\documentclass[ALCO,ThmDefs,Unicode,epreuves]{cedram}
\OneNumberAllTheorems


\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\DeclareMathOperator{\M}{M}
\newcommand{\s}{\sigma}
\DeclareMathOperator{\Irr}{Irr}
\newcommand{\bs}{\boldsymbol}
\newcommand{\expgt}{{}^{t\mkern-3mu}}

%%%%%%%%% a placer avant \begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}


\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Auteur

%%%1
\author{\firstname{Akihide} \lastname{Hanaki}}

\address{Faculty of Science \\
  Shinshu University \\
  3-1-1 Asahi \\
  Matsumoto \\
  390-8621, Japan}

\email{hanaki@shinshu-u.ac.jp}

\urladdr{http://math.shinshu-u.ac.jp/~hanaki}

\thanks{Akihide Hanaki was supported by JSPS KAKENHI Grant Number JP17K05165.
  Masayoshi Yoshikawa was supported by JSPS KAKENHI Grant Number JP17K05173.}


%%%2
\author{\firstname{Masayoshi} \lastname{Yoshikawa}}

\address{Department of Mathematics \\
  Hyogo University of Teacher Education \\
  942-1 Shimokume \\
  Kato \\
  Hyogo \\
  673-1494, Japan}

\email{myoshi@hyogo-u.ac.jp}


%%%%% Sujet

\keywords{Character table, quaternion algebras, association schemes.}

\subjclass{05E30}


%%%%% Gestion

\DOI{10.5802/alco.167}
\datereceived{2019-11-21}
\daterevised{2020-12-20}
\dateaccepted{2020-12-20}


%%%%% Titre et résumé
\title
[A construction of pairs of association schemes]
{A construction of pairs of non-commutative rank 8
  association schemes from non-symmetric rank 3 association schemes}

\begin{abstract}
We construct a pair of non-commutative rank $8$ association schemes
from a rank $3$ non-symmetric association scheme.
For the pair, two association schemes have the same character table
but different Frobenius--Schur indicators.
This situation is similar to that for the dihedral and quaternion groups of order $8$.
We also determine the structures of adjacency algebras of the rank $8$
schemes over the rational number field.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{document}


\maketitle

\section{Introduction}
From a rank $3$ non-symmetric association scheme of order $n-1$,
we construct a pair of association schemes $(\mathcal{D}, \mathcal{Q})$ with the following properties.
\begin{itemize}
  \item $\mathcal{D}$ and $\mathcal{Q}$ are non-commutative, of order $4n$, and of rank $8$.
  \item $\mathcal{D}$ and $\mathcal{Q}$ have the same character tables
  but their Frobenius--Schur indicators are different.
\end{itemize}
These properties are similar to the pair of the dihedral group $D_8$ and the quaternion group $Q_8$ of order $8$.

In the theory of association schemes, the adjacency algebras
are mainly considered over the complex number field.
In this paper, we determine the structures of adjacency algebras of
$\mathcal{D}$ and $\mathcal{Q}$ over the rational number field $\Q$.
We prove 
\begin{align*}
  \mathbb{Q}\mathcal{D}&\cong \mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus \M_2(\mathbb{Q}),\\
  \mathbb{Q}\mathcal{Q}&\cong \mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus\mathbb{Q}\oplus \mathbb{Q}(-1,-n+1),
\end{align*}
where $\M_2(\mathbb{Q})$ is the full matrix algebra of degree $2$ and $\mathbb{Q}(-1,-n+1)$
is the quaternion division algebra.

It is known that a rank $3$ non-symmetric association scheme of order $n-1$
exists if and only if there exists a skew-Hadamard matrix of order $n$ with $n\equiv 0 \pmod{4}$.
There is a conjecture that such a matrix of order $n$ exists for an arbitrary $n\equiv 0\pmod{4}$.

\section{Preliminaries}
\looseness1
For a field $K$, we denote by $\M_n(K)$ the full matrix algebra of degree $n$ over $K$.
For a matrix $M$, the transposed matrix of $M$ will be denoted by $\expgt M$.
By $I_n$, we denote the identity matrix of degree $n$.
By $J_n$, we denote the $n\times n$ matrix with all entries~$1$.

