数理解析研究所講究録
991 巻 1997 年 28-35

28THE ALPERIN AND DADE CONJECTURESFOR SOME FINITE GROUPS

Jianbei An
Department of Mathematics

University of Auckland
Auckland, New Zealand

1. Alperin’s Weight Conjecture

be a finite group, a prime, and $O_{p}(G)$ the largest normal p–subgroup of. In addition, let be a p-block, $R$ a -subgroup of $G,$ an irreducible ordinary
character of the factor group $N(R)/R$ . Then a pairis called a B-weight
Let$G$

$p$

$G$

$B$

$p$

$\varphi$$(R, \varphi)$(character version) if the -defect of$p$$N(R)$ containing$\varphi$$\varphi$

is$0$

and if the block$B(\varphi)$of the normalizer

induces the block $B$ (in the sense of Brauer), where$\varphi$viewed as a charcater of $N(R)$ and the -defect of is the largest integer
thatdivides. A weight is always identified with its conjugates in .
$p$$p^{a}$$\varphi$is also$a$such$G$$\frac{|G|}{\varphi(1)}$Alperin’s Weight Conjecture (1987): The number of $B$ -weights equals the num$ber$of irreducible Brauer characters in the block .
$B$

In 1989, Kn\"orr and Robinson translated the conjecture into one involving only
ordinary irreducible characters.

A -subgroup chain$p$$C:Q_{0}<Q_{1}<\ldots<Q_{n}$of length $|C|=n$ is called a normal

is a proper normal
for $1\leq i\leq n-1$ . Let $N$ be the set of all normal p–chains. Thensubgroup ofacts on $N$ by conjugation, and the stabilizer$p$-chain if each subgroup$Q_{i}$$Q_{n}$$G$$N(C)=\cap in=1N(Q_{i})$of the chain in is called the normffiizer of . Denote by $\mathrm{k}(N(C), B)$ the numberof irreducible characters of $N(C)$ such that the block $B(\psi)$ of $N(C)$ containing
$C$

$G$

$C$

$\psi$

induces the given block$B$$\psi$. Alperin’s weight conjecture is equivalent to the following.

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The Kn\"orr-Robinson Form (1989):-block, we have$m_{enever}c$ isa finite group and $B$ is a$p$$\sum_{c}(-1)^{||}c\mathrm{k}(N(c), B)=0$

where$C$

runs over a set $N/G$ of representatives for$G$,-orbits in $N$ .

2. Dade’s Ordinary Conjecture

A p–subgroupof is called a radical subgroup if is the largest normal
p–subgroup of its normalizer $N(R)$ , that is, $R=O_{p}(N(R))$ .
$R$

$G$

$R$

A -subgroup chain
$p$

$C$

: $P_{0}<P_{1}<\cdots<P_{u}$is called a radical p–chain if it satisfies the following two conditions(a)$P0=o_{p}(G)$(b)$P_{k}$.

is a radical subgroup of the subgroupfor each $1\leq k\leq u$ . Let$\mathcal{R}=\mathcal{R}(G)$$\bigcap_{\mathit{1}=0}^{k-}N1(P_{I})$be the set of all radical -chains of
$p$

$G$.

Given a non-negative integer , a -block of and a radical -chain , let$\mathrm{k}(N(C), B, d)$ be the number of irreducible charactersof the normalizer $N(C)$such that
$d$

$B$

$p$

$G$

$C$

$p$

$\psi$$B(\psi)$induces $B$ and the defect of$\psi$is .
$d$Dade’s Ordinary Conjecture (1990): If $O_{p}(G)=1$ and$B$

is a block with non-trivial defect groups, then for any integer ,
$d$

(2.1)

$\sum_{c}(-1)^{||}C\mathrm{k}(N(c), B, d)=0$

where

$C$

runs over a set$\mathcal{R}/G$of representatives for the$G$-orbits in 7?.

It was shown by Dade [D1] that$\sum_{C\in N/G}(-1)|c_{\mathrm{I}_{\mathrm{k}}}(N(c), B, d)=\sum_{\in C\mathcal{R}/c}(-1)^{1}c|\mathrm{k}(N(C), B, d)$.

Thus Dade’s ordinary conjecture implies the Kn\"orr-Robinson form of Alperin’s

weight conjecture. It is also mentioned in Dade’s paper [D1] that the ordinary

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conjecture is equivalent to the final conjecture if the group has both trivial Schurmultiplier Mult $(G)$ and trivial outer automorphism group Out $(G)$ . These conditions$G$

are satisfied by the following 11 sporadic simple groups:$J_{4},$$J_{1},$$M_{11},$$Co_{2},$$M_{23},$$Co_{3},$$M_{24},$$F_{i_{23}}$$Ly$,, Th, $M$ .

