130

Characters of finite permutation groups and Krein
parameters
Keiji Ito
Graduate School of Information Science, Tohoku University
1

Introduction

A finite permutation group is called multiplicity‐free if the permutation character is a sumof distinct irreducible characters. A transitive permutation group is called regular if every
non‐identity element is fixed point‐free. For a transitive permutation group G on a finite set\Omega , let \Lambda_{0}, \Lambda_{1},\Lambda_{d} be the orbits of G on \Omega\cross\Omega . Then (\Omega, \{\Lambda_{i}\}_{i=0}^{d}) is an association schemeand the adjacency algebra \mathcal{A}=\langle A_{0}, A_{1},A_{d}\rangle_{\mathb {C} is isomorphic to End_{G}(\mathbb{C}[\Omega]) , where \mathbb{C}[\Omega]is the permutation module of G on \Omega.

If G is multiplicity‐free, then \mathcal{A} is commutative and the permutation module \mathbb{C}[\Omega] de‐composes into d+1 non‐isomorphic irreducible G‐modules: \mathbb{C}[\Omega]=V_{0}\oplus V_{1}\oplus\cdots\oplus V_{d} . For
d , let E_{i} be the orthogonal projection from \mathbb{C}[\Omega] onto V_{i} . Then E_{i} are primitive
i=0,1,
idempotents of \mathcal{A} and \{E_{0}, E_{1}, E_{d}\} is a basis of \mathcal{A} . Since \mathcal{A} is closed under the Hadamard

product 0 , there exist scalars q_{l,j}^{k} such that

E_{i} \circ E_{j}=\frac{1}{|\Omega|}\sum_{k=0}^{d}q_{i,j}^{k}E_{k}.
These scalars are called Krein parameters. Let \chi_{i} be the irreducible character corresponding
to the irreducible G‐module V_{i} . Scott’s Theorem reveals the following relations between

Krein parameters and inner products of characters: if q_{i_{J}}^{k}\neq 0 , then (\chi_{i}\chi_{j}, \chi_{k})\neq 0 (seeTheorem 3.3). The converse of this implication is not always true. As a counterexample, wehave the action of the symmetric group S_{2d} on all d‐element subsets of a 2d‐element set (see[2, Section 2.8]).
In this paper, we prove the converse of the implication for transitive permutation groupsof semidirect product type whose regular normal subgroup is abelian. If a permutation group
G has a regular abelian normal subgroup, then G is multiplicity‐free. Under this condition,
we can describe all irreducible characters which appear in the permutation character. We
find that the inner products of characters and Krein parameters are similar in terms of theircomponents (see Theorem 5.4). In particular, the inner products and Krein parameters areequal for some transitive permutation groups.

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2

Commutative association schemes

Let X be a set with |X|=n and R_{i}\subset X\cross X(i=0,1, \ldots, d) . For R_{i}(i=0,1, \ldots, d) ,the i^{th} ‐adjacency matntx A_{i} of \mathcal{X} is the square matrix indexed by X such that (A_{3})_{xy}=1 if(x, y)\in R_{i}, (A_{i})_{xy}=0 otherwise.

Definition 2.1. The pair (X, \{R_{\iota}\}_{i=0}^{d}) is called a commutative association scheme if follow‐
ing hold:(i) A_{0}=I_{n}.
(ii)

J_{n}= \sum_{i=0}^{d}A_{i}.

(iii) For any i\in\{0, d\} , There exists i'\in\{0,1, , d\} such that A_{i}^{T}=A_{x'}.

(iv) There exist p_{\iota,j}^{k} such that A_{i}A_{J}= \sum_{k=0}^{d}p_{i}^{k_{j} ,A_{k} for all i,j\in\{0, d\}.
(v) A_{x}A_{j}=A_{j}A_{i} for all i, j\in\{0, d\}.

