AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS Contents 1. Introduction 1 2. The ...

15 Pages·2012·0.26 MB·English

AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS
ABHINAVSHRESTHA
Abstract. This paper will present an introduction to modular representa-
tions of finite groups via the techniques of block theory, first introduced by
Brauer. Afterdevelopinganunderstandingofblockidempotents,Brauerpairs,
and defect groups, we will present applications to the theory of finite groups
andtheirfusionsystems.
Contents
1. Introduction 1
2. The Brauer Morphism and Relative Trace 3
3. Brauer Pairs 4
4. Defect groups 6
5. Fusion Systems of Groups 10
6. Fusion Systems of Blocks 11
Acknowledgments 15
References 15
1. Introduction
A representation of a finite group over a field k can be thought of as a module
overthegroupalgebrakG. Ifk isafieldofcharacteristiczero, theresultsobtained
are stronger than those obtained in positive characteristic. In particular, we have
the following theorem by Maschke.
Theorem 1.1 (Maschke). Let k be a field, and G a group, such that the order of
G does not divide the characteristic of the field k. Then, if M is a kG-module, M
is completely reducible.
Thus, the irreducible representations of a finite group G are the key to under-
standing arbitrary finite-dimensional representations of G in characteristic zero.
In positive characteristic, irreducible representations are not as fundamental, as
we have representations that cannot be expressed as the direct sum of irreducible
ones. Manyrepresentationscontainsubrepresentationswhosecomplementsarenot
representations themselves, as seen in the following example.
Example 1.2. LetkbethefinitefieldF ,andletG=Z/2Z={g|g2 =e}(written
2
multiplicatively). Then, kG is a 2-dimensional vector space over k, generated by
e and g. The trivial representation, corresponding to the subspace generated by
Date:DEADLINEAUGUST24,2012.
1
2 ABHINAVSHRESTHA
e+g, isanirreduciblesubspace. However, anyother1dimensionalsubspaceisnot
a subrepresentation, as it is not invariant under the action of g ∈Z/2Z.
Instead, we desire to understand representations via their indecomposable sub-
representations. It turns out that while the positive characteristic representations
do not necessarily decompose into irreducible representations, they do decompose
into the sum of indecomposable subrepresentations, via the following lemma.
Lemma 1.3. Let k be a field, G a finite group, and M a finitely generated kG-
module. Then there is a decomposition of M into indecomposable two-sided ideals.
That is,
r
(cid:77)
M = B
i
i=1
where B is an indecomposable two-sided ideal of M.
i
Forconvenience, wewillcallanindecomposabletwo-sidedidealsummandofkG
a block of kG.
There is a correspondence between the blocks of kG and certain central idem-
potents of the group algebra that will prove crucial to our theory. Recall that a
non-zeroelementeinkGisanidempotent ife2 =e. Wesaythattwoidempotents,
e and e(cid:48) are orthogonal if ee(cid:48) = e(cid:48)e = 0. Finally, an idempotent is called central
if lies in the center of kG, and an idempotent is called primitive if it cannot be
written as the sum of two orthogonal idempotents.
A central idempotent will be called primitive if it is primitive in the center of
kG, rather than in kG itself. A central idempotent e therefore may be written as
e = e +e , for two idempotents e and e if they do not lie in the center of kG.
1 2 1 2
The following propositions establish the correspondence between primitive central
idempotents and blocks of kG.
Proposition 1.4. Let R be a Noetherian ring, and M a finitely-generated R-
algebra.
(i) There is a one-to-one correspondence between decompositions of M into two-
sidedidealsanddecompositionsof1asthesumofpairwiseorthogonal, central
idempotents. In particular, a decomposition 1 = e + e + ··· + e is in
1 2 r
correspondence with M = Me ⊕ Me ⊕ ··· ⊕ Me , and a decomposition
1 2 r
M = B +···+B is in correspondence with the decomposition of 1 as an
1 r
element of M in this ideal decomposition.