We define an association scheme in matrix form.
Let $\mathcal{S}=\{A_0,\dots,A_d\}$ be a set of non-zero $n\times n$ matrices with entries in $\{0,1\}$.
Then the set $\mathcal{S}$ is called an \emph{association scheme} of \emph{order} $n$ and \emph{rank} $d+1$ if
\begin{enumerate}
  \item\label{Sect2_1} $A_0=I_n$,
  \item\label{Sect2_2} $\sum_{i=0}^d A_i=J_n$, and
  \item\label{Sect2_3} for any $0\leq i, j\leq d$, $\expgt A_i$ and $A_iA_j$ are linear combinations of
  $A_0,\dots,A_d$.
\end{enumerate}
By definition, all rows of $A_i$ contain the same number of $1$.
We call this number the \emph{valency} of $A_i$ and denote it by $n_i$.
The association scheme $\mathcal{S}$ is said to be \emph{symmetric}
if $\expgt A_i=A_i$ for all $0\leq i\leq d$
and \emph{non-symmetric} otherwise.
The association scheme $\mathcal{S}$ is said to be \emph{commutative}
if $A_iA_j=A_jA_i$ for all $0\leq i, j\leq d$
and \emph{non-commutative} otherwise.
For a field $K$, $K\mathcal{S}\coloneqq \bigoplus_{i=0}^d KA_i$ is a $(d+1)$-dimensional $K$-algebra
by the condition~\eqref{Sect2_3}.
We call $K\mathcal{S}$ the \emph{adjacency algebra} of $\mathcal{S}$ over $K$.
It is known that $K\mathcal{S}$ is semisimple if the characteristic of $K$ is $0$
\cite[Theorem~4.1.3~(ii)]{Zi}.
A \emph{representation} of $\mathcal{S}$ over $K$ means a $K$-algebra homomorphism
from $K\mathcal{S}$ to a full matrix algebra $\M_t(K)$ for some positive integer $t$.

A subset $\mathcal{T}$ of an association scheme $\mathcal{S}$ is called a \emph{closed subset}
if $e_{\mathcal{T}}\coloneqq n_{\mathcal{T}}^{-1}\sum_{A_i\in\mathcal{T}}A_i$ is an idempotent,
where $n_{\mathcal{T}}\coloneqq  \sum_{A_i\in\mathcal{T}}n_i$.
A closed subset $\mathcal{T}$ of $\mathcal{S}$ is said to be \emph{normal}
if $e_{\mathcal{T}}$ is a central element of the complex adjacency algebra $\C\mathcal{S}$.
A closed subset $\mathcal{T}$ of $\mathcal{S}$ is said to be \emph{strongly normal}
if $\expgt A_jA_iA_j\in \bigoplus_{A_\ell\in\mathcal{T}}\C A_\ell$ for $A_i\in \mathcal{T}$ and $A_j\in \mathcal{S}$.
It is known that a strongly normal closed subset is normal.
The \emph{thin radical} of $\mathcal{S}$ is
$O_\theta(\mathcal{S})=\{A_i\mid n_i=1\}$ and is known to be a closed subset of $\mathcal{S}$.
The \emph{thin residue} $O^\theta(\mathcal{S})$ of $\mathcal{S}$ is the intersection of all strongly normal closed subsets of $\mathcal{S}$
and is known to be a strongly normal closed subset of $\mathcal{S}$.
For details, the reader is referred to~\cite{Zi} or~\cite{Hanaki2005}.

\looseness1
Now we consider the adjacency algebra over the complex number field $\C$.
By \hbox{Wedderburn's} theorem~\cite[3.5~Theorem]{Pierce},
$\C\mathcal{S}\cong \bigoplus_{i=1}^\ell \M_{t_i}(\C)$ for some $t_1,\dots,t_\ell$.
The set of projections $\C \mathcal{S}\to \M_{t_i}(\C)$ ($i=1,\dots,\ell$)
is a complete set of representatives of equivalence classes of irreducible representations of $\C\mathcal{S}$.
The matrix trace of a representation is called the \emph{character} of the representation.
By $\Irr(\mathcal{S})=\{\chi_1,\dots,\chi_\ell\}$, we denote the set of irreducible characters of $\C\mathcal{S}$.
The $\ell \times (d+1)$ matrix $(\chi_i(A_j))$
is called the \emph{character table} of $\mathcal{S}$.
Since $\C\mathcal{S}$ is defined as a matrix algebra, the map $A_i\mapsto A_i$ is a representation.
We call it the \emph{standard representation}
and its character the \emph{standard character} of $\mathcal{S}$.
The decomposition of the standard character into the irreducibles is written as
$\sum_{\chi\in\Irr(\mathcal{S})}m_\chi\chi$.
We call the coefficient $m_\chi$ the \emph{multiplicity} of $\chi$. 

The \emph{Frobenius--Schur indicator} $\nu(\chi)$ of $\chi\in\Irr(\mathcal{S})$ is defined by
\[
\nu(\chi) \coloneqq  \frac{m_\chi}{n\ \chi(A_0)} \sum_{i=0}^d \frac{1}{n_i} \chi({A_i}^2).
\]
An irreducible character is said to be of the \emph{first kind}
if it is afforded by a real representation, 
of the \emph{second kind} if it is afforded by a representation
which is equivalent to its complex conjugate but is not of the first kind,
and of the \emph{third kind} if it is not of the first or second kind.
The following theorem is known.