\mathcal{X}=(X, \{R_{l}\}_{i=0}^{d}) be a commutative association scheme with |X|=n and A_{i}(i=d) be the adjacency matrices of \mathcal{X}. k_{i}=p_{i,i}^{0} , is called the valency of R_{l} . The subalgebra\mathcal{A}=\langle A_{0}, A_{1},A_{d}\} of the matrix algebra M_{n}(\mathbb{C}) is called the adjacency algebra of \mathcal{X} . The
Let0,

E_{d} . The integers
adjacency algebra \mathcal{A} has d+1 primitive idempotents E_{0}, E_{1},rank(E_{i}) are called the multiplicities of \mathcal{X}.
Since both of \{A_{0}, A_{1}, , A_{d}\} and \{E_{0}, E_{1}, E_{d}\} are bases of \mathcal{A} , let

m_{i}=

A_{i}= \sum_{j=0}^{r}p_{i}(j)E_{j},E_{i}= \frac{ \imath} {n}\sum_{j=0}^{d}q_{i}(j)A_{j}for

i=0,1,d.

Since \mathcal{A} is closed under the Hadamard product, we may define Krein parameters.
Definition 2.2. Let E_{0}, E_{1},E_{d} be the primitive idempotents of the adjacency algebra \mathcal{A}of \mathcal{X} . Set

E_{i} \circ E_{j}=\frac{1}{n}\sum_{k=0}^{d}q_{\iota,j}^{k}E_{k}for some q_{i,j}^{k}\in \mathbb{C} , where 0 denotes the Hadamard product. The coefficients q_{i,j}^{k} are called the
Krein parameters of \mathcal{X}.

Lemma 2.3 ([2, Theorem 3.6 in Chapter II]). For any i,j, k\in\{0,1, d\},q_{\iota,j}^{k}=\frac{m_{i}m_{J} {n}\sum_{l=0}^{d}\frac{1}{k_{l}^{2} p_{l}(i) p_{l}(j)\overline{p_{l}(k)}.

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3Scott’s Theorem

Let G be a transitive permutation group on a finite set \Omega . We may assume the permutation
character \theta of G decomposes into the sum of distinct irreducible characters: \theta=\chi_{0}+\chi_{1}++\chi_{d} , where \chi_{\iota} are distinct irreducible characters of G for i\in\{0,1, d\} . Note that,
this condition is called multiplicity‐free. Define the action of G on \Omega\cross\Omega by (x, y)^{g}=(x^{g}, y^{g})for x, y\in\Omega and g\in G . Then \mathcal{X}=(\Omega, \{\Lambda_{i}\}_{\iota=0}^{d}) is a commutative association scheme, where\Lambda_{d} are the orbits of G on \Omega\cross\Omega and \Lambda_{0}=\{(x, x)|x\in\Omega\} (see [2, Theorem 1.4\Lambda_{0}, \Lambda_{1},in Chapter II]).
Let H\leq G be the stabilizer of a point a\in\Omega . Identifying H\backslash G with \Omega , we regard
the action of H on \Omega as that on H\backslash G by multiplication of G . Then a\in\Omega , which H
stabilizes, corresponds to H\in H\backslash G . Then the permutation character \theta decompose into

\theta=\chi 0 +\chi ı ++\chi_{d}.

Lemma 3.1 ([2, Corollary ı1.7 in Chapter II]). Let P=(p_{j}(i)) be the first eigenmatrix.

p_{i'}(i)= \frac{1}{|H|}\sum_{x\in Ha_{l}H}\chi_{j}(x),where \{a_{0}, a_{1}, a_{d}\} is the representatives of H\backslash G/H.

Lemma 3.2 ([3, Section 3.4]). Let \rho be the permutation representation of G. Set

E_{i}= \frac{\deg(\chi_{i}) {|G|}\sum_{g\in G}\overline{\chi_{i}(g)}\rho(g)d . Then each E_{i} is the orthogonal projection onto the irreducible G‐for every i=0,1,
module corresponding to \chi_{l} . In other words, E_{0}, E_{1} ,, E_{d} are all primitive idempotents of
the adjacency algebra \mathcal{A} of the commutative association scheme \mathcal{X}.

