数理解析研究所講究録
991 巻 1997 年 16-27

16

CASES OF DADE’S CONJECTURE$arrow\tilde{\hat{\overline{\tau}}}^{g_{)}^{\mathrm{I}}}|$$7\mathrm{X}\mathcal{F}_{\backslash }-\not\subset$$)\Re P-\tau$(Katsuhiro UNO)

We fix a prime . In 1992, Dade published a paper ”Counting characters inblocks, I”, in which he gave a conjecture on the numbers of irreducible characterswith certain heights in a block $B$ and those of irreducible characters with related
heights in p–local blocks related to $B$ . Local blocks are those of the normalizers
of chains of p–subgroups such that they induce the given block $B$ . An importantinvariant of a character is in fact its defect, i.e., the difference between the exponents
of in the order of the subgroup of which it is an irreducible character and in its
degree. For each block, we must count the number of characters of a certain given
defect. The conjecture in this paper is called the ordinary form of the conjecture. In
his later papers, there appeared several other forms, for example, one concerning
the number of characters of twisted group algebras. The ordinary form is thesimplest one among others, and the most complicated one is called the inductive
form, which implies all the other. Dade claimed that, if the inductive form is true
for all finite simple groups, then it is true for all finite groups. So, it is meaningful
to see whether the conjecture holds for simple groups and the related groups. Of
course, it is conceivable also that looking at many examples is important in orderto understand general situation. Anyway, to reach the goal, it seems that we haveto take a much closer look at many cases.
$p$

$p$

In this note, we take a few examples and see what are going on there. As one
might guess, most proofs use concrete computations. However, at least in cases of
infinite families of groups such as finite algbraic groups, one has to figure out generically which characters can be ignored, which must be really taken into account, andso on. (It happens quite often that we may forget about some characters, because
each form is stated in terms of an alternating sum and some characters kill eachother in the alternating sum.) Also, in several situations one encounters phenomena
which resemble each other. In such cases, it is sometimes possible to apply general
assertions to them. It is hoped that some ideas in the proofs in these cases will be
used when verifying more general statements or checking more examples.

Parts of my talk at this conference (and thus the contents of this note) and thatgiven at ”Darstellungstheorie Endlicher Gruppen”. which was held at Oberwolfach,Germany in April 1996 are overlapped. The list of cases where some forms of theconjecture have already been verified may be found in the article of J. An in thisproceeding.

17
1. Dade’s Conjecture

Let

$G$

be a finite group and

$p$

$C$

of p–subgroups$P_{i}’ \mathrm{s}$a prime. By a p–chain we mean a chain: $P_{0}<P1<\cdots<P_{m}$of $G$ . We denote its normalizer by $N_{G}(C)$ which is defined by$N_{G}(c)= \bigcap_{i}Nc(P_{i})$.

The above $m$ is called the length of $C$ , which is denoted by $|C|$ .
The set of p–chains with $P_{0}=O_{p}(G),$$O_{p}(G)$ is the maximal normal psubgroup of , is denoted by $P(G)$ or simply by $P$ . Elements of act on $P$ byconjugation, and we let $P/G$ be a set of representatives of -orbits in $P$ .
It is known that for any p–block of $N_{G}(C)$ , the induced blockin the senseof Brauer is defined. For an irreducible character of , we denote bytheexponent of the p–part of $|G|/\chi(1)$ . It is called the defect of . For a p–block $B$ of$G,$
is called the defect of $B$ .

.ere$\mathrm{w}\mathrm{h}$

$G$

$G$

$G$

$b$

$b^{G}$$\chi$$G$$d(\chi)$$\chi$$\max_{\chi \mathrm{r}\mathrm{r}}\mathrm{I}(B)d\in(\chi)$Given a p–chainby$C$, a p–block $B$ of$G$

and a non-negative integer , let us denote$d$$k(N_{G}(c), B, d)$the number of irreducible characters of $N_{G}(C)$ such that belong to a p-blockof $N_{G}(C)$ with $b^{G}=B$ and that $d(\theta)=d$ . The following is called the ordinary
form of the Dade conjecture.
$\theta$$\theta$$b$

Conjecture O. Assume that $O_{p}(G)=\{1\}$ and that $d(B)>0$ . Then is it truethat$\sum_{C\in P/c}(-1)^{1}c|k(Nc(c), B, d)=0$?

Now we explain the other forms briefly. Suppose that is a normal subgroup ofsome finite group $E$ . Then $E$ acts on $P$ , and for any $C$ in $P$ its normalizer $N_{G}(C)$inis a normal subgroup of its normalizer $N_{E}(C)$ in $E$ . Thus for an irreduciblecharacter of $N_{G}(C)$ , its stabilizer $T(\theta)=T_{N_{E}(C}()\theta)$ in $N_{E}(C)$ is defined. Let $F$
be a subgroup of $E$ containing . Then, for $B,$ , and $F$ as above, we denote by
$G$

$G$

$\theta$$G$$d$$k(Nc(c), B, d, F)$

the number of i.rreducible characters of $N_{G}(C)$ such that belong to a p–blockof $N_{G}(C)$ with $b^{G}=B,$ $d(\theta)=d$ , and that $T(\theta)G=F$ . The following is called theinvariant ordinary form of the conjecture.
$\theta$$\theta$$b$

Conjecture I-O. Assume that $O_{p}(G)=\{1\}$ and that $d(B)>0$ . Then is it truethat$\sum_{c\in P/c}(-1)|C|k(N_{G(}C),$$B,$ $d,$$F)=0$ ?