\goodbreak
\begin{thm}[{\cite[(7,5), (7.6)]{Higman1975}}]\label{thm:FS}\ \samepage
  \begin{enumerate}
    \item\label{theo2.1_1} For $\chi\in\Irr(\mathcal{S})$,
    \[
\nu(\chi) = \begin{cases}
      1 & \text{if $\chi$ is of the first kind,} \\
      -1 & \text{if $\chi$ is of the second kind,} \\
      0 & \text{if $\chi$ is of the third kind.}\end{cases}
\]
     \item\label{theo2.1_2} $\sum_{\chi\in\Irr(\mathcal{S})}\nu(\chi)\chi(A_0)=\sharp\{A_i\in \mathcal{S}\mid \expgt A_i=A_i\}$.
  \end{enumerate}
\end{thm}

\section{Construction}
Let $\{A_0=I_{n-1}, A_1, A_2= \expgt A_1\}$ be a non-symmetric rank $3$ association scheme of order $n-1$.
Remark that $n\equiv 0\pmod{4}$ for such an association scheme to exist.
We will construct a pair $(\mathcal{D}, \mathcal{Q})$ of
association schemes with the properties described in the Introduction.

Set $a \coloneqq  n-1$ and $b\coloneqq (n-2)/2$.

The following lemma is well-known.

\begin{lemm}\label{lem3.1}\ \samepage
  \begin{enumerate}
    \item\label{lezmma3.1_1} $A_1^2=\frac{b-1}{2}A_1+\frac{b+1}{2}A_2$.
    \item\label{lezmma3.1_2} $A_2^2=\frac{b+1}{2}A_1+\frac{b-1}{2}A_2$.
    \item\label{lezmma3.1_3} $A_1A_2=A_2A_1=bA_0+\frac{b-1}{2}A_1+\frac{b-1}{2}A_2$.
  \end{enumerate}
\end{lemm}

Set $x\coloneqq (1,2,3,4)$, $y\coloneqq (1,2)(3,4)$, permutations of degree $4$,
and $G\coloneqq \langle x, y\rangle\cong D_8$.
We identify the elements of $G$ with the corresponding permutation matrices,
namely
\[
x = \begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\end{pmatrix},
\quad
y = \begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix},
\]
and so on.
The next lemma is clear by definition.

\begin{lemm}\label{lem3.2}
  As matrices, $1+x^2=xy+x^3y$.
\end{lemm}

We keep the above notations in the rest of this article.

\subsection{The association scheme \texorpdfstring{$\mathcal{D}$}{D}}
We define $n\times n$ matrices by
\[
E\coloneqq \left(\begin{array}{c|ccc}
  0&1&\dots&1\\ \hline 1&&&\\ \vdots&&O&\\ 1&&&
  \end{array}\right),\quad
  \tilde{A}_i\coloneqq \left(\begin{array}{c|ccc}
  0&0&\dots&0\\ \hline 0&&&\\ \vdots&&A_i&\\ 0&&&
  \end{array}\right)\quad(i=1,2).
\]

By Lemma~\ref{lem3.1} and direct calculations, we have the following lemma.

\begin{lemm}\label{lem3.3}\ \samepage
  \begin{enumerate}
    \item\label{lezmma3.3_1} $E^2+\tilde{A}_1\tilde{A}_2+\tilde{A}_2\tilde{A}_1=aI_n+b\tilde{A}_1+b\tilde{A}_2$.
    \item\label{lezmma3.3_2} $\tilde{A}_1^2+\tilde{A}_2^2=b\tilde{A}_1+b\tilde{A}_2$.
    \item\label{lezmma3.3_3} $E\tilde{A}_1+\tilde{A}_2E=E\tilde{A}_2+\tilde{A}_1E=bE$.
  \end{enumerate}
\end{lemm}

We consider a subgroup $H\coloneqq C_G(y)=\{1,x^2,y,x^2y\}\cong C_2\times C_2$ of $G$.
It is easy to see that $\sum_{h\in H}h=\sum_{g\in G\setminus H}g =J_4$.
We define $4n\times 4n$ matrices by
\[
\sigma_h\coloneqq I_n\otimes h \quad \text{for $h\in H$}
\]
and
\begin{align*}
  \mu_g&\coloneqq E\otimes g+\tilde{A_1}\otimes gy+\tilde{A_2}\otimes gx^2y\\
       &=(I_n\otimes g)(E\otimes 1+\tilde{A_1}\otimes y+\tilde{A_2}\otimes x^2y) \\
       &=(E\otimes 1+\tilde{A_1}\otimes x^2y+\tilde{A_2}\otimes y)(I_n\otimes g) \quad
           \text{for $g\in G\setminus H$.}
\end{align*}
We will show that $\mathcal{D}\coloneqq \{\sigma_h\mid h\in H\}\cup \{\mu_g\mid g\in G\setminus H\}$
forms an association scheme.