Theorem 3.3 (Scott’s Theorem, [2, Theorem 8.1 in Chapter II], [5, Theorem 3]). Let G

be a transitive permutation group on a finite set \Omega and \mathcal{X}=(\Omega, \{\Lambda_{t}\}_{\iota=0}^{d}) be the association
scheme defined by the action of G on \Omega\cross\Omega , where \Lambda_{i} are the orbits of G on \Omega\cross\Omega . Assume+\chi_{d} , where \chi_{i} are irre‐
the permutation character \theta is multiplicity‐free: \theta=\chi_{0}+\chi_{1}+ducible characters of G and \chi_{0}=1_{G} . Then, for any i, j, k\in\{0,1, d\} , if q_{i,j}^{k}\neq 0 , then(\chi_{i}\chi_{j}, \chi_{k})\neq 0.

The detailed proof of this theorem is written in the references. In this paper, we mention
a sketch of the proof. To prove this theorem, for i,j, k\in\{0,1, d\} , we construct \sigma_{i,j}^{k}\in
Hom_{G}(V_{i}\otimes V_{j}, V_{k}) such that \sigma_{i,j}^{k}=0 if and only if q_{i,j}^{k}=0 , where V_{\iota} is an irreducible G‐
module with its irreducible character \chi\iota for i\in\{0,1, d\} . If q_{i,j}^{k}\neq 0 , then \sigma_{i,j}^{k}\neq 0 and, by
Schur’s lemma, it implies that V_{i}\otimes V_{j} contains the irreducible G‐module which is isomorphic
toV_{k}.

To investigate the converse of Theorem 3.3, we suppose q_{i,j}^{k}=0 . Then, we obtain \sigma_{l}^{k_{J} ,=0by \sigma_{i,j}^{k}=0 if and only if q_{\iota,j}^{k}=0 . This does not always imply (\chi_{i}\chi_{j}, \chi_{k})=0 . Indeed, thesymmetric group S_{2d} on the Johnson Scheme J(2d, d) is an example (see [2, Section 2.8]).

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4Transitive permutation groups of semidirect producttype

Let G be a transitive permutation group on a finite set \Omega . We assume that G=H\ltimes N,
where H is the stabilizer of a point x\in\Omega, N is a regular abelian permutation group and \ltimes
denotes the semidirect product. I.e.
\bullet\{1\}\cross N is a normal subgroup of G , where 1\in H is the identity element of H,e

For any g\in G , there exist unique h\in H and unique n\in N such that g=(h, n) ,e

The multiplication of G is defined as (h_{1}, n_{1})(h_{2}, n_{2})=(h_{1}h_{2}, n_{1}^{h_{2}}+n_{2}) .

In this case, we may identify N with \Omega , where the identity element

0\in N corresponds tothe fixed point x\in\Omega.

When we regard N as a normal subgroup of G , we should write {1} \cross N instead of N.
However we will identify {1} \cross N with N for brevity. Similarly, we will identify H\cross\{0\}with H.

By the definition of the semidirect product, it is clear that H\backslash G=\{H(1, n)|n\in N\}.
Let Irr(N) be the set of all irreducible characters of N . Since N is abelian, irreduciblecharacters of N are also irreducible representations of N . Define an action of H on Irr(N)by\varphi^{h}(n)=\varphi(n^{h^{-1}})

forh\in H, \varphi\in Irr(N) and n\in N . Let

H_{\varphi}=\{h\in H|\varphi^{h}=\varphi\}\varphi\in Irr(N) . This means that H_{\varphi} is the stabilizer of \varphi\in Irr(N) and H_{\varphi}\leq H .
extension of \varphi\in Irr(N) to H_{\varphi}\ltimes N , denoted by ex(\varphi) , is defined as

for

ex

The

(\varphi)(h, n)=\varphi(n)for h\in H_{\varphi} and n\in N (see [6, Chapter 17]).
Lemma 4.1. For each \varphi\in Irr(N), ex(\varphi) is an irreducible representation of H_{\varphi}\ltimes N.
Proof. Since

ex(\varphi)((h, n)(h', n'))=ex(\varphi)(hh', n^{h'}+n')=\varphi(n^{h'}+n')=\varphi^{h^{J-1}}(n)\varphi(n')=\varphi(n)\varphi(n')=ex(\varphi)(h, n)ex(\varphi)(h', n')
for h, h'\in H_{\varphi} and n, n'\in N, ex(\varphi) is a representation of H_{\varphi}\ltimes N . By the definition of theextension, \deg(ex(\varphi))=\deg(\varphi) , and ex(\varphi) is irreducible.