18
Assume that an irreducible character of $N_{G}(C)$ satisfies $T(\theta)G=F$ . Then we
have $F/G\cong T(\theta)/N_{G}(C)$ . Hence, if we take a-module which gives ,the endomorphism algebra End
overof the induced module$F/G$
is isomorphic to a twisted group algebra ofover C. Recall that isomorphismclasses of twised group algebras are determined by elements of the second cohomology groupof $F/G$ over the multiplicative group. Then, for $B,$ ,$F$ and, we denote by$\theta$$V$$\mathrm{C}N_{G}(C)$$\theta$$V^{T(\theta)}$$(V^{T(\theta}))$$\mathrm{H}^{2}(F/G, \mathrm{C}^{*})$$\mathrm{C}T(\theta)$$d$$\mathrm{C}^{*}$$\alpha\in \mathrm{H}^{2}(F/G, \mathrm{C}^{*})$$k(N_{G}(C), B, d, F, \alpha)$

the number of irreducible characters of $N_{G}(C)$ such that belong to a p–blockof $N_{G}(C)$ with $b^{G}=B,$ $d(\theta)=d,$ $T(\theta)G=F$ , and that the above endomorphism
algebra End
gives . The following is called the extended ordinary form of
the conjecture.
$\theta$$(V^{T(\theta)})$$b$$\theta$$\alpha$Conjecture E-O. Assume that $O_{p}(G)--\{1\}$ and that $d(B)>0$ . Then is it truethat
$C \in P/\sum_{c}(-1)|C|k(NG(c), B, d, F, \alpha)=0$?

In every form, we can consider characters of twisted group algebras, i.e., projective characters instead of ordinary characters. That is, fixing a twisted groupalgebra of , we take a p–block $B$ of projective irreducible characters and nonnegative integer . Now it is possible to formulate the projective form of theconjecture (Conjecture P), whose statement itself is completely the same as thatof Conjecture O. For the invariant form, we begin with a twisted group algebra of$E$ . The
twisted group algebra of which we must deal with is obtained by taking
the subalgebra generated by the bases elements parametrized only by the elementsof . We can define
for a projective irreducible character of $N_{G}(C)$ , too,and $k(N_{G}(c), B, d, F)$ makes sense for a subgroup $F$ of $E$ containing . Thus wehave the invariant projective form (Conjecture I-P). Furthermore, the extended
projective form (Conjecture E-P) may also be formulated since End
isagain a twisted group algebra of $F/G$ .
$G$$\mathrm{a}\vee$

$d$

$G$

$G$

$T(\theta)$$\theta$$G$$(V^{T(\theta)})$So far, we have six forms. However, Dade gives one more, which he calls theinductive form. Here we do not explain its detail, which can be found in [D3]. A
rough sketch of the idea is as follows. We must start with a p–modular system$(K, R, k)$ , whereis a large complete discrete valuation ring, is its residue classfield of characteristic , and $K$ is the field of fractions of . Everything is considered overat the beginning. That is, we take an $RN_{G}(C)$ -lattice which gives$K$over . Twisted group algebras and endomorphism algebras, which is denoted by, are also taken over , and then over . by factoring out by the uniqueEndmaximal ideal of . Let $B,$ $F,$ be as in the extended projective form. Recall that
it is necessary to look at only those blocks that induce the given block of .
Thus
and are related via the Brauer homomorphism. Moreover, if satisfies$T(\theta)G=F$ , then
is $F$-invariant and is $N_{F}(C)$ -invariant. Hence and give
twisted group algebras of $F/G$ over the center $Z(B)$ of and over the center $Z(b)$of , respectively. Denote the former algebra by $Z(B)^{F}$ . By composing the Brauer
$R$

$k$

$R$

$p$

$R$

$\theta$$(\theta^{T(\theta}))$

$R$

$R$

$k’$

$\alpha$

$B$

$b$

$B$

$b$

$B$

$B$

$b$

$B$

$b$

$G$

$\theta$$b$

19homomorphism with the natural embedding of $Z(b)$ to the endomorphism algebra
End , we have a -algebra homomorphism from $Z(B)$ to End . Moreover, we
obtain naturally a -algebra homomorphism from $Z(B)^{F}$ to End, which isalso a twisted group algebra of $F/G$ over . Let us fix a twisted group algebra$k^{*}(F/G)$ of $F/G$ . Thus we fix. For any given -algebra homomorphism of twisted group algebras of $F/G$ from $Z(B)^{F}$ to $k^{*}(F/G)$ , we countthe number $k(N_{G}(c), B, d, F, \iota)$ of those with the same conditions concerning $B$ ,$F$ as above,
and with End
such that there is an isomorphism of twistedgroup algebras from End
to $k^{*}(F/G)$ with which composing the Brauer homorphism and the natural embedding give . The inductive form is of course as
follows.
$k$$(\theta)$$(\theta)$$k$$(\theta^{T(\theta)})$