\begin{lemm}\label{lem3.4}
The set $\mathcal{D}$ is closed under the transposition and
  \[
  \sum_{h \in  H}\sigma_h + \sum_{g \in  G\setminus H}\mu_g  =  J_{4n}.
  \]
\end{lemm}

\begin{proof}
  We have
  $\expgt\sigma_h=\sigma_h$ for $h\in H$,
  $\expgt \mu_{xy}=\mu_{xy}$,
  $\expgt \mu_{x^3y}=\mu_{x^3y}$
  and $\expgt \mu_{x}=\mu_{x^3}$.
  Thus~$\mathcal{D}$ is closed under the transposition.

  Since $\sum_{h\in H}h=\sum_{g\in G\setminus H}g=J_4$ and $E+\tilde{A}_1+\tilde{A}_2=J_n-I_n$,
  we have $\sum_{h\in H}\sigma_h+\sum_{g\in G\setminus H}\mu_g=J_{4n}$.
\end{proof}

\begin{lemm}\label{lem3.5}\ \samepage
  \begin{enumerate}
    \item\label{lemma3.5_1} $\sigma_h\sigma_{h'}=\sigma_{hh'}$ for $h, h'\in H$.
    \item\label{lemma3.5_2} $\sigma_h\mu_g=\mu_{hg}$ and $\mu_g\sigma_h=\mu_{gh}$ for $h\in H$ and $g\in G\setminus H$.
    \item\label{lemma3.5_3} $\mu_g\mu_{g'}=a\sigma_{gg'}+b\mu_{gg'x}+b\mu_{gg'x^3}$ for $g, g'\in G\setminus H$.
  \end{enumerate}
\end{lemm}

\begin{proof}
  It is easy to show that~\eqref{lemma3.5_1} and~\eqref{lemma3.5_2} hold.

  Suppose $g, g'\in G\setminus H$.
  Remark that $gg'\in H$.
  By $g'x^2=x^2g'$, $yg'=g'x^2y$, Lemma~\ref{lem3.2} and Lemma~\ref{lem3.3}, we have
  \begin{align*}
    \mu_g\mu_{g'}&= (I_n\otimes g)(E\otimes 1+\tilde{A_1}\otimes y+\tilde{A_2}\otimes x^2y)
                (E\otimes 1+\tilde{A_1}\otimes x^2y+\tilde{A_2}\otimes y)(I_n\otimes g')\\
                &= (I_n\otimes g)(aI_n\otimes 1+b(\tilde{A}_1+\tilde{A}_2)\otimes(1+x^2)
                    +bE\otimes (y+x^2y))(I_n\otimes g')\\
                 &= a\sigma_{gg'}+b(\tilde{A}_1+\tilde{A}_2)\otimes g(1+x^2)g'
                    +bE\otimes g(y+x^2y)g'\\
                 &= a\sigma_{gg'}+b(\tilde{A}_1+\tilde{A}_2)\otimes gg'(1+x^2)
                    +bE\otimes gg'x^2(y+x^2y)\\
                 &= a\sigma_{gg'}+b(\tilde{A}_1+\tilde{A}_2)\otimes gg'(xy+x^3y)
                    +bE\otimes gg'x^2y(xy+x^3y)\\
                 &= a\sigma_{gg'}+b(\tilde{A}_1+\tilde{A}_2)\otimes gg'(xy+x^3y)
                    +bE\otimes gg'(x+x^3)\\
                 &= a\sigma_{gg'}+b\mu_{gg'x}+b\mu_{gg'x^3}.
  \end{align*}
  Now~\eqref{lemma3.5_3} holds.
\end{proof}

\begin{thm}\label{thm3.6}
  The set $\mathcal{D}=\{\sigma_h\mid h\in H\}\cup \{\mu_g\mid g\in G\setminus H\}$ forms an association scheme.
\end{thm}

\begin{proof}
  This is clear by Lemma~\ref{lem3.4} and Lemma~\ref{lem3.5}.
\end{proof}

We summarize basic properties of the association scheme $\mathcal{D}$ by Lemma~\ref{lem3.4} and Lemma~\ref{lem3.5}.