\square134
Lemma 4.2. For \varphi\in Irr(N), h\in H andn\in N,

Ind_{H_{\varphi}\ltimesN}^{G}ex(\varphi)(h,n)=\sum_{\psi\n\varphi^{H}, \psi^{h}=\psi}\psi(n).

Proof. For h\in H, n\in N,

Ind_{H_{\varphi}\ltimesN}^{G}ex(\varphi)(h,n)=\sum_{H_{\varphi} ex(\varphi) (l^{-1}hl,n^{l})=\sum_{H_{\varphi},l^{-1}hl\inH_{\varphi} \varphi(n^{l})= \sum_{\psi^{h}=\psi}\psi(n)l\inH/l\inH/\psi\n\varphi^{H}.
Since there is a bijection from H/H_{\varphi} to G/(H_{\varphi}\ltimes N) defined by lH_{\varphi}\mapsto(l, 0)H_{\varphi}\ltimes N , we canchoose l\in H/H_{\varphi} instead of G/(H_{\varphi}\ltimes N) .

\squareLemma 4.3.

(i) Induced characters Ind_{H_{\varphi}\cdot N}^{G}ex(\varphi) are irreducible characters of G for all \varphi\in Irr(N) .
(ii) For \varphi, \varphi'\in Irr(N), Ind_{H_{\varphi}\ltimes N}^{G}ex(\varphi)=Ind_{H_{\varphi},\ltimes N}^{G} ex(\varphi') if and only if there exists h\in Hsuch that \varphi^{h}=\varphi'.

Proof. By Lemma 4.2,

(Ind_{H.\ltimesN}^{G}ex(\varphi),Ind_{H_{\varphi},\ltimesN}^{G}ex(\varphi') _{G}=\frac{1}|HN|}\sum_{h\inH}\sum_{n\i N}(\sum_{\psi^{h}=\psi}\psi(n) (\sum_{\psi^{h}=\psi},\overline{\psi'(n)}=\frac{1}{|H}\sum_{\psi\n\varphi^{H}\sum_{\psi'\in\varphi^{\primeH} |H_{\psi}\capH_{\psi'}|(\frac{1}{|N}\sum_{n\inN}\psi(n)\overline{\psi'(n)}= \frac{1}{|H|}\sum_{\psi\ n\varphi^{H} \sum_{\psi'\in\varphi^{;H} |H_{\psi} \cap H_{\psi'}|(\psi, \psi')_{N}=\frac{1}{|H}\delta_{\varphi^{H},\varphi^{\primeH}\sum_{\psi, \psi\n\varphi^{H}|H_{\psi}\capH_{\psi'}|\delta_{\psi,\psi'}=\delta_{\varphi^{H},\varphi^{\prime H}}. \square

Remark that Lemma 4.3(i) is also given in [4, (6.11)Theorem].
Let \varphi_{0} , \varphi ı,
In other words,\varphi_{d} be representatives of the orbit partition of H on

Ir (N)=\prod_{i=0}^{d}\varphi_{l}^{H}.

Irr(N) , where \varphi_{0}=1_{N}.

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Lemma 4.4. Let 1_{H}^{G} be the permutation character of G. Then

Ind_{H}^{G}1_{H}=\sum_{i=0}^{d}Ind_{H_{\varphi_{i} \cdot N}^{G}ex(\varphi_{i}).