(ii) If B is a block of kG, with M =B⊕B(cid:48), and 1=e+e(cid:48) as the corresponding
decomposition, then e is a primitive central idempotent. Conversely, if e is a
primitive central idempotent, then Me is a block of M.
Proposition 1.5. Let R be a Noetherian ring, and M a finitely generated R-
algebra, and let M =Me ⊕···⊕Me be a decomposition of M into blocks. Then
1 r
if B =Me is a block of M, then B =Me , for some i.
i
Givenadecompositionasintheprevioustheorem,wecallthee theblock idem-
i
potents of M. Now that we know more of the theory, we can provide a more
interesting example.
Example 1.6. Let k be a field of characteristic 2, and let G = S . We want to
3
find the decomposition of kG into indecomposable representations by computing
AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS 3
block idempotents. Recall that the center of the group algebra Z(kG) is generated
by class sums of conjugacy classes of G. Since the block idempotents are central,
this is where we will begin searching. The conjugacy classes of S correspond
3
exactly with the cycle types of the permutations. Thus, Z(kG) is generated by
1, (1 2)+(1 3)+(2 3), and (1 2 3)+(1 3 2). A computation yields that b =
0
1+(1 2 3)+(1 3 2) and b =(1 2 3)+(1 3 2) are both block idempotents.
1
Note that in general, the group algebra kG will not contain all indecomposable
representations as subrepresentations. Indeed, it is often the case that there are
infinitely many indecomposable representations.
Finally, we will state the following lemma.
Lemma 1.7 (Rosenberg’s Lemma). Let R be a Noetherian ring, and M a finitely-
generated R-module, and let e be a primitive idempotent of M. If e lies in a sum
of two-sided ideal of M, then e lies in at least one of them.
For the remainder of the paper, we will let G be a finite group, p be a prime
dividing the order of G, and k an algebraically closed field of characteristic p.
2. The Brauer Morphism and Relative Trace
We now define the Brauer morphism and relative trace map. Despite the fact
thatthedefinitionsofthesemapsarefairlystraightforward, theywillbeincredibly
useful in the future.
Definition 2.1. Let P be a p-subgroup of G. The Brauer morphism, Br :kG→
P
kC (P) is the surjective k-linear map given by the following:
G
(cid:88) (cid:88)
α g (cid:55)→ α g
g g
g∈G g∈CG(P)
TheBrauermorphismistheprojectionofkGontothekC (P). Thisisnotak-
G
algebra homomorphism in general. For example, consider kS and let P =(cid:104)(1 2)(cid:105).
3
The 3-cycles are both mapped to zero, but since they are inverses in S , their
3
product is not mapped to zero. However, if we restrict our domain, then Br is
P
an algebra homomorphism. Let H be a subgroup of the group G acting on G by
conjugation. This action can be extended linearly to an action on kG. The Brauer
mapisanalgebrahomomorphismifwerestrictthedomaintotheP-stableelements
of kG, that is, the elements of kG fixed under the action of P.
Notation 2.2. The H-stable elements of an arbitrary set X will be denoted by
XH. In the case that X =kG, we write (kG)H.
Proposition2.3. LetP beap-subgroupofG. ThemapBr ismultiplicativewhen
P
restricted to (kG)P, the P-stable elements of kG.
Corollary 2.4. Let P be a p-subgroup of G.
(i) If e is a central idempotent of kG, then Br (e) is either zero or a central
P
idempotent of kC (P).
G
(ii) If b is a block idempotent of kG, such that Br (b) (cid:54)= 0, then Br (b) = b +
P P 1
···+b is a sum of block idempotents of kC (P).
r G
(iii) If e is a block idempotent of kC (P), then Br (b)e (cid:54)= 0 if and only if e = b
G P i
for some i. In this case, Br (b)b =b .
