AssociativeScheme

Let $G$ be a group and let $H$ be a subgroup of $G$. Denote by $\Omega=G/H$ the collection of right cosets of $H$ in $G$. Then $G$ acts transitively on $\Omega$ from the right via $(Ha)\cdot g=Hag$ for all $a,g\in G$. This induces an action of $G$ on $\Omega\times\Omega$ via $(Ha,Hb)\cdot g=(Hag,Hbg)$ for all $a,b,g\in G$. Let $R$ be the orbits of $G$ on $\Omega\times\Omega$. Then we have

• (A1) $1_\Omega=\{(\alpha,\alpha)\mid \alpha\in\Omega\}\in R$;
• (A2) for all $r\in R$, $r^*=\{(\alpha,\beta)\mid(\beta,\alpha)\in r\}\in R$;
• (A3) for all $d,e,f\in R$, $\#\{\gamma\in\Omega\mid (\alpha,\gamma)\in d$, $(\gamma,\beta)\in e\}=a_{def}$ is a constant whenever $(\alpha,\beta)\in f$. This is equivalent to
• (A3') $$ \forall d,e\in R,\quad A_dA_e=\sum_{f\in G}a_{def}A_f, $$

where $A_r$ is defined to be the adjacency matrix of $r$.

Let $X$ be a finite set, $R$ a partition of $X\times X$. We say that $(X,R)$ is an association scheme if (A1), (A2) and (A3) are fulfilled. We say that $(X,R)$ is of "group type" if there is a finite group $G$ and a subgroup $H$ of $G$ such that $(X,R)=(G/H,R_{G/H})$.