Proof. It suffices to prove (Ind_{H}^{G}I_{H}, Ind_{H_{\varphi}\ltimes N}^{G}ex(\varphi_{i}))_{G}=1 and \sum_{i=0}^{d}\deg(Ind_{H_{\varphi}\ltimes N}^{G}ex(\varphi_{i}) =|G/H| . Since N is abelian, \deg(\psi)=1 for any \psi\in Irr(N) . By Lemma 4.2 and Frobenius

reciprocity theorem (see [1, Theorem 2 in Section 6.16]),(Ind_{H}^{G}I_{H}, Ind_{H_{\varphi_{i} \cdot N}^{G}ex(\varphi_{\iota}) _{G}=(1_ {H}, {\rm Res}_{H}^{G}Ind_{H_{\varphi}\cdot N}^{G}ex(g_{t}) _{H}=\frac{1}{|H}\sum_{h\inH}\sum_{\psi\n\varphi^{H},\psi^{h}=\psi}\psi(0)= \frac{1}{|H}\sum_{\psi\n\varphi_{\iota}^{H}|H_{\psi}|\deg(\psi)=1.
Moreover,\sum_{i=0}^{d}\deg Ind_{H_{\varphi_{\iota}\cdotN}^{G}(

5

ex (\varphi_{i}) )=\sum_{i=0}^{d}\frac{|H}{|H_{\varphi_{l} | \deg(ex (\varphi_{i}) )= \sum_{i=0}^{d}\frac{|H|}{H_{\varphi_{i} | =\sum_{i=0}^{d}|\varphi_{i}^{H}|= |N|.

\squareThe converse of Scott’s Theorem

In this section, we assume that a transitive permutation group G=H\ltimes N on \Omega is same asthat of the previous section. By Lemma 4.4, 1_{H}^{G} is multiplicity‐free: 1_{H}^{G}=\chi_{0}+\chi_{1}+\cdots+\chi_{d},where \chi_{\iota}=ex(\varphi_{i})^{G} . Then we can construct the association scheme \mathcal{X}=(\Omega, \{\Lambda_{i}\}_{i=0}^{d}) , where\Lambda_{\iota} are orbits of G on \Omega\cross\Omega and \Lambda_{0}=\{(x, x)|x\in\Omega\} . Let \{n_{0}, n_{1}, n_{d}\} be a set ofrepresentatives of H\backslash G/H . We can assume n_{i}\in N for all i\in\{0,1, d\} and n_{0}=0 , i.e.

H\backslash G/H=IIH\cross n_{i}^{H}.
Then the association scheme \mathcal{X} has valencies k_{i}=|H|/|Stab_{H}(n_{t})| and multiplicities m_{i}=|H|/|H_{\varphi_{t}}|.

Lemma 5.1. For any i,j, k\in\{0,1, d\},q_{i,j}^{k}= \frac{|H_{\varphi_{k} |R_{i,j}^{k}| {|H|},
where

R_{l}^{k_{j} ,=\{(\psi_{1}, \psi_{2}, \psi_{3})\in\varphi_{i}^{H}\cross\varphi_ {J}^{H}\cross g_{k}^{H}|\psi_{1}\psi_{2}=\psi_{3}\}.

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Proof. By Lemma 3.1 and Lemma 4.2,

p_{s'}(i)= \frac{1}{|H|}\sum_{g\in H\cros n_{s}^{H} Ind_{H_{\bul et}\ltimes N} ^{G}ex(\varphi_{z})(g)= \frac{1}{|H|}\frac{1}{|Stab_{H}(n_{s})|}\sum_{h_{1},h_{2}\in H} Ind_{H_{\varphi_{t} \ltimes N}^{G}ex(\varphi_{i})(h_{1}, n_{s}^{h_{2} )=\frac{1}|HStab_{H}(n_{s})|\sum_{h_{1},h_{2}\inH} \sum_{\psi\n\varphi_{l}^{H},\psi^{h_{1}=\psi}\psi(n_{s}^{h_{2})=\frac{|H_{\varphi_{l} |H|}{H|Stab_{H}(n_{s})| \sum_{\psi\n\varphi_{i}^{H} }\psi(n_{s})=\frac{|H_{\varphi_{l} | {Stab_{H}(n_{s})| \sum_{\psi\n\varphi_{l}^{H} \psi(n_{s}).