P i i
4 ABHINAVSHRESTHA
The next step is to define the relative trace map.
Definition 2.5. Let G be a finite group acting on an abelian group X, and let H
be a subgroup of G. The relative trace, denoted TrG :XH →XG, is the map
H
(cid:88)
TrG(x)= xg
H
g∈T
where T is a right transversal to H in G.
Note that the relative trace is well-defined, since, using the notation above, if
Hg =Hh, and x∈X, then xg =xhg−1g =xh.
Notation 2.6. Given a group M with a G-action, then we can consider the group
algebra kM as an abelian group under addition with a G-action by extending
linearly. Wewrite(kM)G todenotetheimageof(kM)H underthetracemapTrG.
H H
We will require several properties of the relative trace, but we omit the proofs
as they are tangential to the goal of this paper.
Theorem 2.7. Let e be a block idempotent of kG. There exists a p-subgroup D
such that, for any subgroup H of G, we have that e ∈ (kG)G if and only if H
H
contains a conjugate of D. The subgroup D is unique, up to G-conjugacy.
Lemma 2.8. Let H be a subgroup of G, and let P be a p-subgroup of H. Then,
(cid:88)
Ker(Br )∩(kG)H = (kG)H
P Q
Q∈Q
where Q is the set of all subgroups of H not containing a subgroup H-conjugate to
P.
3. Brauer Pairs
WenowdefinetheconceptofaBrauerpair, whichwillbeapowerfultoolinour
study of fusion systems of blocks.
Definition 3.1. Let G be a finite group and let p be a prime dividing the order
of the group G. A Brauer pair consists of a p-subgroup Q of G, and a block
idempotent e of kC (Q). We write (Q,e) to denote such a Brauer pair.
G
SinceC (Qg)=C (Q)g,thisgivesabijectionbetweentheblockidempotentsof
G G
kC (Q) and kC (Qg). Thus, G acts on the Brauer pairs by conjugation, sending
G G
Q to Qg and e to eg.
Lemma 3.2. Let G be a finite group, and let Q(cid:69)R be p-subgroups of G. If e is a
block idempotent of kC (R), then there is a unique R-stable block idempotent f of
G
kC (Q) such that
G
Br (f)e=e.
R
If f(cid:48) is any other R-stable block idempotent of kC (Q), then Br (f(cid:48))e=0.
G R
Proof. We first prove the second part of the lemma. Suppose f is a block idem-
potent of kC (Q) such that Br (f)e = e, and let f(cid:48) (cid:54)= f be a an R-stable block
G R
idempotent. Then, the result follows from the following calculation:
Br (f(cid:48))e=Br (f(cid:48))(Br (f)e)=Br (f(cid:48)f)e=0.
R R R R
AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS 5
To show existence, first note that if g ∈ G, and r ∈ R, then Br (gr) = Br (g),
R R
since g ∈ C (R) if and only if gr ∈ C (R). As a result, Br (fr) = Br (f). So,
G G R R
the group R acts on the block idempotents of kC (Q) via conjugation, and any
G
idempotent that is not R-stable belongs to an orbit whose length is a multiple of
p. Since k has characteristic p, this gives us that
(cid:88) (cid:88)
1=Br (1)= Br (b)= Br (b)
R R R
b∈B(Q) b∈B(Q)R
where B(Q) is the set of block idempotents of kC (Q), and B(Q)R is the R-stable
G
subset of B(Q). By multiplying on the right by some e∈B(R), we notice
(cid:88)
e=1·e= Br (b)e.
