数理解析研究所講究録
第 1687 巻 2010 年 152-156

152

Eigenvalues of Cartan matrices forgroup algebras of finite groups
豊 (Yutaka Yoshii)
Graduate School of Science, Chiba University

千葉大学大学院理学研究科 吉井

1Introduction

Let be a finite group, an algebraically closed field of characteristic $p>0$ ,and let $B$ be a block of the group algebra $kG$ with defect group 1). Let
be the largest eigenvalue ofbe the Cartan matrix of , and let
andbe ;he sets ofwhich is called Frobenius-Perron eigenvalue. Let
all eigenvalues and all elementary divisors (with multiplicity) respectively.
Let $B_{0}(G)$ be the principal block of $kG$ . Let $O_{p’}(G)$ be the largest normal
subgroup of $G$ whose order is prime to , and let $O^{p’}(G)$ be the smallest
normal subgroup of $G$ whose index is prime to .
byThc eigcnvalucs of Cartan matrices of finite groups liave bceri
some people around T.Wada. The following conjecture is given in [4]:
$G$

$k$

$C_{B}$$B$$C_{B}$$\rho(C_{B})$$R_{B}$$E_{B}$$\dagger$

$p$

$p$

$\llcorner\sigma itudicd$

Conjecture 1.1 4, Questions 1 and 2$[$$]$

The following conditions areequivalent:(a).
(b) $\rho(C_{B})=|D|$ .
(c) $R_{B}=E_{B}$ .
$\rho(C_{B})\in \mathbb{Z}$It is clear that $(c)\Rightarrow(b)\Rightarrow(a)$ . This conjecture is known to be $t:rue$ in thefollowing cases:$\bullet$$\bullet$$\bullet$$D$

a cyclic group (see [3, Corollary 4.8] and [4, Proposition 3]),

isomorphic to a dihedral, a semidihedral, or a generalized quaternion
group (see [4, Proposition 4]),$D$

The principal block with $D$ abelian when $p=2$ (see [7, Theorem]),153$\bullet$

The principal block with $D$ an elementary abelian group of order 9 (see[9, Theorem ] .
$C$$)$

Moreover, in these cases, the conditions in Conjecture 1.1 are equivalent tothe condition(d) $B$ is Morita equivalent to the Brauer correspondent$b$

in $N_{G}(D)$ .

It is known that (d) implies (c) in general (see [4, Proposition 2]), but theconverse is not always true (see [4, Example]). Nevertheless, if $D$ is abelian,then all of these conditions are expected to be equivalent:

Conjecture 1.2 If $D$ is abelian, then the above conditions $(a)-(d)$ are equivalent.

2Results

We assert that Conjecture 1.2 is true for the principal block of an arbitrarygroup with an abelian Sylow 3-subgroup, which is a generalization of Wada’s
result [9, Theorem$C$]. This is a joint work with Shigeo Koshitani.

Theorem 2.1 (see [6, Theorem 1.4]) Let $G$ be a finite group with anand let $B$ be the principal 3-block of $kG$ . Thenabelian Sylow 3-subgroupthe following conditions are equivalent.
$D_{f}$.
(a)(b) $\rho(C_{B})=|D|$ .
(c) $R_{B}=E_{B}$ .
(d) $B$ is Morita equivalent (even stronger Puig equivalent) to the Brauer
correspondent in $N_{G}(D)$ .
(e) Set $\tilde{G}=O_{3’}(G/O^{3’}(G))$ . Then$\rho(C_{B})\in \mathbb{Z}$$b$$\tilde{G}\cong S\cross G_{1}\cross\cdots\cross G_{r}$for some positive integer , where is an abelian 3-group and eachis one of the following nonabelian simple groups:$r$$S$(1) nonabelian simple groups with cyclic Sylow 3-groups,(2) $PSU_{3}(q^{2})$ with $3|q+1$ and $3^{2}(q+1$ ,(3) $PSp_{4}(q)$ with $3|q-1$ ,$G_{i}$

154
(4)
(5)

with $3|q+1$ and $3^{2}\{q+1$ ,$PSU_{4}(q^{2})$ with $3|q-1$ ,(6) $PSU_{5}(q^{2})$ with $3|q-1$ .
$PSL_{5}(q)$The proof is essentially similar to that of [9, Theorem ] foror[7, Theorem] for $p=2$ . But in our case, some additional tools aretoprove $(a)\Rightarrow(e)$ . The following lemma is useful to reduce the condition (a).
to$C$$D\cong C_{\text{ノ}}^{v_{3}}\cross C_{3}$$:\mathfrak{c}ieeded$$B_{0}(\tilde{G})$Lemma 2.1 (see [8, Theorem 1.1]) Let $G$ be a finite group and let $H$ bea normal subgroup of $G$ whose index is not divisible by . Let be a $(p-)block$of $H$ and let $B$ be a block of $G$ covering . Then $\rho(C_{B})=\rho(C_{b})$ .
$b$

$p$

$b$

Moreover, sincelemma enables us to reduce the condition (a) for$\rho(C_{B_{0}(\overline{G})})=|S|\cross\rho(C_{B_{0}(G_{1})})\cross\cdots\cross\rho(C_{B_{0}(G_{r})})$$B_{0}(\tilde{G})$, the following

to each$B_{\mathfrak{c}1}(G_{i})$.