|H_{\varphi_{\iota}}|=|H_{\psi}| holds for all \psi\in\varphi_{i}^{H}.
Since k_{l}=|H|/|Stab_{H}(n_{i})| and m_{i}=|H|/|H_{\varphi_{l}}| , by Lemma 2.3,

Remark that

q_{i,j}^{k}= \frac{m_{l}m_{J} {|X}\sum_{s=0}^{d}\frac{1}{k_{s}^{2} p_{s'}(i)p_ {s'}(j)\overline{p_{s'}(k)}=\frac{|H^{2}{|H_{\varphi_{l}|H_{\varphi_{J}|N}\sum_{s=0}^{d} \frac{|Stab_{H}(n_{s})|^{2}{|H^{2}(\frac{|H_{\varphi_{t}|{Stab_{H}(n_{6}) |}\sum_{\psi_{1}\in\varphi_{l}^{H}\psi_{1}(n_{s})(\frac{|H_{\varphi_{J}|{Stab_{H}(n_{s})|\sum_{\psi_{2}\in\varphi_{J}^{H} \psi_{2}(n_{s})(\frac{|H_{\varphi_{k}|{Stab_{H}(n_{s})|\sum_{\psi_{3} \in\varphi_{k}^{H}\overline{\psi_{3}(n_{s})=\frac{|H_{\varphi_{k}|{N|}\sum_{s=0}^{d}\frac{\imath}{|Stab_{H}(n_{s})| \sum_{\psi_{1\in\varphi_{}^{H} \sum_{\psi_{2\in\varphi_{j}^{H} \sum_{\psi_{3\in\varphi_{k}^{H} \psi_{1}(n_{s})\psi_{2}(n_{s})\overline{\psi(n_ {s})=\frac{|H_{\varphi_{k}|{H|N}\sum_{s=0}^{d}\sum_{\psi_{1}l\inH/Stab_{H} (n_{s})\in\varphi_{l}^{H\sum}\psi_{2}\\sumin\varphi_\sum\psi{j}^{H}\psi_{3}\in\_v{1}(n_{s})\psi
arphi_{k}^{H} _{2}(n_{s})\overline{\psi_{3}(n_{s})}=\frac{|H_{\varphi_{k}|{H|N}\sum_{s=0}^{d}\sum_{(l\inH/Stab_{H}n_{8}) \psi_{1\in\varphi_{l}^{H\sum}\psi_{2}\\sumin\varphi_\sum\psi{J}^{H}\psi_{3}\_in{\1va}r(pn_{hi_{ks}^}{H^}{l-1})\psi_{2}(n_{s}^{l-1})\overline{\psi_{3} (n_{s}^{l-1})}=\frac{|H_{\varphi_{k}|{H|}\sum_{\psi_{1}\in varphi_{l}^H}\sum_{\psi_{2} \in varphi_{j}^H}\sum_{\psi_{3}\in varphi_{k}^{H}(\frac{1}|N}\sum_{n\i N} \psi_{1}\psi_{2}\overline{\psi_{3}(n)137= \frac{|H_{\varphi_{k} |}{|H|}R_{\iota,j}^{k}|.
\squareLemma 5.2. For any i,j, k\in\{0,1, d\},( \chi_{i}\chi_{j}, \chi_{k})_{G}=\frac{1}{|H|}\sum_{(\psi_{1},\psi_{2}, \psi_{3})\in R_{i,j}^{k} |H_{\psi_{1} \cap H_{\psi_{2} \cap H_{\psi_{3} |.
Proof. By Lemma 4.2,(\chi_{i}\chi_{J}, \chi_{k})_{G}=(Ind_{H_{\varphi_{i} \ltimes N}^{G}ex(\varphi_ {z})Ind_{H_{\varphi_{J} \cdot N}^{G}ex(\varphi_{j}), Ind_{H_{\varphi_{k} \cdot N}^{G}ex(\varphi_{k}) _{G}=\frac{1}|G\sum_{h\inH}\sum_{n\i N}(\sum_{\psi_{1}\in varphi_{}^H} \psi_{1}(n) (\begin{ary}l \Sigma psi_{2}(n)\psi_{2}n\varphi_{J}^H \psi_{2^h}=\psi_{2} \end{ary})(\begin{ary}l \Sigma overlin{\ps_3}(n) \psi_{ n\varphi_{k}^H \psi_{3^h}=\psi_{3} \end{ary})=\frac{1}{|G}\sum_{\psi_{1\in\varphi_{l}^{H} \sum_{\psi_{2\in\varphi_{J}^{H} }\sum_{\psi_{3\in\varphi_{k}^{H} |H_{\psi_{1}\capH_{\psi_{2}\cap H_{\psi_{3}|\sum_{n\inN}\psi_{1}\psi_{2}\overline{\psi_{3}(n)=\frac{1}|H}\sum_{\psi_{1\in\varphi_{l}^{H} \sum_{\psi_{2\in\varphi_{J}^{H} }\sum_{\psi_{3\in\varphi_{k}^{H} (\frac{1}|N}\sum_{n\inN}\psi_{1}\psi_{2} \overline{\psi_{3}(n)|H_{\psi_{1}\capH_{\psi_{2}\capH_{\psi_{3}|=\frac{1}{|H}\sum_{g}|H_{\psi_{1} \capH_{\psi_{2} \capH_{\psi_{3} |(\psi_{1},\psi_{2},\psi_{3})\inR_{l}^{k},\cdot\squareLemma 5.3 ( [4, (6.32)Theorem] ) . Let G be a group and H be a group which acts on Irr(G)and the set of conjugacy classes of G , where Irr(G) is the set of irreducible characters of G.
If \chi^{h}(g^{h})=\chi(g) for all g\in G, h\in H, \chi\in Irr(G) , then the number of irreducible characterswhich h fixes is equal to the number of conjugacy classes which h fixes for any h\in H.
Theorem 5.4. Let G=H\ltimes N be a transitive permutation group on \Omega, H be the stabilizer
of a point x\in\Omega and N be an abelian normal subgroup. Then the permutation character ismultiplicity‐free: Ind_{H}^{G}1_{H}=\chi_{0}+\chi_{1}+\cdots+\chi_{d} , where \chi_{0} is the identity character. Moreover,the following hold.