R
b∈B(Q)R
In particular, Br (b)e (cid:54)= 0 for some b. Letting f be such a block idempotent, we
R
apply Corollary 2.4 to see that Br (f)e=e. (cid:3)
R
We can define a partial ordering on the set of Brauer pairs of a group G. We
firstdefinethenon-transitiverelation(cid:69)asfollows. Wesay(Q,f)(cid:69)(R,e)ifQ(cid:69)R,
the block idempotent f is R-stable, and Br (f)e=e. We take the partial order ≤
R
on the Brauer pairs to be the transitive closure of the relation (cid:69). The lemma we
just proved shows that given Q(cid:69)R, and a block idempotent e of kC (R), there is
G
a unique Brauer pair (Q,f) such that (Q,f)≤(R,e). It turns out we have a much
stronger result.
Theorem 3.3. Let Q ≤ R be p-subgroups of G. If (R,e) is a Brauer pair, then
there is a unique Brauer pair (Q,f) such that (Q,f)≤(R,e).
In order to prove this theorem, we require the following lemma.
Lemma 3.4. Let R be a p-subgroup of G. Let P and Q be normal subgroups of R
with P ≤ Q. Suppose that e is a block idempotent of kC (R), and let f and f
G 1 2
be the unique R-stable block idempotents of kC (P) and kC (Q) respectively, such
G G
that Br (f )e=e, and Br (f )e=e. If f is the unique Q-stable block idempotent
R 1 R 2
of kC (P) with Br (f)f =f , then f =f .
G Q 2 2 1
As a result, if P (cid:69)R, and e is a block idempotent of kC (R), then there is a
G
unique Brauer pair (P,f) such that (P,f)≤(R,e).
Proof. First, we show that f is indeed R-stable. Let x ∈ R, and note that fx is
a block idempotent of kC (P) and is Q-stable, since Q(cid:69)R. Thus, Br (fx) =
G Q
Br (f)x. But, f is R-stable, and so
Q 2
Br (fx)f =Br (f)xfx =fx =f .
Q 2 Q 2 2 2
Thus, fx =f, and f is R-stable by Lemma 3.2.
Now, we want to check that Br (f)e=e, via the following computation:
R
Br (f)e=Br (f)Br (f )e=Br (Br (f))e
R R R 2 R Q
=Br (Br (f)f )e=Br (f )e=e.
R Q 2 R 2
Thus, f =f . This shows that if (P,f)(cid:69)(Q,f )(cid:69)(R,e), and (P,f )(cid:69)(R,e), then
1 2 1
(P,f)=(P,f ),andthatif(P,f )(cid:69)(Q,f )(cid:69)(R,e)andP(cid:69)R,then(P,f )(cid:69)(R,e).
1 1 2 1
Now suppose we have another Brauer pair (P,f(cid:48))≤(R,e), with P (cid:69)R, via the
chain
(P,f(cid:48))(cid:69)(Q ,f(cid:48))(cid:69)···(cid:69)(Q ,f(cid:48))(cid:69)(R,e).
1 1 r r
6 ABHINAVSHRESTHA
We can apply the previous process inductively on (P,f(cid:48))(cid:69)(Q ,f(cid:48))(cid:69)(Q ,f(cid:48)), and
1 1 2 2
onwards to see that (P,f(cid:48))(cid:69)(R,e). (cid:3)
Proof of Theorem. We induct on the index [R:Q]. The base case, when [R:Q]=
p,isproveninthelemma,sinceinthiscase,Q(cid:69)R. Furthermore,givenasubnormal
series of subgroups of R, that is a chain of subgroups Q=P (cid:69)P (cid:69)···(cid:69)P =R,
1 2 r
we have a unique chain of inclusions of Brauer pairs.
We now consider the case of some arbitrary subgroup Q of R. We can exhibit a
subnormal series by taking the iterated normalizers. To see that this does create a
subnormalseries,firstnoticethatbytheclassequation,anyp-grouphasnon-trivial
center. Now suppose for contradiction that N (S) = S for some proper subgroup
R
S of R. Then S = S/Z(S) is a subgroup of R = R/Z(S) with normalizer
1 1
N (S)/Z(S) = S/Z(S). Continuing inductively for S = S /Z(S ) and R =
R i i−1 i−1 i
R /Z(S ), we get that |S | < |R |, for all i. Then, S must be the trivial
i−1 i−1 i i i
group for some i. For this i, R must also be the trivial group, since otherwise
i
N (S )(cid:54)=S , which is a contradiction.