$k$

$k$

$\alpha\in \mathrm{H}^{2}(F/G, \mathrm{C}^{*})$$\iota$$\theta$$d,$$(\theta^{T(\theta)})$$(\theta^{T(\theta)})$$\iota$Conjecture IND. Assume that $O_{p}(G)=\{1\}$ and that $d(B)>0$ . Then is it truethat
$C \in P/\sum_{c}(-1)^{|c}|k(NG(c), B, d, F, \iota)=0$?

Finally we remark the following.
Remark. All the invariants $k(N_{G}(C), B, d, ***)$ is zero if the final subgroupof $C$ is not contained in any defect group of $B$ .

$P_{m}$

2. p-chains

Since there are so many p-chains, even up to -conjugate, many computations
are necessary to check the conjecture. However, it is known that we may concentrateon certain families of p–chains, which consisit of fewer p-chains.
$G$

Definition. If a p-chain: $P_{0}<P_{1}<\cdots<P_{m}$ satisfies $P_{0}=O_{p}(G)$ andis elementary abelian for a.ll , then it is called an elementary chain. If $C$ satisfies$P_{0}=O_{p}(G)$ andfor all ,$C$$P_{i}$$i$$P_{i}=O_{p}(\mathrm{n}_{j0}^{i}Nc(=)P_{j})$$i$then it is called a radical chain. The sets of elementary chains and radical chainsare denoted by and 72, respectively. The sets and are -invariant (and also
E-invariant).
$\mathcal{E}$$\mathcal{E}$$\prime \mathcal{R}$$G$

One of the important and useful results is as follows.
Proposition. In any form of the conjecture, the sum can be replaced by the sum
taken over eitherorinstead of.
$\mathcal{E}/G$$\mathcal{R}/G$$\mathcal{R}/G$

The above is observed by Kn\"orr and Robinbson [KR] and Dade [D1]. In [KR],they consider more general situations and some other families of chains. We canuse those families whenever they seem to be suitable for the cases we treat.

If a p–subgroup $P$ ofsatisfies $P=O_{p}(N_{G}(P))$ , then it is called a radical
subgroup of . Thus, a: $P_{0}<P_{1}<\cdots<P_{m}$ is radical if and only if$P_{0}=O_{p}(G)$ and $P_{i+1}/P_{i}$ is radical infor all $i\geq 1$ . Therefore, toobtain dical chains, we look for radical subgroups at each stage.
$G$

$G$

$\mathrm{g}\succ \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{n}c$$\bigcap_{j=0}^{i}NG(P_{j})/P_{i}$$\mathrm{r}.\mathrm{a}$20
Example. Let $S(n)$ be the symmetric group on letters. Radical subgroups of$S(n)$ are determined by Alperin and Fong [AF]. For a non-negative integer , let
be the regular elementary abelian group of order . For example, if $p=3$ , wehave
$A_{2}\cong<(123)(456)(789),$ $(147)(258)(369)>$
in $S(9)$ .
$n$

$c$

$A_{c}$$p^{c}$For a sequence$\underline{c}=(c_{1}, c_{2}, \cdots)$of positive integers, let$A_{\underline{c}}$be the wreath product$A_{c_{1}}lA_{C_{2}}l\cdots$

Alperin and Fong proved that, if $P$ is a radical subgroup of $S(n)$ , then it follows
that $P$ is $S(n)$ -conjugate to$\prod_{i}(A_{\underline{c}_{i}})^{\dot{m}_{i}}$for some sequences
necessarily true.)$\underline{c}_{i}’ \mathrm{s}$and positive integers

Subgroups of the above type are calledshowed that, if $P$ is as above, then$(*)$.
$m_{i}’ \mathrm{s}$. (Note : The converse is not

ic subgroups. Furthermore, they$\mathrm{b}\mathrm{a}s$$N_{S(n)}(P)/P \cong\prod(GL(c_{i,1},p)\iota’\cross\cdots\cross GL(c_{i,n},p)i)lS(mi)$.

