Association schemes of conjugate matrices (Q2581817)

The author discusses a family of association schemes defined on the set of \(n\times n\) matrices which are roots of a monic, irreducible polynomial \(f\) over the field \(F_q\). If \(A,B\in M(n,q)\), then \(\text{orb}(A,B)\) denotes the orbit of the induced action of \(\text{GL}(n,q)\times \text{GL}(n,q)\) on \(M(n,q)\times M(n,q)\). Let \(f\) be a monic, irreducible polynomial over the field \(F_q\) of degree \(n\). If \(\text{CM}(f,q)=\{A\in M(n,q)\mid f(A)=O_n\}\) and \({\mathcal R}=\{\text{orb}(A,B)\mid (A,B)\in \text{CM}(f,q)\times \text{CM}(f,q)\}\), then \((\text{CM}(f,q),{\mathcal R})\) is an association scheme (Theorem 3). Further, the association scheme \((\text{CM}(f,q),{\mathcal R})\) depends only on \(q\) and \(n\) (Theorem 4). This association scheme is symmetric if and only if \(n=2\) (Theorem 7). Intersection numbers in the case \(n=2\) and \(q\) odd are determined (Theorem 10).