\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}