Formal self-duality and numerical self-duality for symmetric association schemes

Abstract

Let $X=(X,{\left\{R_i\right\}}_{i=0}^d)$ denote a symmetric association scheme. Fix an ordering ${\left\{E_i\right\}}_{i=0}^d$ of the primitive idempotents of $X$, and let P (resp. Q) denote the corresponding first eigenmatrix (resp. second eigenmatrix) of $X$. The scheme $X$ is said to be formally self-dual (with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$) whenever $P=Q$. We define $X$ to be numerically self-dual (with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$) whenever the intersection numbers and Krein parameters satisfy $p_{i,j}^h=q_{i,j}^h$ for $0\le h,i,j\le d$. It is known that with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$, $X$ is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that $X$ is P-polynomial. We show that the following are equivalent: (i) $X$ is formally self-dual with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$; (ii) $X$ is numerically self-dual with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$. Assume that the ordering ${\left\{E_i\right\}}_{i=0}^d$ is Q-polynomial. We show that the following are equivalent: (i) $X$ is formally self-dual with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$; (ii) $X$ is numerically self-dual with respect to the ordering ${\left\{E_i\right\}}_{i=0}^d$.