provided that
if $i\neq j$ . Here, we write. Thus,when we look for a radical chain $C$ with $|C|\geq 2$ , we must find radical subgroups of$\prod_{i}(GL(c_{i},1,p)\cross\cdots\cross GL(Ci,n_{i}’ p))lS(m_{i})$ . For this purpose,the following results
are of great use.
$\underline{c}_{i}=(c_{i,1}, ci,2, \cdots, c_{i,n_{i}})$$\underline{c}_{i}\neq\underline{c}_{j}$Lemma. (1) Let $P$ be a radical subgroup of the direct productof twoand.
Thenthere are radical subgroups
finiteandof and, respectively, such that $P=P_{1}\cross P_{2}$ .
(2) Let be a finite group, and let $P$ be a radical subgroup of the wreath product$GlS(n)$ ofwith $S(n)$ . Then
-conjugate to, whereis aradicalof and is a basic subgroup of $S(n)$ .
$G_{1}\cross G_{2}$$group_{\mathit{8}}G_{1}$$G_{2}$$P_{1}$$P_{2}$$G_{1}$$G_{2}$$G$$G$$\mathit{8}ubgroup$$\tilde{P}\rangle\triangleleft S$$Pi\mathit{8}G\mathrm{t}S(n)$$G^{n}$$\tilde{P}$$S$

By using the above, one shows that our third subgroupsatisfies$(**)$$P_{2}/P_{1}\cong\square Qi\nu Rii$

$P_{2}$

in a radical chain

,

where
is a radical subgroup of $(GL(c_{i,1,p})\cross\cdots\cross GL(\alpha_{n_{i}},,p))mi$ andis a basic
subgroup of $S(m_{i})$ . Here notice that radical subgroups of finite algebraic groups
are known to be the normalizers of parabolic subgroups (when is the defining
characteristic), and that, if
in the above is trivial for all , thenalso basic
in $S(n)$ . .
$Q_{i}$$R_{i}$$p$$Q_{i}$$i$$P_{2}.\mathrm{i}\mathrm{s}$In the rest of this note, we see several examples for which some form of theconjecture is verified.

21
3. Case 1: Symmetric groups
Here we sketch the proof of the ordinary form of the conjecture for $S(n)(n\geq 4)$
for an odd prime , following [OU2].
$p$

STEP 1.
Let $C$ : $P_{0}<P_{1}<\cdots<P_{m}$ be a radical chain of $S(n)$ . Then $P_{0}=O_{p}(S(n))=$, and if $m\geq 1$ , thenis
. Moreover, by the results in the previous section,
we can define
so thatare basic subgroups of $S(n)$ andisnot. Then we havethe form$\{1\}$$P_{1}$$\ell\geq 1$$\mathrm{b}\mathrm{a}s\mathrm{i}\mathrm{c}$$P_{1},$$P_{2},$$\cdots,$$P_{\ell}$$P_{\ell+1}$$\bigcap_{i=0S}^{\ell 1}+N((n)P_{i})/P\ell$$\prod_{i}(GL(ci,1,p)\cross\cdots\cross GL(_{\mathrm{G},,p}n_{i}))\iota S(mi)$.

Also, by the choice ofhas ”-trivial” $GL$-part. In case$s\geq\ell$$P_{\ell}/P_{\ell}=\{1\},$
let with
be such that
contained in the GL-part$\ell,$$P_{\ell+}1/P_{\ell}$$\mathrm{a}$$\mathrm{n}\mathrm{o}\mathrm{n}$$s$$P_{\ell+1}/P_{\ell},$$P_{\ell+2}/P_{\ell},$$\prod_{i}(GL(c_{i,1},p)\cross\cdots\cross GL(c_{i},n_{i}’ p))^{m_{i}}$$\mathrm{o}\mathrm{f}|C|\geq P+1$$\cdots,$$P_{s}/P_{\ell}$

,

are

,

and
is not. Then
have the formwith
all $R$ trivial, and we may, whereis a radical subgroup:$(GL(c_{i},1,p)\cross\cdots\crossGL(c_{i,n_{i}},p))^{m}$ofandis a basic subgroup of $S(m_{i})$ , andmoreover, for at least one
is not trivial. Notice also that someis not trivial
by the choice of .
Now consider the radical chains $C$ for which the above satisfies $|C|\geq s+1$ .
Then define $\tau(C)$ for such a $C$ as follows.
$P_{s+1}/P_{\ell}$$P_{\ell+1}/P_{\ell},$$P_{\ell+2}/P_{\ell},$$\cdots,$$(^{**})$$P_{s}/P_{\ell}$$P_{s+1}/P_{\ell} \cong\prod_{i}\tilde{Q}_{i}\mathrm{x}\tilde{R}_{i}$$\tilde{Q}_{i}$$\tilde{R}_{i}$$i,\tilde{R}_{i}$$\tilde{Q}_{i}$$\ell$$s$$\tau(C)$:$P_{0}<\cdots<P_{S-1}<P_{s+1}<P_{s+2}<\cdots$if$P_{s}/P_{\ell}= \prod_{i}\tilde{Q}i$$P_{0}<\cdots<P_{s-1}<P_{s}<Q<P_{s+1}<P_{s+2}<\cdots$if$P_{s}/P_{\ell} \neq\prod_{i}\tilde{Q}_{i}$$\{$where is the inverse image ofin.
Then it is not difficult to show that $\tau(\tau(C))--C,$ $|\tau(C)|=|C|\pm 1$ , and that
$N_{S(n)}(\tau(C))=N_{s_{(n)()}}C$ .
Thus gives an involutive bijection from the set of$C$
the chains with $|C|\geq s+1$ to itself, which keeps the normalizers of the chains
invariant, but changing the parity of their length. Since the number $k(N_{G}(c), B, d)$depends only on $N_{G}(C)$ for given $B$ and , two chains
and $\tau(C)$ contributenothing to the alternating sum. Therefore, we may consider only those chains $C$
with $m=|C|=s$ , that is,are contained in the
GL-part.
$Q$$\prod_{i}\tilde{Q}_{i}$$P_{s+1}$$\tau$$C$$d$$P_{\ell}/P_{\ell}=\{1\},$$P_{\ell+1}/P_{\ell},$$\cdots,$$P_{m}/P_{\ell}$

