2 =1. Hence Λ′1,c ∼=kQ1/Jc1,1 for a certain c1 ∈k. Since the k-algebra
homomorphism
Λ =kQ /I −→ kQ /J
1,0 1 1,0 1 c1,1
z modulo I 7→ z modulo J for z ∈{1 ,1 ,α,β},
1,0 c1,1 0 1
γ modulo I 7→ γ−c αγ(βαγ)2d−2−1 modulo J
1,0 1 c1,1
is a k-algebra isomorphism for all c1 ∈ k, it follows that Λ′1,c ∼= Λ1,0. Therefore, the parameter c
must be equal to zero.
Suppose next that i=2. As seen in the second paragraph after Hypothesis 1, the radical series
length of the projective Λ -module cover of T (resp. T ) is 4 (resp. 4 if d = 3 and 2d−2+1 if
2,c 0 1
′
d > 3). Using the Morita equivalence between Λ and Λ induced by π , it follows that the
2,c 2,c Λ
′
radical series length of the projective Λ -module cover of S (resp. S ) is also 4 (resp. 4 if d= 3
2,c 0 1
and 2d−2 +1 if d > 3). Since both the radical of Λˆ and the radical of Λ′ are generated by
2,cˆ 2,c
arrows,all paths in kQ starting at vertex 0 of length greater than or equal to 4 modulo Iˆ must
2 2,cˆ
lie in Ker(π ). Also, if d = 3 (resp. if d > 3), then all paths in kQ starting at vertex 1 of length
Λ 2
DIHEDRAL BLOCKS WITH TWO SIMPLE MODULES 9
greater than or equal to 4 (resp. 2d−2+1) modulo Iˆ must lie in Ker(π ). Thus we can argue
2,cˆ Λ
similarly to (5), using the description of Iˆ in Figure 3, that
2,cˆ
(11) α(γβα),β(αγβ),α2,ηβ,γη,γ(βαγ),η4 modulo Iˆ if d=3
2,cˆ
(12) (resp. α(γβα),β(αγβ),α2,ηβ,γη,βγ,η2d−2+1 modulo Iˆ if d>3)
2,cˆ
all lie in Ker(π ). We first show that for d = 3, we also have that βγ and η2d−2+1 modulo Iˆ lie
Λ 2,cˆ
in Ker(π ). Using that Λ /soc(Λ ) is a string algebra,we see that if d=3, there is no uniserial
Λ 2,c 2,c
Λ -module with descending composition factors T ,T ,T . Because of the Morita equivalence
2,c 1 1 1
′ ′
between Λ and Λ induced by π , there is then also no uniserial Λ -module with descending
2,c 2,c Λ 2,c
compositionfactorsS1,S1,S1. However,wecandefinetheuniserialΛˆ2,cˆ-moduleMη2 corresponding
tothepathη2 inkQ withsuchdescendingcompositionfactors. Sinceβγ =η3 moduloIˆ ifd=3,
2 2,cˆ
we can use (11) to argue similarly to (6) that there are constants b ,b in k such that
1 2
(13) η2 + b βαγ + b η3 modulo Iˆ
1 2 2,cˆ
lies in Ker(π ) if d=3. Multiplying (13) on the left with η modulo Iˆ and using that ηβ and η4
Λ 2,cˆ
modulo Iˆ lie inKer(π ) by (11), wesee thatη3, andthus βγ, modulo Iˆ lie in Ker(π )if d=3.
2,cˆ Λ 2,cˆ Λ
Thus it follows from (11) and (12) that for all d≥3,
(14) α(γβα),β(αγβ),α2,ηβ,γη,βγ,η2d−2+1 modulo Iˆ
2,cˆ
all lie in Ker(π ). Using that Λ /soc(Λ ) is a string algebra, we see that there is no uniserial
Λ 2,c 2,c
Λ -module with descending composition factors T ,T ,T ,T . Because of the Morita equivalence
2,c 1 0 0 1
′ ′
between Λ and Λ induced by π , we see that there is also no uniserial Λ -module with
2,c 2,c Λ 2,c
descending composition factors S ,S ,S ,S . On the other hand, we can define the uniserial Λˆ -
1 0 0 1 2,cˆ
module M corresponding to the path βαγ in kQ with such descending composition factors.
βαγ 2
Thus we canargue similarly to (6) and (8), using (14), that there exist constants a2,...,a2d−2 in k
such that
2d−2
(15) βαγ + a ηj modulo Iˆ
X j 2,cˆ
j=2
lies in Ker(π ). Suppose that not all a are zero, and let 2 ≤ j ≤ 2d−2 be minimal with a 6= 0.
Λ j 0 j0
If j <2d−2 then d>3. Since ηβ and η2d−2+1 modulo Iˆ lie in Ker(π ) by (14), we can multiply
0 2,cˆ Λ
(15) on the left with η2d−2−j0 modulo Iˆ to see that η2d−2 modulo Iˆ lies in Ker(π ). But this
2,cˆ 2,cˆ Λ
impliesthatallpathsoflength2d−2 moduloIˆ lie inKer(π ),andhence rad2d−2(Λ′ )=0,which
2,cˆ Λ 2,c
is a contradiction. Thus a =0 for 2≤j <2d−2, and we obtain
j
(16) π βαγ−c η2d−2 modulo Iˆ =0
Λ(cid:16) 1 2,cˆ(cid:17)
for some c ∈k. Similarly, one sees that
1
(17) π η2d−2 −c βαγ modulo Iˆ =0
Λ(cid:16) 2 2,cˆ(cid:17)
forsomec ∈k. Ifoneofc ,c werezero,wecouldalsochoosetheotheronetobezero. Thiswould
2 1 2
′
imply that the radical series length of the projective Λ -module cover of S is at most 3 (resp.