Ri i i
Suppose we have a subnormal series
Q=R (cid:69)···(cid:69)R (cid:69)R.
0 s
We assume that the R are distinct. Furthermore, since R ≤ N (Q), we apply
i 1 R
induction to see that there is a unique Brauer pair (R ,f(cid:48))≤(R,e), and a unique
1
Brauer pair (N (Q),f(cid:48)(cid:48))≤(R,e). Thus, by the lemma,
R
(Q,f)(cid:69)(N (Q),f(cid:48)(cid:48))(cid:69)(R,e).
R
Since both (R ,f(cid:48)) and (N (Q),f(cid:48)(cid:48)) are less than (R,e), we apply the inductive
1 R
hypothesis to get that
(R ,f(cid:48))≤(N (Q),f(cid:48)(cid:48)).
1 R
Therefore, (Q,f)(cid:69)(N (Q),f(cid:48)(cid:48)) and (R ,f(cid:48)) ≤ (N (Q),f(cid:48)(cid:48)), so (Q,f)(cid:69)(R ,f(cid:48)).
R 1 R 1
(cid:3)
TheBrauerpairscontainedin(P,e)thereforeformaposetidenticaltotheposet
of subgroups of P. Note that although if given a Brauer pair (P,e) and Q ≤ P,
there is a unique Brauer pair (Q,e ) ≤ (P,e), this does not hold in the opposite
Q
direction. Furthermore, the pair (Q,e ) may be contained in another Brauer pair
Q
(P,f) with f (cid:54)=e.
4. Defect groups
ThemachineryofBrauerpairsputsusonestepclosertodevelopingthenotionof
thefusionsystemofablock. Wecontinueonthispathbyintroducingthefollowing
definition.
Definition 4.1. Let G be a finite group and let b be a block idempotent of kG. A
b-Brauer pair is a Brauer pair (R, e) such that Br (b)e=e. A maximal b-Brauer
R
pair is a b-Brauer pair (D,e) such that |D| is maximal. We call the subgroup D a
defect group of the block b.
Notethatwecanalsocharacterizeb-Brauerpairsinthefollowingtwoways. First,
a Brauer pair (R,e) is a b-Brauer pair if e is a term of Br (b), when expressed as a
R
sum of block idempotents of kC (R). Second, a Brauer pair (R, e) is a b-Brauer
G
pair if (1, b) ≤ (R, e).
AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS 7
Example 4.2. Let p=2. Let V =Z/3Z×Z/3Z, the elementary abelian group of
order 9, with generators u and v. Let Q = (cid:104)x,y|x4 = y4 = 1,xyx−1 = y3(cid:105) be the
quaternion group. We let Q act on V on the left as follows:
x·u=u2 y·u=u
x·v =v y·v =v2
Let G=V (cid:111)Q. Given g ∈G, denote the class sum of g, that is, the sum of all
elements of the conjugacy class of g, by gˆ. The group algebra kG has 4 blocks:
b =1+uˆ+vˆ+uˆv b =vˆ+uˆv
0 1
b =uˆ+uˆv b =uˆv
2 3
The subgroups of Q and their centralizers are as follows:
R≤Q C (Q)
G
1 G
Z(Q)=(cid:104)x2(cid:105)=(cid:104)y2(cid:105) G
(cid:104)x(cid:105) (cid:104)y(cid:105) (cid:104)xy(cid:105) (cid:104)v(cid:105)×(cid:104)x(cid:105) (cid:104)u(cid:105)×(cid:104)y(cid:105) (cid:104)xy(cid:105)
Q Z(Q)
Under the Brauer morphism, the blocks map to the following idempotents
Br (b ) Br (b ) Br (b )
R 0 R 1 R 3
b b b
0 1 3
b b b
0 1 3
(1+v+v2) (1+u+u2) 1 (v+v2) 0 0 0 0 0
1 0 1
Thecaseforb isomittedsinceitisanalogoustob . Thisdescribestheb-Brauer
2 1
pairs contained in Q.