STEP 2.
Recall that in
the factor group of the normalizer contains a direct products
GL’s,ofand by Lemma and Step 1, the factor groups,
are direct products as well. Thus by using an argument simila.r to the one$(^{*})$$P_{\ell}/P_{\ell}=\{1\},$$P_{m}/P_{\ell}$$P_{\ell+1}/P_{\ell},$$\cdots$

22in Step 1, we may concentrate on only those chains such that every has only one
part. More precisely, instead of using semi-direct products, we use direc.t products,
and can find similarly cancellations whenever some products h.ave
than one$\underline{c}_{i}$more.

factors.
Here we remark that for those arguments in Steps 1 and 2, the fact that $p\neq 2$
is necessary to conclude that $\tau(C)$ is again a radical chain.
From now on, we assume that every has only one part and write
for
all .
$\underline{c}i=(c_{i})$$\underline{c}_{i}$$i$STEP 3.
By Step 2,
are contained in $\prod_{i}GL(C_{i,p)}m_{i}$ .
In fact, they give a radical chain of $\prod_{i}GL(Ci,p)\iota S(mi)$ . In [OU1], it is shown thatthe ordinary form of the conjecture is true for the general linear groups $GL(n, q)$
and their direct products in the defining characteristic. This also yields that it istrue for $\prod_{i}GL(C_{i},p)\iota S(mi)$ as well.
these results, one can show that thealternating sum vanishes on, wheredenotes the set of radical chains$P_{\ell}/P_{\ell}=\{1\},$$P_{\ell+1}/P_{\ell},$$P_{m}/P_{\ell}$$\cdots,$$\mathrm{W}\mathrm{i}\dot{\mathrm{t}}\mathrm{h}$$\mathcal{R}_{1}/G$$C$$\mathcal{R}_{1}$: $\{1\}<P_{1}<P_{2}<\cdots<P_{m}$with $|C|=s$ ( is defined as in Step 1), andbeing contained in the GL-part$\prod_{i}cL(C_{i},p)m_{i}$ such thatfor some . Here we need the condition ”
for”some because of the following reason. We know that the ordinary form is true for$GL(n, q)$ and for the other related groups such as $\prod_{i}GL(ci,p)\iota S(mi)$ . However, wemust consider also $V\mathrm{x}GL(n, q)$ and, for example, where $V$$V’$andare vector spaces on which $GL(n, q)$ and $\prod_{i}GL(ci,p)\iota s(m_{i})$ , respectively,
act naturally, since we have to deal with.
and not
But, it is known that the alternating sum in the conjecture does not always vanish
if $O_{p}(G)\neq\{1\}$ . In fact, the ordinary form is true for $V\cross GL(n, q)$ only when
$n=\dim V\geq 2$ . Since we use these facts, the above condition on
is necessary.
Now remaining radical chains are those $C$ such that $|C|=s$ andiscontained in the $GL$-part $\prod_{i}GL(1,p)mi$ . But, since $GL(1,p)$ is a $p’$ -group, it follows
that $m=|C|=P$ . Namely,
are all basic subgroups isomorphic to$P_{m}/P_{\ell}$$s$$c_{i}\geq 2$$c_{i}\geq 2$

$i$

$i$

$V’ \rangle\triangleleft\prod_{i}cL(Ci,p)\iota S(mi)$$\bigcap_{i}^{m}=0^{N_{S(n)()}}P_{i}$$\bigcap_{i=0^{N_{S(n)()}}}^{m}P_{i}/P_{\ell}$$c_{i}$$P_{m}/P_{\ell}$$P_{1},$$P_{2},$$\cdots,$$P_{m}$$A_{1}\cross A_{1}\cdots\cross A_{1}$.

STEP 4.
By Step 3, we may consider only the radical chains of the following type.
$C$

where
integer , let$\ell_{1},$$t$$\ell_{2},$$\cdots,$$\mathcal{R}(t)$: $\{1\}<(A_{1})^{\ell_{1}}<(A_{1})^{\ell_{2}}<\cdots<(A_{1})^{l_{m}}$ ,are positive integers with $p_{1}<p_{2}<\ldots<p_{m}$ . For any positive$pm$

denote the set of radical chains of type$C$

: $\{1\}<(A_{1})^{t}<(A_{1})^{\ell_{2}}<\cdots<(A_{1})^{\ell_{m}}$ .