\begin{prop}\label{propD}
  For the association scheme $\mathcal{D}\Mk = \Mk \{\s_1\mk,\mk \s_{x^2}\mk ,\mk \s_y\mk ,\mk \s_{x^2y}\mk ,\mk \mu_x\mk ,\mk \mu_{x^3}\mk ,\mk \mu_{xy}\mk ,\mk \mu_{x^3y}\}$,
  the following properties hold.
  \begin{enumerate}
    \item\label{prop3.7_1} The valencies of $\s_1$, $\s_{x^2}$, $\s_y$, $\s_{x^2y}$ are $1$
    and the valencies of $\mu_x$, $\mu_{x^3}$, $\mu_{xy}$, $\mu_{x^3y}$ are $n-1$.
    \item\label{prop3.7_2} The thin radical is $O_\theta(\mathcal{D})=\{\s_1,\s_{x^2},\s_y,\s_{x^2y}\}$,
    and is normal in $\mathcal{D}$.
    \item\label{prop3.7_3} The thin residue is $O^\theta(\mathcal{D})=\{\s_1,\s_{x^2},\mu_{x},\mu_{x^3}\}$.
    \item\label{prop3.7_4} The matrices $\s_1$, $\s_{x^2}$, $\s_y$, $\s_{x^2y}$, $\mu_{xy}$, $\mu_{x^3y}$ are symmetric
    and $\mu_{x}$, $\mu_{x^3}$ are non-symmetric.
  \end{enumerate}
\end{prop}

\subsection{The association scheme \texorpdfstring{$\mathcal{Q}$}{Q}}
We define $n\times n$ matrices by
\[
\hat{A}_1\coloneqq \left(\begin{array}{c|ccc}
  0&1&\dots&1\\ \hline 0&&&\\ \vdots&&A_1&\\ 0&&&
  \end{array}\right),\quad
  \hat{A}_2\coloneqq \left(\begin{array}{c|ccc}
  0&0&\dots&0\\ \hline 1&&&\\ \vdots&&A_2&\\ 1&&&
  \end{array}\right).
\]

By Lemma~\ref{lem3.1} and direct calculations, we have the following lemma.

\begin{lemm}\label{lem3.7}\  \samepage
  \begin{enumerate}
    \item\label{lemma3.8_1} $\hat{A}_1^2+\hat{A}_2^2=b(J_n-I_n)$.
    \item\label{lemma3.8_2} $\hat{A}_1\hat{A}_2+\hat{A}_2\hat{A}_1=bJ_n+(a-b)I_n$.
  \end{enumerate}
\end{lemm}

We consider a subgroup $K\coloneqq C_G(x)=\{1,x,x^2,x^3\}\cong C_4$ of $G$.
It is easy to see that $\sum_{k\in K}k=\sum_{g\in G\setminus K}g =J_4$.
We define $4n\times 4n$ matrices by
\[
\sigma_k\coloneqq I_n\otimes k \quad \text{for $k\in K$}
\]
and
\begin{align*}
  \tau_g&\coloneqq \hat{A_1}\otimes g+\hat{A_2}\otimes gx^2\\
       &=(I_n\otimes g)(\hat{A_1}\otimes 1+\hat{A_2}\otimes x^2) \\
       &=(\hat{A_1}\otimes 1+\hat{A_2}\otimes x^2)(I_n\otimes g) \quad
           \text{for $g\in G\setminus K$.}
\end{align*}
We will show that $\mathcal{Q}\coloneqq \{\sigma_k\mid k\in K\}\cup \{\tau_g\mid g\in G\setminus K\}$ forms an association scheme.

\begin{lemm}\label{lem3.8}
  The set $\mathcal{Q}$ is closed under the transposition and
  $\sum_{k\in K}\sigma_k+\sum_{g\in G\setminus K}\tau_g=J_{4n}$.
\end{lemm}

\begin{proof}
  We have
  ${}^t \sigma_1=\sigma_1$,
  ${}^t \sigma_{x^2}=\sigma_{x^2}$,
  ${}^t \sigma_x=\sigma_{x^3}$,
  ${}^t \tau_{y}=\tau_{x^2y}$,
  and ${}^t \tau_{xy}=\tau_{x^3y}$.
  Thus $\mathcal{D}$ is closed under the transposition.

  Since $\sum_{k\in K}k=\sum_{g\in G\setminus K}g=J_4$ and $I_{n}+\hat{A}_1+\hat{A}_2=J_n$,
  we have $\sum_{k\in K}\sigma_k+\sum_{g\in G\setminus K}\tau_g=J_{4n}$.
\end{proof}

\begin{lemm}\label{lem3.9}\ \samepage
  \begin{enumerate}
    \item\label{lemma3.10_1} $\sigma_k\sigma_{k'}=\sigma_{kk'}$ for $k, k'\in K$.
    \item\label{lemma3.10_2} $\sigma_k\tau_g=\tau_{kg}$ and $\tau_g\sigma_k=\tau_{gk}$ for $k\in K$ and $g\in G\setminus K$.
    \item\label{lemma3.10_3} $\tau_g\tau_{g'}=a\sigma_{gg'x^2}+b\tau_{gg'xy}+b\tau_{gg'x^3y}$ for $g, g'\in G\setminus K$.
  \end{enumerate}
\end{lemm}

\begin{proof}
  It is easy to show that~\eqref{lemma3.10_1} and~\eqref{lemma3.10_2} hold.

