数理解析研究所講究録 1440 巻 2005 年 34-40

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The intersection of normal closed subsets of association scheme is
not always normal吉川

信州大学大学院工学系研究科
昌慶 (Masayoshi Yoshikawa)

Department of Mathematical Sciences, Faculty of Science,
Shinshu University

Three questions on normal closed subsets of association schemes:
1. The intersection of normal closed subsets of an association
scheme is normal?

2. When a simple association scheme is primitive?

3. If $N$ andare normal closed subsets of such that $G//N$
and $G//N’$ are commutative, then $G//N\cap N’$ is commutative?
$N^{t}$$G$

Motivation
1. Ql is a natural question for non-commutative schemes.
Association schemes have closed subsets corresponding to subgroups, and normal closed subsets and strongly normal closed subsets corresponding to normal subgroups.
Closed subsets and strongly normal closed subsets are closed on
the intersection as well as subgroups and normal subgroups. However we have not known for normal closed subsets.
We have Hanaki’s result for group-like schemes but we do notknow for general schemes.
I had to calculate normal closed subsets for Q2 and Q3.
2. We want to consider the solvability of association schemes.
However we do not have the suitable definition. So I want to knowwhat condition occurs when to be simple is to be primitive

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An association scheme is simple iff the scheme has no non-trivial
normal closed subsets.
An association scheme is primitive iff the scheme has no nontrivial closed subsets.
A finite solvable simple group has no non-trivial subgoups. So
primitive is the one of characteristics of finite solvable
simplegroups.
I will verify the validity by considering Burnside’stheorem
and Feit-Thompson theorem.
$\Leftrightarrow$$p^{a}q^{b}$

3. We want to consider the normal closed subsets corresponding
to the derived subgroup.
We have the strongly normal closed subsets corresponding to thederived subgroup. However the factor scheme by a strongly normalclosed subsets is essentially a finite group.
Normal Closed Subsets and Character Table (Hanaki)
We can find all normal closed subsets of from its character
table.
Let be a character of . We set
$G$

$G$

$\eta$$K(\eta):=\{\mathit{9} \in G|\eta(g)=n_{g}\eta(1)\}$.

Generally $K(\eta)$ is not always a normal closed subset.
Proposition 1. For a normal closed subset $N$ ofcharacter of such that $N=K(\eta)$ .
$\eta$$G$, there exist$G$

For a character$\eta$

of$G$, we set}.
{ $\chi\in Irr(G)|\chi(g)=n_{g}\chi(1)$ for all
Theorem 2. For a character of G,
if and only if$I(\eta)$$:=$$g$$\eta$$K(\eta)\triangleleft G$$\sum_{\chi\in I(\eta)}m_{\chi}\chi(1)=\frac{n_{G}}{n_{K(\eta)}}$.

$\in K(\eta)$$a$3GTheorem 3. Let$\chi$

ancl$\eta$

be characters of G.

$K(\chi+\eta)=K(\chi)\cap K(\eta)$.

Group-like association scheme (Hanaki)
We define a relation on .
$G$$\sim$

For$g$, $h\in G$ , we write$\chi\in It$ $r(G)$$g$$\sim h$if $n_{g}^{-1}\chi(g)=n_{h}^{-1}\chi(h)$ for any.

Then this relation is an equivalent relation. Let
equivalent classes.

$T_{1}$,$\ldots$,$T_{s}$be$\sim$

We say $(X, G)$ is group-like if $s=|Irr(G)|$ .
If$G$is a finite group, then this relation is the conjugacy relation.

Theorem 4. We set ATi$= \sum_{g\in T_{i}}A_{g}$.

$Z(\mathbb{C}G)=\oplus \mathbb{C}A_{T_{i}}1\leq i\leq s$.

For the question 1, we know the following theorem.
Theorem 5.
, G) be a group-like association scheme. Thenthe intersection of normal closed subsets is also normal.
$Let\backslash Xr$Ql. The intersection of normal closed subsets of an association scheme is also normal?

From Theorem 5, it is enough that we checked only non-grouplike schemes.
I found three couterexamples for this question.
(as16[186], as18[79], as24[452]37

as16[186],,$K(\chi\tau)=[0, 2,5, 7]$ $K(\chi_{8})=[0,3,4,7]$ $K(\chi_{7}+\chi_{8})=[0,7]$$I(\chi_{7})=\{\chi_{1}, \chi_{2}, \chi_{7}\}$,$I(\chi_{8})=\{\chi_{1}, \chi_{2}, \chi_{8}\}$,$I(\chi_{7}+\chi_{8})=\{\chi_{1}, \chi_{2}, \chi_{7}, \chi_{8}\}$$m_{\chi_{1}}\chi_{1}(A_{0})+m_{\chi_{2}}\chi_{2}(A_{0})+m_{\mathrm{X}7}\chi\tau(A_{0})=4=16/4=n_{G}/n_{K(\chi_{7})}$$m_{\chi_{1}}\chi_{1}(A_{0})+m_{\chi_{2}}\chi_{2}(A_{0})+m_{X\mathrm{s}}\chi_{8}(A_{0})=4=16/4=n_{G}/n_{K(\chi_{8})}$$m_{\chi_{1}}\chi_{1}(A_{0})+m_{\chi_{2}}\chi_{2}(A_{0})+m_{\chi_{7}}\chi_{7}(A_{0})+m_{\chi_{8}}\chi \mathrm{s}(A_{0})=6$$n_{G}/n_{K(\chi_{7}+\chi \mathrm{s})}=8$.