Then the subset of
consisting of all the remaining chains is precisely the
disjoint union
and the subset consisting only of the trivial chain. On the$\mathcal{R}$$\cup \mathcal{R}(t)$23other hand, for any p–block $B$ of $S(n)$ , we can define its weight $w(B)$ as follows.
Let $D$ be a defect group of $B$ . Then divides $n-f$ , where is the number of the
fixed points of $D$ . Then $w(B)$ is defined by $w(B)=(n-f)/p$ .
Now by induction on $|C|$ and by some local analysis of rblocks of $S(n)([\mathrm{O}2])$ ,one can show that, for any with $t<w(B)$ , the alternating sum vanishes on.
On the other hand, again by some local analysis, one can prove that, if $P_{m}>w(B)$ ,then the number $k(N_{S(n})(C),$ $B,$ is zero. (This is also related to the final remark
in \S 1.) Thus, we may concentrate only on chains with $t\geq w(B)$ and $\ell_{m}\leq w(B)$ .
But, since $t\leq p_{m}$ , we have $t=P_{m}=w(B)$ . This is the chain: $\{1\}<(A_{1})^{w(B)}$ .
Its normalizer has only the principal block
which satisfies $b_{1}^{S(n)}=B$ . Because
we have now only the trivial chain and , it suffices to check that
$f$

$p$

$t$

$\mathcal{R}(t)$$d)$$C_{1}$$b_{1}$$C_{1}$$k(S(n), B, d)=k(N_{S()}n(C1), B, d)$holds for all . However, the above has already been proved by Olsson in [O1].
Therefore, we can conclude that the ordinary form is true for $S(n)$ when is odd.
$d$

$p$

4. Case 2 : Cyclic defect group cases
In this section, we consider the case where a defect group $D$ of a given p-block
is cyclic. The ordinary form is true in this case
. Moreover, the invariant
form is also true. We give an outline of the proof to it. (See [D4].)$B$$([\mathrm{D}1])$

We use instead of . Since $D$ is cyclic, $D$ has the unique cyclic subgroup $P$of order . By the final remark in \S 1, it suffices to show that$\mathcal{E}$$\mathcal{R}$$p$$k(G, B, d, F)=k(N_{G}(P), B, d, F)$

for all and $F$ . Here, of course, we assume that is a normal subgroup of some
$E$ andthat $F$ is a subgroup of $E$ containing . It is known that there is only one
block of $N_{G}(P)$ such that $d(b)=d(B)$ and $b^{G}=B$ . Furthermore, letting be
the inertia index of $B$ , we have
$d$

$G$

$G$

$b$

$e$

$k(G, B, d(B))=k(N_{G}(P), B, d(b))=e+ \frac{|D|-1}{e}$ ,and $k(G, B, d)$ and $k(N_{G}(P), B, d)$ for any other are zero. (See for example [F].)Thus the ordinary form holds.
Let $d=d(B)=d(b)$ . Some irreducible characters in $B$ and are called exceptional characters and the numbers of exceptional characters arefor both $B$
and . The actions of $E$ and $N_{E}(P)$ on, respectively,
and onpreserve exceptional characters, and moreover, examining their character values, it
is not difficult to see how the actions on them are.
On the other hand, $B$ and both have irreducible Brauer characters. Let be
the Green correspondence which sends indecomposable $kG$-modules with vertex $D$
to those $kN_{G}(P)$ -modules with the same vertex. (Note: $N_{G}(D)\leq N_{G}(P).$ ) It is
known that irreducible $kG$-modules in $B$ have vertex $D$ . Using the fact that
$d$

$b$

$\frac{|D|-1}{e}$$b$$\mathrm{I}\mathrm{r}\mathrm{r}(B)$$b$$D^{g}\cap D=\{1\}$$\mathrm{I}\mathrm{r}\mathrm{r}(Nc(P))$

$f$

$e$

for all$g\in G\backslash N_{G}(P)$,24
one can show that, if

is an irreducible $kG$-module with vertex $D$ , then the
maximal semisimple submodule of $f(V)$ is irreducible, and that this gives a bijection
from the set of irreducible Brauer characters of belonging to $B$ to that of $N_{G}(P)$belonging to . This bijection is compatible with the actions of $E$ and of $N_{E}(P)$ .
Recall that $B$ and have non-exceptional characters and irreducible Brauer
characters. By looking at the decomposition matrices, it is possible to see therelation between the actions of $E$ and of $N_{E}(P)$ on non-exceptional charactersthrough those on irreducible Brauer characters. Finally we can conclude that the
numbers of non-exceptional characters in $B$ and whose inertia subgroups give $F$
are the same. This completes the proof of the invariant form in the case where $D$is cyclic.
$V$