We now analyze the conjugacy action of G on its b-Brauer pairs. We begin with
a few useful lemmas. The proof of the first is omitted [3].
Lemma 4.3. Let b be a block idempotent of kG, and D be a defect group of b.
Then, b∈(kG)G, i.e. there is some a∈(kG)D such that b=TrG(a).
D D
Lemma 4.4. Let G be a finite group, let b be a block idempotent of kG, and let Q
be a p-subgroup of G.
(i) There exists a b-Brauer pair (Q,e) if and only if Br (b)(cid:54)=0.
Q
(ii) The subgroup Q is a defect group if and only if Br (b) (cid:54)= 0, and for all p-
Q
subgroups R properly containing a conjugate of Q, we have Br (b)=0.
R
Proof. If Br (b) = 0, then Br (b)e = 0, and (Q,e) is not a b-Brauer pair, for
Q Q
any block idempotent e of kC (Q). Conversely, if Br (b) (cid:54)= 0, then we can write
G Q
Br (b) = b +b +···+b , where the b are block idempotents of kC (Q), and
Q 1 2 r i G
Br (b)e=e when e=b for some i.
Q i
8 ABHINAVSHRESTHA
Suppose there is some g ∈ G and subgroup R of G such that Qg is properly
contained in R, with Br (b) (cid:54)= 0. Then, (Q,f) is a b-Brauer pair if and only
R
if (Qg,fg) is a b-Brauer pair. Furthermore, (Qg,fg) ≤ (R,e). Then (R,e) is a
b-Brauer pair with strictly greater order than Q, so Q cannot be a defect group.
Now suppose that Q is not a defect group of b, and (Q,f) is a b-Brauer pair.
Then, thereissomedefect group R andsome g ∈G suchthat Qg ≤R. ByLemma
4.3, b ∈ (kG)G, that is b = TrG(a) for some a ∈ (kG)R. Let Cl denote the set
R R R
of conjugacy classes of G containing an element x such that R contains a Sylow
p-subgroup of C (x). The set of class sums of Cl , that is {Xˆ|X ∈ Cl }, is a
G R R
k-basis for (kG)G. Thus, we can write
R
b= (cid:88) α Xˆ, α ∈k
X X
X∈ClR
BecauseBr (b)(cid:54)=0,thereissomeX ∈Cl suchthatBr (Xˆ)(cid:54)=0,andX∩C (Q)
Q R Q G
is not empty, and there is some x ∈ X that commutes with Q. In particular,
Q ≤ C (x), and thus Q is contained in some Sylow p-subgroup of C (x), and
G G
hence conjugate to a subgroup of R. (cid:3)
Theorem 4.5. Let b be a block idempotent of kG. Suppose D is a minimal p-
subgroup such that b ∈ (kG)G. Then, D is a defect group of b. Furthermore, G
D
acts transitively by conjugation on the set of defect groups of b.
Proof. Let D denote a minimal p-subgroup of G such that b ∈ (kG)G. We claim
D
that given a p-subgroup Q, Br (b) (cid:54)= 0 if and only if Q is contained in D. First,
Q
suppose that Br (b)=0. By Lemma 2.8, b∈ (cid:80) (kG)G, where R is the collection
Q R
R∈R
of all p-subgroups, R, not containing a conjugate of Q. By Rosenberg’s Lemma
(1.7), b lies in (kG)G, for some R∈R. Thus, Q is not contained in D.