2,c 1
2d−2) if d = 3 (resp. if d > 3), which is a contradiction. Thus c ,c have to be non-zero. Using
1 2
thatΛ /soc(Λ )isastringalgebra,weseethatthereisnouniserialΛ -modulewithdescending
2,c 2,c 2,c
composition factors
(T ,T ,T ,T ) (resp. (T ,T ,T ,T )).
0 0 1 0 0 1 0 0
′
Because of the Morita equivalence between Λ and Λ induced by π , we see that there is also
2,c 2,c Λ
′
no uniserial Λ -module with descending composition factors
2,c
(S ,S ,S ,S ) (resp. (S ,S ,S ,S )).
0 0 1 0 0 1 0 0
10 FRAUKEM.BLEHER
On the other hand, we candefine a uniserial Λˆ -module M with w=γβα (resp. w=αγβ) with
2,cˆ w
such descending composition factors. Hence we obtain in a similar way to (16) and (17)
(18) π γβα−c αγβ modulo Iˆ = 0,
Λ(cid:16) 3 2,cˆ(cid:17)
(19) π αγβ−c γβα modulo Iˆ = 0
Λ(cid:16) 4 2,cˆ(cid:17)
′
for certain c ,c ∈ k. Moreover, since the radical series length of the projective Λ -module cover
3 4 2,c
of S is 4, we can argue as above that c ,c have to be non-zero. Therefore, there exist certain
0 3 4
c ,c ∈k∗ such that Ker(π ) contains the image Jˆ in Λˆ of the ideal
1 3 Λ c1,c3 2,cˆ
J =hα2,ηβ,γη,βγ,βαγ−c η2d−2,γβα−c αγβi ⊂ kQ .
c1,c3 1 3 2
Wecancomputethek-dimensionofkQ2/I2,c =Λ2,c (resp. ofkQ2/Jc1,c3 ∼=Λˆ2,cˆ/Jˆc1,c3)bycounting
k-linearly independent paths in kQ modulo I (resp. modulo J ). Since these k-dimensions
2 2,c c1,c3
are equal, it follows that Ker(πΛ)=Jˆc1,c3 and thus Λ′2,c ∼=kQ2/Jc1,c3. Because Λ′2,c is symmetric,
we canarguein a similarwayas for Λ′1,c that c3 =1. Hence Λ′2,c ∼=kQ2/Jc1,1 for a certainc1 ∈k∗.
Since the k-algebra homomorphism
Λ =kQ /I −→ kQ /J
2,0 2 2,0 2 c1,1
z modulo I 7→ z modulo J for z ∈{1 ,1 ,α,β,η},
2,0 c1,1 0 1
γ modulo I 7→ c−1γ modulo J
2,0 1 c1,1
is a k-algebra isomorphism for all c1 ∈ k∗, it follows that Λ′2,c ∼= Λ2,0. Therefore, the parameter c
must be equal to zero. (cid:3)
Corollary 3. Let k be an algebraically closed field of characteristic 2, and let d≥3 be an integer.
Suppose Gˆ is a finite group with generalized quaternion Sylow 2-subgroups of order 2d+1 such that
Gˆ has no normal subgroups of odd order. Let Z(Gˆ) denote the center of Gˆ and define G=Gˆ/Z(Gˆ).
Then G has dihedral Sylow 2-subgroups of order 2d. If the principal block B of kG has precisely two
isomorphism classes of simple modules, then B is Morita equivalent to either Λ or Λ .
1,0 2,0
Proof. By a result of Brauer and Suzuki [5], the center of Gˆ is generated by an element of order 2.
Since the principal block B of kG is contained in the image of the principal block Bˆ of kGˆ under
the natural projection π :kGˆ →kG, Corollary 3 follows from Theorem 2. (cid:3)
Corollary 4. Let k be an algebraically closed field of characteristic 2. Suppose q is an odd prime
power and B is the principal block of kPGL (F ). Further, let 2d ≥ 8 be the maximal 2-power
2 q
dividing (q2 −1). Then B is Morita equivalent to Λ (resp. Λ ) if q ≡ 1 mod 4 (resp. q ≡ 3
1,0 2,0
mod 4).
Proof. By aresultofDickson(see forexample [10, HauptsatzII.8.27]),the projectivespeciallinear
groupPSL2(Fq2)containsasubgroupHq thatisisomorphictoPGL2(Fq). Considertheshortexact
sequence of groups
(20) 1→{±I}→SL2(Fq2)−→σ PSL2(Fq2)→1
where I is the 2×2 identity matrix in SL2(Fq2) and σ is the natural surjection. Define Gˆq to be
the full preimage in SL2(Fq2) of Hq under the map σ. Then Gˆq/{±I} is isomorphic to PGL2(Fq).
Moreover, the center of Gˆq is {±I}, since the center of Hq ∼= PGL2(Fq) is trivial. The order of
Gˆq is twice the order of PGL2(Fq), and thus equal to 2·q(q2 −1). Since the order of SL2(Fq2)
is equal to q2(q2−1)(q2+1), it follows that the Sylow 2-subgroups of SL2(Fq2) and the Sylow 2-
subgroups of Gˆq have order 2d+1. Because SL2(Fq2) has generalized quaternion Sylow 2-subgroups
(see for example [10, Satz II.8.10]),it follows that the Sylow 2-subgroupsofGˆ arealso generalized
q
quaternion. Hence Corollary 4 is a consequence of Corollary 3. (cid:3)