  Suppose $g, g'\in G\setminus K$.
  Remark that $gg'\in K$.
  By Lemma~\ref{lem3.2} and Lemma~\ref{lem3.7}, we have
  \begin{align*}
    \tau_g\tau_{g'}&= (I_n\otimes g)(\hat{A_1}\otimes 1+\hat{A_2}\otimes x^2)^2(I_n\otimes g')\\
                     &= (I_n\otimes gg')(\hat{A_1}^2\otimes 1+(\hat{A_1}\hat{A_2}+\hat{A_2}\hat{A_1})\otimes x^2
                         +\hat{A_1}^2\otimes 1)\\
                     &= (I_n\otimes gg')(b(J_n-I_n)\otimes 1+(bJ_n+(a-b)I_n)\otimes x^2)\\
                     &= (I_n\otimes gg')(aI_n\otimes x^2+b(J_n-I_n)\otimes(1+x^2))\\
                     &= (I_n\otimes gg')(aI_n\otimes x^2+b(\hat{A_1}+\hat{A_2})\otimes(1+x^2))\\
                     &= (I_n\otimes gg')(aI_n\otimes x^2+b(\hat{A_1}+\hat{A_2})\otimes(xy+x^3y))\\
                     &= a\sigma_{gg'x^2}+b\tau_{gg'xy}+b\tau_{gg'x^3y}.    
  \end{align*}
  Now~\eqref{lemma3.10_3} holds.
\end{proof}

\begin{thm}\label{thm3.10}
  The set $\mathcal{Q}=\{\sigma_k\mid k\in K\}\cup \{\tau_g\mid g\in G\setminus K\}$ forms an association scheme.
\end{thm}

\begin{proof}
  This is clear by Lemma~\ref{lem3.8} and Lemma~\ref{lem3.9}.
\end{proof}

We summarize basic properties of the association scheme $\mathcal{Q}$ by Lemma~\ref{lem3.8} and Lemma \ref{lem3.9}.

\begin{prop}\label{propQ}
  For the association scheme $\mathcal{Q}\Mk =\Mk \{\s_1\mk ,\mk \s_{x^2}\mk ,\mk \s_x\mk ,\mk \s_{x^3}\mk ,\mk \tau_{xy}\mk ,\mk \tau_{x^3y}\mk ,\mk \tau_{y}\mk ,\mk \tau_{x^2y}\}$,
  the following properties hold.
  \begin{enumerate}
    \item\label{prop3.12_1} The valencies of $\s_1$, $\s_{x^2}$, $\s_x$, $\s_{x^3}$ are $1$
    and the valencies of $\tau_{xy}$, $\tau_{x^3y}$, $\tau_{y}$, $\tau_{x^2y}$ are $n-1$.
    \item\label{prop3.12_2} The thin radical is $O_\theta(\mathcal{Q})=\{\s_1,\s_{x^2},\s_x,\s_{x^3}\}$,
    and is normal in $\mathcal{Q}$.
    \item\label{prop3.12_3} The thin residue is $O^\theta(\mathcal{Q})=\{\s_1,\s_{x^2},\tau_{xy},\tau_{x^3y}\}$.
    \item\label{prop3.12_4} The matrices $\s_1$, $\s_{x^2}$ are symmetric
    and $\s_x$, $\s_{x^3}$, $\tau_{xy}$, $\tau_{x^3y}$, $\tau_{y}$, $\tau_{x^2y}$ are non-symmetric.
  \end{enumerate}
\end{prop}

\section{The character tables of \texorpdfstring{$\mathcal{D}$}{D} and \texorpdfstring{$\mathcal{Q}$}{Q}}
In this section, we will determine the character tables of $\mathcal{D}$ and $\mathcal{Q}$.
Consequently, we can see that the tables are the same.
Moreover, we will show that their Frobenius--Schur indicators are different.