$G$

$b$

$b$

$e$

$e$

$b$

5. Case 3 : Dihedral defect group cases$B$

In this section, we consider the case where a defect group $D$ of a given p-blockis dihedral$<x,$ $y|x^{2^{n-1}}=y^{2}=1,$ $y^{-11}xy=x^{-}>$of orderwith $n\geq 3$ . Thus we have $p=2$ . We assume that $B$ is of typein the sense of Brauer [B]. Namely, we suppose thatand $xy$ are conjugate
in . Note that, in particular, $<z,$ $y>$ and $<z,xy>$ are not -conjugate. The
invariant form is proved in [U], and we give its idea here. Again assume that is anormal subgroup of $E$ and $F$ is a subgroup of $E$ containing . Ler $z=x^{2^{n-2}}$ . Thenthe center $Z(D)$ of $D$ and the following are representatives of -orbits inwhose final subgroups are contained in D. (See also the final remark in \S 1.)$2^{n}$$(\mathrm{a}\mathrm{a})$$x^{2^{n-2}},$

$y$

$G$

$G$

$G$

$G$

$<z>\mathrm{i}\mathrm{s}$$G$$\mathcal{E}${1},$\{1\}<Z(D)$ ,$\{1\}<<z,$ $y>$ ,$\{1\}<Z(D)<<z,$ $y>$ ,$\{1\}<<z,$ $xy>$$\{1\}<Z(D)<<z,$ $xy>$

Notice that all the local blocks of the normalizers of the above chains havedihedral defect groups. Hence by taking pairs$G$$N_{G}(<z, y>)$$N_{G}(<z, xy>)$

and
and
and

$N_{G}(Z(D))$$N_{G}(<z, y>)\cap N_{G}(Z(D))$$N_{G}(<z, xy>)\cap N_{G}(Z(D))$ ,it suffices to show that$k(G, B, d.F)=k(N_{G}(z(D)), B, d, F)$$F$ and any
holds for any
blocks $B$ with dihedral defect groups. Note that, infact, we have to prove the above equation for blocks with defect groups of othertypes (ab) and, because local blocks appearing in the above (even in the casewhere the given $B$ itself is of typemay be of those types. However, here we
give the idea to show the above equation, only in the case where $B$ is of type.
For the other cases, see [U].
$d,$$(\mathrm{b}\mathrm{b})$$(\mathrm{a}\mathrm{a}))$$(\mathrm{a}\mathrm{a})$25
It is known by [B] that there is the unique block. Moreover, has also $D$ as a defect group. and$b^{G}=B$

of $Nc(z(D,))$ such that

$b$

$b$

$k(G, B, d)=k(N_{G}(z(D)), B, d)=$Thus the ordinary form follows from the above.
For the invariant form, we may assume thatis $E$-invariant. Thus is also
$N_{E}(Z(D))$ -invariant. This yields that $E=N_{E}(D)G=N_{E}(Z(D))c$ by the Frattini
argument. On the other hand, the actions of $E$ and $N_{E}(Z(D))$ on characterswith $d(\chi)=n-1$ are easy to see, since those characters are mostly determinedby their values on . Hence, the above observations imply that these actions of$E$ and of $N_{E}(Z(D))$
on characters of defect $n-1$ are permutation isomorphic. In
particular, their inertia groups coincide ”in $E$ modulo ”. This means that
$B$

$b$

$\chi$$x^{j}$$G$$k(G, B, n-1, F)=k(N_{G}(Z(D)), B, n-1, F)$

for all $F$ . However, all the characters with $d=n$ are rational valued, and in general,
it is hard to see how those are distributed into $E$-orbits or $N_{E}(Z(D))$ -orbits. But, $E$
can act on the column index set of the generalized decomposition matrix of . The
same is true for $N_{E}(Z(D))$ and . A column index of the generalized decomposition
matrix looks like, where is an element inand is an irreducible Brauer$C_{G}(g)$character of $C_{G}(g)$ belonging to a blockofwhich induces . The set ofcolumn indices consists of representatives of -conjugate classes of such pairs. The
row index set is, on the other hand, consists of irreducible ordinary characters in $B$ .
Brauer proved also in [B] that $B$ has three irreducible Brauer characters, say ,, and has only one, say . Letand.
be irreducible characters$B$inwhose defects are . Also letbe those in . Then we can findthe following parts in the generalized decomposition matrices of and .
$B$

$b$

$(g, \eta)$

$D$

$g$

$\eta$$B$$b_{g}$$G$$\varphi_{1}$$\varphi_{2}$$\varphi_{3}$$b$$\varphi$$n$$\theta_{1},$$\theta_{2},$$\chi_{1},$$\chi_{2}$$\theta_{3},$$\theta_{4}$$\chi_{3},$$\chi_{4}$

$b$

$B$

$B$

$(1,\varphi_{1})*$$(1,\varphi_{2})*$

$b$

$b$

$(1,\varphi_{3})*$$(z,\varphi)*$$(1..\varphi)*$$(z,\varphi)*$$(y,\varphi_{y})*$$(xy,\varphi_{xy})*$$\theta_{1}$$\chi_{1}$$\chi_{2}$