R
Now, supposethatBr (b)(cid:54)=0. Then, againbyLemma2.8, bdoesnotlieinany
Q
(kG)G, where R does not contain any conjugate of Q. Thus, if b∈(kG)G for some
R S
p-subgroup S, then S does contain a conjugate of Q. Since b∈(kG)G, we see that
D
D contains a conjugate of Q as claimed.
Next,Theorem2.7tellsusthatminimalp-subgroupsP ofGsuchthatb∈(kG)G
P
must be conjugate. Thus, G acts transitively on the defect groups of b. (cid:3)
It turns out that this conjugation action is compatible with the the Brauer
pairs. That is, the group G acts transitively not only on the defect groups of a
block idempotent b, but on the maximal b-Brauer pairs. We will first show this for
the case of normal defect groups, but omit the proof of the first claim.
Proposition 4.6. Let P be a normal p-subgroup of G, and let b be a block idempo-
tent of kG. Then defect groups of b contain P, and b∈(kC (P))G. Furthermore,
G
write b = b +b +···+b , where b is a block idempotent of kC (P), and let
1 2 r i G
D be a defect group of b , and H be the stabilizer in G of b . Then, for each i,
i i i i
we have that b = TrG (b ). Moreover, if b has defect group P, then the Brauer
Hi i
pairs (P,b ) are the maximal b-Brauer pairs, and G acts transitively on the set of
i
maximal b-Brauer pairs.
Proof. LetT beatransversaltoH inG. Ift∈T ,thenbt =band((cid:80)b )t =(cid:80)b .
j j j i i
Moreover, bt is a block of kC (P), since C (P)t = C (Pt) = C (P) and hence
i G G G G
conjugation by t induces an automorphism of kC (P). We know H stabilizes b .
G j j
AN INTRODUCTION TO THE FUSION SYSTEMS OF BLOCKS 9
Additionally, G acts transitively on the block idempotents, since otherwise, there
is a j such that for all i and g ∈ G, bg (cid:54)= b , and b = (cid:80)bg − (cid:80)b , which is
i j j i i
i(cid:54)=j
impossiblesinceb isprimitive. Therefore, wecanarrangeeachT ={t ,t ,...t },
j i 1 2 r
where bti =b . Then, b=TrG (b ).
j i Hj j
NowsupposebhasdefectgroupP. Sinceb∈kC (P),Br (b)=b. Inparticular,
G P
(cid:80)
given the decomposition b= b into block idempotents, we have that the (P,b )
i i
are maximal b-Brauer pairs. Since G acts transitively on the b , it acts transitively
i
on the pairs. (cid:3)
The following is a major result due to Brauer. The proof is omitted, but can be
found in David Craven’s Theory of Fusion Systems[1], or most standard introduc-
tory texts in modular representation theory.
Theorem 4.7 (Brauer’s First Main Theorem). Let P be a p-subgroup of G. The
map Br induces a bijection between the blocks of kG with defect group P and the
P
blocks of kN (P) with defect group P.
G
Corollary 4.8. If b is a block idempotent of kG, then G acts transitively on the
set of maximal b-Brauer pairs.
Proof. We’ve already shown that G acts transitively on the defect groups of b in
Theorem 4.5. Fix such a defect group D. By Brauer’s First Main Theorem, we
know Br induces a bijection between the block idempotents of kG with defect
D
groupD andtheblockidempotentsofkN (D)withdefectgroupD. Furthermore,
G
D is normal in N (D), so we can apply Proposition 4.6 to see that N (D) acts
G G
transitively on the maximal Br (b)-Brauer pairs. However, since Br is a pro-
D D
jection, the maximal Br (b)-Brauer pairs are exactly the maximal b-Brauer pairs,
D
proving the corollary. (cid:3)
Therelationshipbetweenrepresentationsandthedefectgroupisnotcompletely
understood and is the source of many conjectures in modern group representation
theory,includingBrauer’sk(B)ConjectureandBroue’sAbelianDefectConjecture.