Lemma 2.2 (see [6, Lemma 2.3]) Let$f(x)=x^{m}+a_{m-1}x^{m-1}+\cdots+a_{1}x+a_{0}$ ,$g(x)=x^{n}+b_{n-1}x^{n-1}+\cdots+b_{1}x+b_{0}$be two -polynomials with $m,$ $n\geq 1$ . Suppose thatandbe the roots of $f(x)$ and $g(x)$ . Moreover, we assume that all roots of $f(x)$and $g(x)$ are real and$\mathbb{Z}$$\alpha_{1},$$\alpha_{1}>\alpha_{2}\geq\cdots\geq\alpha_{m}>0$$\cdots,$$\alpha_{1}\in \mathbb{Z}$$\{i_{1},$$\cdots,$$\beta_{n}$and $\beta_{1}>\beta_{2}\geq\cdots\geq\beta_{n}>0$(but if $m=1$ (resp. $n=1$ ) this means $\alpha_{1}>0$ (resp.
.
thenand$\alpha_{1}\beta_{1}\in \mathbb{Z}_{f}$$\alpha_{m}$$\beta_{1}>0)$). Then, if$\beta_{1}\in \mathbb{Z}$In general, it is known that for a finite group$G$with nontrivial abelian Sylow

3-subgroup such that $O_{3’}(G)=1$ and $O^{3’}(G)=G$ , the form of $G$ is$G=S\cross G_{1}\cross\cdots\cross G_{r}$,where is an abelian 3-group and eachis one of the nonabelian simple
groups in a list (see [5, Proposition 1.1]). So it suffices to check the condition(a) for the groups in the list. But for almost all of the groups the condition(a) is checked in [9], so we only have to check for the principal blocks of thefollowing groups:$S$$G_{i}$

155$\bullet$the O’Nan simple group $O^{l}N$ ,$\bullet$$PSL_{4}(q)$with $3|q+1$ ,$\bullet$$PSL_{5}(q)$with $3^{2}|q+1$ ,$\bullet$$PSp_{4}(q)$with $3|q+1$ ,$\bullet$$PSL_{2}(q)$with $q=3^{n},$ $n=3,4,5,$$\cdots$.

It is easy to check it for the principal block of $O’N$ . Indeed, we can checkdirectly that the condition (a) does not hold for the block. So we only haveto check it for the other groups. In fact, by the result of Alperin [1] and Dade[2] we can reduce to $GL_{4}(q),$ $GL_{5}(q),$ $Sp_{4}(q)$ , and SL $2(q)$ respectively. Now
we have the following result:
Theorem 2.2 Let
p-subgroup of $G$ .

$B_{0}$

be the principal block of $kG$ , and let $D$ be a Sylow(1) If $G=GL_{4}(q),$$p$

is odd and $p|q+1$ , thenIf $G=GL_{5}(q),$$p$

is odd and $p|q+1$ , then$\rho(C_{B_{0}})\not\in \mathbb{Z}$is odd and $p|q+1$ , then$\rho(C_{B_{0}})\not\in \mathbb{Z}$(2)$\rho(C_{B_{0}})\not\in \mathbb{Z}$.
except when $p=3$

and $3^{2}(q+1$ .

(3) If $G=Sp_{4}(q),$$p$(4) If $G=$ SL2 $(q)$ , with $q=3^{n}$ with $n\geq 2$ , then.

$\rho(C_{B_{0}})\not\in \mathbb{Z}$.

References[1] J.L.Alperin, Isomorphic blocks, J. Algebra 43, 694-698 (1976).
[2] E.C.Dade, Remarks on isomorphic blocks, J.Algebra 45, 254-257 (1977).
[3] M.Kiyota and T.Wada, Some remarks on eigenvalues of the Cartan matrix in finite groups, Comm. Algebra 21, 3839-3860 (1993).

[4] M.Kiyota, M.Murai and T.Wada, Rationality of eigenvalues of Cartan
matrices in finite groups, J. Algebra 249, 110-119 (2002).

156[5] S. Koshitani, Corrigendum, Conjectures of Donovan and Puig for principal“3-blocks with abelian defect groups” by Shigeo Koshitani, Comm. Algebra
31, 2229-2243 (2003), Comm. Algebra 32, 391-393 (2004).

[6] S.Koshitani and Y.Yoshii, Eigenvalues of Cartan matrices of principal
3-blocks of finite groups with abelian Sylow 3-subgroups, in preparation,
2009.
[7] N.Kunugi and T.Wada, Eigenvalues of Cartan matrices of principa12blocks with abelian defect groups, J. Algebra 319, 4404-4411 $1’2008$ ).
[8] T.Okuyama and T.Wada, Eigenvalues of Cartan matrices of block in.
finite groups, preprint,$200^{(}J$[9] T.Wada, Eigenvector matrices of Cartan matrices for finite g)roups, J.
Algebra 308, 629-640 (2007).
[10] Y.Yoshii, On the Frobenius-Perron eigenvalues of Cartan matrices forsome finite groups, preprint, 2009.