\begin{prop}\label{prop:charD}
  The character table of $\mathcal{D}$ is 
  \[
\begin{array}{c|cccccccc|c}
      &\s_1&\s_{x^2}&\s_y&\s_{x^2y}&\mu_x&\mu_{x^3}&\mu_{xy}&\mu_{x^3y}&m_{\chi_i}\\
      \hline
      \chi_1&1&1&1&1&n-1&n-1&n-1&n-1&1\\
      \chi_2&1&1&-1&-1&n-1&n-1&-n+1&-n+1&1\\
      \chi_3&1&1&1&1&-1&-1&-1&-1&n-1\\
      \chi_4&1&1&-1&-1&-1&-1&1&1&n-1\\
      \chi_5&2&-2&0&0&0&0&0&0&n\\
  \end{array}
\]
  The Frobenius--Schur indicators are 
  $\nu(\chi_i)=1$ ($i=1,2,3,4, 5$).
\end{prop}

\begin{proof}
  \looseness-1
  By Proposition~\ref{propD}~\eqref{prop3.7_2},
  the thin radical $O_\theta(\mathcal{D})=\{\s_1,\s_{x^2},\s_y,\s_{x^2y}\}$ is a normal closed subset of $\mathcal{D}$.
  By~\cite[Theorem~3.5]{Hanaki2003}, we can determine $\chi_1$ and $\chi_3$.
  By Proposition~\ref{propD}~\eqref{prop3.7_3},
  the thin residue $O^\theta(\mathcal{D})=\{\s_1,\s_{x^2},\mu_{x},\mu_{x^3}\}$ is a strongly normal closed subset of $\mathcal{D}$
  and $\chi_2$ is determined.
  By~\cite[Theorem~3.5]{Hanaki2005}, $\chi_4\coloneqq \chi_2\chi_3$ is an irreducible character.
  Now, by $\sum_{i=1}^5 m_{\chi_i}\chi_i(\rho)=0$ for $\s_1\ne \rho\in \mathcal{D}$~\cite[Chap.~4]{Zi},
  we can determine $\chi_5$.

  Since $\chi_i$ ($i=1,2,3,4$) are rational characters of degree $1$, they are of the first kind.
  There are $6$ symmetric matrices by Proposition~\ref{propD}~\eqref{prop3.7_4}. %the proof of Lemma~\ref{lem3.4}.
  By Theorem~\ref{thm:FS}~\eqref{theo2.1_2},
  \[
6=\sum_{i=1}^5 \nu(\chi_i)\chi_i(1)=1+1+1+1+2\nu(\chi_5)
\]
  and we have $\nu(\chi_5)=1$.
\end{proof}

\begin{prop}\label{prop:charQ}
  The character table of $\mathcal{Q}$ is 
  \[
\begin{array}{c|cccccccc|c}
      &\s_1&\s_{x^2}&\s_x&\s_{x^3}&\tau_{xy}&\tau_{x^3y}&\tau_{y}&\tau_{x^2y}&m_{\varphi_i}\\
      \hline
      \varphi_1&1&1&1&1&n-1&n-1&n-1&n-1&1\\
      \varphi_2&1&1&-1&-1&n-1&n-1&-n+1&-n+1&1\\
      \varphi_3&1&1&1&1&-1&-1&-1&-1&n-1\\
      \varphi_4&1&1&-1&-1&-1&-1&1&1&n-1\\
      \varphi_5&2&-2&0&0&0&0&0&0&n\\
  \end{array}
\]
  The Frobenius--Schur indicators are 
  $\nu(\varphi_i)=1$ ($i=1,2,3,4$) and $\nu(\varphi_5)=-1$.
\end{prop}

\begin{proof}
  By Proposition~\ref{propQ}~\eqref{prop3.12_2},
  the thin radical $O_\theta(\mathcal{Q})=\{\s_1,\s_{x^2},\s_{x},\s_{x^3}\}$ is a normal closed subset of $\mathcal{Q}$.
  We can determine $\varphi_1$ and $\varphi_3$.
  By Proposition~\ref{propQ}~\eqref{prop3.12_3},
  the thin residue $O^\theta(\mathcal{Q})=\{\s_1,\s_{x^2},\tau_{xy},\tau_{x^3y}\}$
  is a strongly normal closed subset of $\mathcal{Q}$
  and $\varphi_2$ is determined.
  Now the character table can be calculated in the same  way as in Proposition~\ref{prop:charD}.

  Since $\varphi_i$ ($i=1,2,3,4$) are rational characters of degree $1$, they are of the first kind.
  There are $2$ symmetric matrices by Proposition~\ref{propQ}~\eqref{prop3.12_4}. 
    By Theorem~\ref{thm:FS}~\eqref{theo2.1_2},
  \[
2=\sum_{i=1}^5 \nu(\varphi_i)\varphi_i(1)=1+1+1+1+2\nu(\varphi_5)
\]
  and we have $\nu(\varphi_5)=-1$.
\end{proof}

\section{Irreducible representations and rational adjacency algebras}
We will determine irreducible representations and the structures of rational adjacency algebras
of $\mathcal{D}$ and $\mathcal{Q}$, respectively.