Thefinaltheoremofthissectionwillgiveasampleofhowthedefectgroupcanyield
information about the representation. We provide a proof of one direction of the
theorem. Theconverseisomitted,butcanbefoundinRadhaKessar’sIntroduction
to Block Theory [3].
Theorem 4.9. Let b be a block idempotent of kG and let P be a defect group of b.
Then, P is trivial if and only if kGb is isomorphic to a matrix algebra over k.
Proof. Suppose that P is trivial. Then b = TrG(a) for some a ∈ kG by Theorem
1
4.5. Let M and N be kG-modules that are also kGb-modules, and let ϕ:M →N
be a surjective map of kG-modules. Furthermore, let ψ : N → M be a k-linear
map (not necessarily a module homomorphism) that is a splitting of ϕ. For each
y ∈ N, define ψˆ(y) = (cid:80) xψ(ax−1y). Then ψˆ is a kG-linear splitting of ϕ by the
x∈G
10 ABHINAVSHRESTHA
following calculation:
(cid:32) (cid:33)
ϕ(ψˆ(y))=ϕ (cid:88)xψ(ax−1y)
x∈G
= (cid:88)x(cid:0)ϕ(ψ(ax−1y)(cid:1)
x∈G
(cid:88)
= xax−1y
x∈G
=by =y.
Thus,everykGb-moduleisprojective,andhence,kGbissemisimple(completely
decomposable). Since kGb is also indecomposable, kGb must be a simple algebra.
By Wedderburn’s Theorem on classifying semisimple rings, kGb must therefore be
isomorphic to a matrix algebra over k. (cid:3)
5. Fusion Systems of Groups
We now have enough machinery to be able to define fusion systems of blocks,
but first we will explore fusion systems on finite groups.
Definition5.1. Afusionsystem F ofafinitep-groupP isacategorywhoseobjects
are all subgroups of P, and given two subgroups Q and R, the set of morphisms
Hom (Q,R) is a subset of allinjective homomorphismsfrom Q to R satisfying the
F
following properties.
(i) For each g ∈ P with Qg ≤ R, the associated conjugation map c : Q → R is
g
in Hom (Q,R)
F
(ii) Foreachϕ∈Hom (Q,R),theassociatedisomorphismobtainedbyrestricting
F
the codomain to the image of ϕ lies in Hom (Q,ϕ(Q))
F
(iii) If ϕ ∈ Hom (Q,R) is an isomorphism, then its inverse ϕ−1 : R → Q lies in
F
Hom (R,Q).
F
The main examples of fusion systems are those of the Sylow p-subgroups of a
group G. The fusion system of G on a Sylow p-subgroup P, denoted F (G), is the
P
category with the subgroups of P as its objects. Given two subgroups R and Q of
P, the set Hom (Q,R) consists of maps induced by conjugation by elements g
FP(G)
of G such that Qg ≤ R. Note that these maps do not have to be surjective. The
fusion system of a group G thus encapsulates the information of how p-subgroups
are embedded in the group G.
Example 5.2. Recall the group G=V (cid:111)Q, as in Example 4.2. We let p=2, and
consider the fusion system of the Sylow 2-subgroup Q. Since the subgroups 1 and
Z(Q) are central, Hom(1,R) and Hom(Z(Q),R) are trivial, for any subgroup R
of Q. The subgroup (cid:104)x(cid:105) has one non-trivial automorphism, which sends x to x−1,
and is induced by conjugation by y, y3, xy, or (xy)3. This extends to the unique
non-trivial map from (cid:104)x(cid:105) to Q. The situation for the other subgroups of order 4
is analogous. Finally, there are 3 non-trivial automorphisms of Q in the category,
induced by conjugation by x, y, and xy respectively. All other sets of morphisms
are empty. Since the full automorphism group Aut(Q) has order 4 and |V|=4, no
non-trivial morphisms in the fusion system are induced by elements in V.