.

Calculation
We assume that we have the setof adjacency matrices of anassociation scheme.
, $A_{s}=$We set $e_{s}=n_{s}^{-1}A_{s}$ for $s\subseteq G$ , where.
$G$$n_{s}= \sum_{A_{i}\in s}n_{i}$$\sum_{A_{i}\in s}A_{i}$We use the following facts.
$H\subseteq G$$H\subseteq G$is a closed subset iff$e_{H}$is a normal closed subset iff

1. We make the set$S$

is an idempotent of$e_{H}$$\mathbb{C}G$.

is a central idempotent of$=\{s\subseteq G|A_{0}\in s\}$.

$\mathbb{C}G$.

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2. We make the set $C=\{s\in S |e_{s}^{2}=e_{s}\}$ .
Thus is the set of all closed subsets of$C$$G$.

3. We make the set $N=$ { $s\in C|\mathrm{A}.e_{s}=e_{s}A_{i}$ for $A_{i}\in G$ }.
Thus $N$ is the set of all normal closed subsets of .
$G$

We can get relation matrices of all association schemes with$|X|\leq 29$ from Hanaki’s. This method is very simple but it
is enough for assocation schemes with $|X|\leq 29$ . To be enoughmeans that the computation time is not very long.
$\mathrm{H}\mathrm{P}$When a simple scheme is primitive?
I searched for association schemes with $|X|\leq 29$ .

as20[51], as20[66], as21[19], as28[123] are simple and imprimitive.

All of them are non-group-like association schemes.

We consider under the following conditions:
1. We use the cardinality of $X$ instead of the order of the finite group.

2. We define the solvability of assoication schemes as simple$arrow$

Burnside’s$p^{a}q^{b}$theoremFeit-Thompson theorem$\Leftrightarrow$primitive.

$\mathrm{x}$$\mathrm{x}$If we consider only group-like association schemes with $|X|\leq 29$ ,both of them hold.
Future Tasks

1. Under above conditions, two theorems hold for group-like association schemes39
2. Under above conditions, two theorems hold for -power orderassociation schemes?
Hirasaka, Ponomarenko and Zieschang approach to this problem
by restriction of valencies.
We consider the Frame number of an association scheme.
fff(G)$=|X|^{|G|} \frac{\prod_{g\in G}n_{g}}{\prod_{\chi\in Irr(G)}m_{\chi}^{\chi(1)^{2}}}$

We know the following theorem.

Theorem 6. Let be a prime number artd
$p$

$p$

.
$p$

$F$

a field of characteristic{$(G) iff $FG$ is semisimple.

We may use the Frame number instead of the order of the finitegroup.

ff (as20 [51])$=2^{11}3^{4}5^{6}$$\mathfrak{F}(\mathrm{a}\mathrm{s}20[66])=2^{13}3^{4}5^{7}$fff$(\mathrm{a}\mathrm{s}21 [19])=2^{4}3^{5}7^{6}$

fff(as2S[123])$=2^{12}3^{5}7^{6}$

Thus they do not satisfy the assumption of two theorem. However I
think that the Frame number is very large for the order of the finitegroup.
Q3. If N,

such that $G//N$ , $G//N’$ are commutative,$G//N\cap N’$ is commutative?
$N’\triangleleft G$

Generally the question holds if $N$ , $N’$ are strongly normal
Actually we can define a strongly normal closed subset corresponding to the derived subgroup.

Let

$R$

be the thin residue of$G$.

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Then $G//R$ is essentially the finite group.
We can obtain the derived subgroup $D(G//R)$ of $G//R$ .
By the Homomorphis theorem, we can obtaincorresponding to $D(G//R)$ .
$D(G)$ is the intersection of strongly normal closed subsets bywhich the factor schemes is an abelian group.
We can check by using the following fact. We assume thatand $\chi\in Irr(G)$ . Then$D(G)\triangleleft^{\mathfrak{g}}G$$N\triangleleft G$$\chi\in Irr(G//N)$

iff $\chi(h)=n_{h}\chi(1)$ for any $h\in N$ .

ResultWe do not have the normal closed subset corresponding to thederived subgroup even for group-like association schemes,(as16[158],as16[170])

Acknowledgement
The author thanks to A. Hanaki for valuable suggestions andcomments.

References[1] E. Bannai and T. Ito, Algebraic Combinatorics. I. Association
Schemes, Benjamin-Cummings, Menlo Park,1984.
$\mathrm{C}\mathrm{A}_{;}$[2] A. Hanaki and I. Miyamoto,$\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{p}://\mathrm{k}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{e}$.shinshu-u.

$\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}/\mathrm{a}\mathrm{s}/$.

[3] A. Hanaki and I. Miyamoto, “Classification of Association
Schemes with 16 and 17 Vertices”, Kyushu J. Math., vol. 52,1998, 383-395.
[4] A. Hanaki, “Characters of association schemes and normal closedsubsets”, Graphs Comb., 19(3) 363-369, 2003.

[5] P.-H. Zieschang, An Algebraic Approach to Association Schemes(Lecture Notes in Math. 1628), Springer, Berlin-HeidelbergNew York, 1998,