(i) For any i,j, k\in\{0,1, d\} , if (\chi_{i}\chi_{j}, \chi_{k})_{G}\neq 0 , then q_{i,j}^{k}\neq 0.

(ii) If any g\in G(g\neq 1_{G}) fixes less than 3 points in \Omega , thenq_{i,j}^{k}=|H_{\varphi_{k} |(\chi_{i}\chi_{j}, \chi_{k})_{G}
for distinct i,j, kE\{0,1, d\}.

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Proof. (i) By Lemma 5.1, if q_{l}^{k_{J}}=0 , then R_{l}^{k_{j} ,=\emptyset . By Lemma 5.2,

( \chi_{i}\chi_{j}, \chi_{k})_{G}=\frac{1}{|H|}\sum_{(\psi_{1},\psi_{2}, \psi_{3})\in R_{i,g}^{k} |H_{\psi_{1} \cap H_{\psi_{2} \cap H_{\psi_{3} |=0.
This means that we proved the contrapositive of the assertion.

(ii) By Lemma 5.3, if any g\in G(g\neq 1) fixes less than 3 points in \Omega , then |H_{\psi_{1}}\cap H_{\psi_{2}}\cap\squareH_{\psi_{3}}|=1 for distinct \psi_{1}, \psi_{2}, \psi_{3}\in Irr(N) . Thus we obtain the result.

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1995.

[2] E. Bannai and T. Ito, Algebraic Combinatrics I: Association Schemes, Ben‐
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[3] W. Fulton and J. Harris, Representation Theory. A first course, Springer‐Verlag, New
York, 1991.

[4] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York‐London,
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[5] L. L. Scott, Some properties of character products, J. Algebra 45 (1977), no. 2, 259‐265.
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