3. Dade’s Invariant Conjectureis trivial. Then we can identifywith its
inner automorphism group Inn $(G)$ . So the automorphism groupacts
naturally on each -chain , and moreover, the stabilizer $N_{A}(C)$ of inacts
on each irreducible character of $Nc(C)$ . So $N_{G}(C)$ is a normal subgroup of thestabilizer $N_{A}(C, \psi)$ of in $N_{A}(C)$ . The factor group $N_{A}(C, \psi)/N_{G}(C)$ is isomorphic
to the subgroup of an outer automorphism groupof .
Suppose the center $Z(G)$ of$G$$G$$A=\mathrm{A}\mathrm{u}\mathrm{t}(G)$

$C$

$p$

$C$

$A$

$\psi$$\psi$$G$$O=\mathrm{o}_{\mathrm{u}\mathrm{t}}(G)$, a non-negative integer , andGiven a radical -chain , a -blocka subgroup, let $\mathrm{k}(N(C), B, d, U)$ be the number of irreducibleofcharacters of $N_{G}(C)$ such that the block of $N(C)$ containing induces the block, the defect of is , and $N_{A}(C, \psi)/N_{G}(C)=U$ . The Dade invariant conjectureis stated as follows:
$C$

$p$

$U$

$p$

$d$

$B\in \mathrm{B}\mathrm{l}\mathrm{k}(G)$$O=\mathrm{O}\mathrm{u}\mathrm{t}(G)$$\psi$$\psi$$B$$\psi$$d$Dade’s Invariant Conjecture [D3]: If $Z(G)=O_{p}(G)=1$ and $B$ is a p-block

ofwith non-trivial defect group, then for any $\dot{\iota}ntegerd\geq 0$ and any subgroup$G$$U\leq \mathrm{O}\mathrm{u}\mathrm{t}(c)$,$\sum_{C\in R/G}(-1)^{1}c_{1}\mathrm{k}(N(c), B, d, U)=0$

where$\mathcal{R}/G$

is a set of representatives for the$G$-orbits in,$\mathcal{R}$.

Dade’s invariant conjecture is equivalent to his final conjecture whenever$G$

has trivial Schur multiplier Mult $(G)$ and an outer automorphism group all of whose
Sylow subgroups are cyclic. A lot of finite simple groups satisfy these conditions,for example,
He, $HN,$ $R_{1}(q),$ $R_{2}(q),$ $3D_{4}(q)$ ,$G_{2}(q)$(with $q\neq 3,4$ ), $F_{4}(q)$ (with $q\neq 4$ ), $E_{8}(q)$ .

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4. Current Works
1.

Alperin’s weight conjecture has been verified for the following groups andblocks:

Blacks:(a) Cyclic and tame blocks (by Dade, Uno).
(b) Abelian defect blocks with small inertial index (by Puig and Usami).
(c) Abelian defect principal 2-blocks (by Fong and Harris).

(d) Abelian defect unipotent blocks of a finite reductive group (by Brou\’e, Malleand Michel).
Groups:(a) -solvable groups (by Okuyama, Isaacs, Navarro, Gres, Barker).
$p$(b) Groups of Lie type in the defining characteristic (by Alperin, Cabanes, andreproved by Lehrer and Th\’evenaz).

(c)$S_{n}$(by Alperin and Fong).

(d) CIassical groups in non-defining characteristics (by Alperin, Fong, Conderand An). In this case, the numbers of irreducible Brauer characters for blocks ofsymplectic and even-dimensional orthogonal groups are unknown (when $p\neq 2$ ).
(e)$s_{\mathcal{Z}}(2^{2n+1}),$(f) $M_{11,1}M2,$$2G_{2}(q^{2}),$$M_{2}2,$$2F4(q2),$ $G2(q),$ $3D_{4}(q)$ (by Dade, An).

$M_{2}s,$$M_{24}$(g) The covering groups of$S_{n}$, He,$Co_{3},$$J_{1}$(by Dade, Conder, An).

and $A_{n}(p\neq 2)$ (by Michler and Olsson).

(h) Wrath product groups $GlS_{n}$ provided the conjecture holds for that finitegroup$G$(by Ewert)

2. Dade’s final conjecture has been verified for the following cases:(a) 10 sporadic simple groups:$M_{11},$$M_{12},$$M_{22},$$M_{23},$$M_{24}$, He,$J_{1},$$J_{2},$$J_{3},$$\mathrm{C}\mathrm{o}_{3}$(by Dade, Huang, Kotlica,

Schwartz, Conder, An).
(b)

$L_{2}(q),$$L_{3}(q)(p|q),$ $s_{z}(q),$ $2G_{2}(3^{2}n+1)(p\neq 3),$ $G_{2}(q)$the Tits group (by Dade, An).
(c) Cyclic blocks (by Dade).

( $p\neq 3$ and$p\parallel q\neq 4$),

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3. The invariant conjecture has been verified for all tame blocks, and for thegroup(by Uno, Murray).
$\mathrm{M}\mathrm{c}\mathrm{L}(p\neq 2)$

4. The ordinary conjecture has been verified for the following cases:(a)$\mathrm{G}\mathrm{L}_{n}(q)(p|q),$(b)$S_{n}$$2F_{4}(2^{2+1}n)(p\neq 2),$ $G_{2}(q)(p \int q)$(by Olsson and Uno when$p$(by Olsson, Uno, An).

odd, An when $p=2$ ).

(c) $Ru$ (by Dade).
(d) Unipotent abelian defect blocks (by Brou\’e, Malle and Michel).
(e) Abelian defect principal 2-blocks (by Fong and Harris).
(f) All abelian defect blocks with small inertia index (by Usami).

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