$*$

$*$

$*$

$*$

$\theta_{2}$

$*$

$*$

$*$

$*$

$\chi_{3}$

$*$

$*$

$*$

$*$

$\theta_{3}$

$*$

$*$

$*$

$*$

$\chi_{4}$

$*$

$*$

$*$

$*$

$\theta_{4}$

$*$

$*$

$*$

$*$

Here

and
are the unique irreducible Brauer characters of $C_{G}(y)$ and$C_{G}(xy)$ , respectively, which belong to blocks inducing $B$ .
Now, it is clear that$E$
is -invariant. Also, for , the columns
andare $N_{E}(Z(D))-$$B$
invariant. Moreover, by looking at the decomposition matrix of , it follows thatat least one ofis $E$-invariant. Say thatis $E$-invariant. Since the above
parts of the generalized decomposition matrices are invertible and since the actionsof $E$ and of $N_{E}(Z(D))$ on the complements of the above subsets of column and$\varphi_{y}$$\varphi_{xy}$$(z, \varphi)$$b$$\varphi_{i}$$(1, \varphi)$$\varphi_{3}$$(z, \varphi)$26
row indices are permutation isomorphic (the actions on those subsets are the sameas those on $\{x^{j}|1\leq j\leq 2^{n-2}-1\})$ , we can conclude, by using the Brauer’s
permutation lemma, that$k(G, B, n, F)=$
where$\tau_{!}$

is the inertia group of$\varphi_{1}$in $E$ , and that$k(N_{G}(z(D)), B, n, F)=$
where $T_{2}=c_{N_{G}(z(D))}(y)G$ . Hence, it suffices to prove that $T_{1}=T_{2}$ .
Now we look at periodic indecomposable $kG$-modules. There are those modules
in $B$ with period three and vertices $<z,$ $y>$ or $<z,$ $xy>$ . For some of thosemodules, their structures are quite well known. (See [E].) In particular, their composition series show that, an element of $E$ sends those modules with vertices, i.e.,
andconjugation, if and only if
those with $<z,$$<z,$$T_{1}=N_{E}(<z,y>)G$
implies
.
This
Thus,
are not -invariant.
it follows that
. On the other hand, the fact that $<z,$ $y>$ and $<z,$ $xy>$ are notthat-conjugate implies that
. This completes the proof.
$g$$\varphi_{1}^{g}=\varphi_{2}$$xy>\mathrm{b}\mathrm{y}$$y>\mathrm{t}\mathrm{o}$$\varphi_{1}$$g$$\varphi_{2}$$T_{1}\geq T_{2}$$G$$T_{1}\leq T_{2}$RemarkTo check certain forms such as invariant forms, one may have to use some information on indecomposable modules. (See \S 4 and \S 5.) It is probably the case that,for proofs of general statements on those forms, information not only on characters but also on block algebras would be necessary. One must study the relations
between the structures of algebras (and indecomposable modules over them) of agiven block and of the related local blocks.

References[AF] J. Alperin, P. Fong : Weights for symmetric and general linear groups, J.
Algebra 131 (1990) 2-22.
[B] R. Brauer: On 2-blocks withdefect groups, Symposia Mathematica,
13 (1974) 367-393.
[D1] E. C. Dade : Counting characters in blocks, I, Invent. math. 109 (1992)

di.hedral

187-210.
[D2] E. C. Dade: Counting characters in blocks, II, J. reine angew. Math. 448(1994) 97-190.
[D3] E. C. Dade: Counting characters in blocks, 2.9, preprint.
[D4] E. C. Dade : Counting characters in blocks with cyclic defect groups, I,
preprint.
[E] K. Erdmann : Blocks of tame representation type and related algebras,
Lecture Notes in Mathematics 1428 (1990) Springer Verlag.

27[F] W. Feit: The Representation Theory of Finite Groups, (1982).
[KR] R. Kn\"orr, G. Robinson : Some remarks on a conjecture of Alperin, J.
London Math. Soc. 39 (1989) 48-60.
[O1] J. B. Olsson: McKay numbers and heights of characters, Math. Scand. 38(1976) 25-42.
[O2] J. B. Olsson : Lower defect groups in symmetric groups, J. Algebra 104(1986) 37-56.
[OU1] J. B. Olsson, K. Uno: Dade’s conjecture for general linear groups in thedefining characteristic, Proc. London Math. Soc. 72 (1996) 359-384.
[OU2] J. B. Olsson, K. Uno : Dade’s conjecture for symmetric groups, J. Alg.
176 (1995) 534-560.
[U] K. Uno : Dade’s conjecture forblocks, Osaka J. Math. 31 (1994)$\mathrm{N}_{0}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}$747-772.

tame.

Addendum
In the beginning of February 1997, several -mails were flying about. They said
that there were counterexamples to some forms of the conjecture. However, later,
errors were found in the arguments or in the computations, and it turned out thatthey were not counterexamples at all. So, there seem to be no counterexamples
now. As of Monday February 10th, 1997, Dade wrote what had been happening
in those days. (In what Dade wrote on 10th, there was still a possibility of acounterexample. But, after a while, at least, an error was found in the argument.)
This note was completed on Riday February 14th.