\begin{prop}\label{repD}
  The map $T:\mathcal{D}\to \M_2(\C)$ defined by
  \[
\sigma_{x^2} \mapsto \begin{pmatrix}-1&0\\0&-1\end{pmatrix},\quad
  \sigma_{y} \mapsto \begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad
  \mu_{x} \mapsto \left(\begin{array}{cc}0&-1\\n-1&0\end{array}\right)
\]
  is an irreducible representation of $\mathcal{D}$ affording $\chi_5$.
\end{prop}

\begin{proof}
  We can check all products in Lemma~\ref{lem3.5}.
\end{proof}

Similarly, we have the following proposition.

\begin{prop}\label{repQ}
  The map $T':\mathcal{Q}\to \M_2(\C)$ defined by
  \[
\sigma_{x} \mapsto \begin{pmatrix}\sqrt{-1}&0\\0&-\sqrt{-1}\end{pmatrix},\quad
  \tau_{y} \mapsto \begin{pmatrix}0&-1\\n-1&0\end{pmatrix}
\]
  is an irreducible representation of $\mathcal{Q}$ affording $\varphi_5$.
\end{prop}

Now we can determine the structures of rational adjacency algebras.
To describe the result, we define (generalized) quaternion algebras~\cite[1.6]{Pierce}.
For a field $F$ and $r,s\in F\setminus\{0\}$, a \emph{quaternion algebra} $F(r,s)$ is a four dimensional
$F$-algebra with basis $1$, $\bs{i}$, $\bs{j}$, and $\bs{k}$ with the products
\[
\bs{i}^2=r,\quad \bs{j}^2=s,\quad \bs{i}\bs{j}=-\bs{j}\bs{i}=\bs{k}.
\]
If $r$ and $s$ are negative rational numbers, then $\Q(r,s)$ is a division algebra.

\begin{prop}
  As $\Q$-algebras, we have the following isomorphisms.
  \begin{enumerate}
    \item\label{prop5.3_1} $\Q\mathcal{D}\cong \Q\oplus\Q\oplus\Q\oplus\Q\oplus \M_2(\Q)$.
    \item\label{prop5.3_2} $\Q\mathcal{Q}\cong \Q\oplus\Q\oplus\Q\oplus\Q\oplus \Q(-1,-n+1)$.
    ($\Q(-1,-n+1)$ is a division algebra.)
  \end{enumerate}
\end{prop}

\begin{proof}
  Since $\chi_i$ ($i=1,2,3,4$) have rational values on $\mathcal{D}$, we can determine
  $\Q\oplus\Q\oplus\Q\oplus\Q$.
  For the representation $T$ in Proposition~\ref{repD},
  it is easy to see that $T(\Q\mathcal{D})=\M_2(\Q)$.
  Thus~\eqref{prop5.3_1} holds.

  Since $\varphi_i$ ($i=1,2,3,4$) have rational values on $\mathcal{Q}$, we can determine
  $\Q\oplus\Q\oplus\Q\oplus\Q$.
  For the representation $T'$ in Proposition~\ref{repQ}, the set consisting of 
  \begin{align*}
  T'(\s_1)&=\begin{pmatrix}1&0\\0&1\end{pmatrix},
  &T'(\s_x)&=\begin{pmatrix}\sqrt{-1}&0\\0&-\sqrt{-1}\end{pmatrix},\\
  T'(\tau_y)&=\begin{pmatrix}0&-1\\n - 1&0\end{pmatrix}, 
  &T'(\tau_{xy})&=\begin{pmatrix}0&-\sqrt{-1}\\-(n-1)\sqrt{-1}&0\end{pmatrix}
  \end{align*}
  is a $\Q$-basis of $T'(\Q\mathcal{Q})$.
  We can see that
  \begin{gather*}
    T'(\s_x)^2=-T'(\s_1), \qquad T'(\tau_y)^2=-(n-1)T'(\s_1),\\
    T'(\s_x)T'(\tau_{y})=-T'(\tau_{y})T'(\s_x)=T'(\tau_{xy}).
  \end{gather*}
  This shows that $T'(\Q\mathcal{Q})\cong \Q(-1,-n+1)$ and~\eqref{prop5.3_2} holds.
\end{proof}

\section{Remark}
Our pair $(\mathcal{D}, \mathcal{Q})$ has similar properties to the pair $(D_8, Q_8)$,
the dihedral and quaternion groups of order $8$.
Our association schemes have order $4n$ and $n\equiv 0\pmod{4}$,
and thus $(D_8,Q_8)$ is not obtained by our construction.
If we set $n=2$ and $A_1=A_2=O$ and apply our construction,
then we can construct the pair $(D_8,Q_8)$ and all arguments are valid for it.

\longthanks{%
The authors would like to thank the anonymous referees for their helpful comments.}

%\nocite{*}
\bibliographystyle{amsplain-ac}
\bibliography{ALCO_Hanaki_537}

\end{document}
