Modular adjacency algebras, standard representations, and \(p\)-ranks of cyclotomic association schemes
We say that two association schemes are algebraically isomorphic if their intersection numbers coincide. There are well-known examples of association schemes which are not isomorphic, but are in fact algebraically isomorphic. Thus, tools for distinguishing algebraically isomorphic association schemes from one another would be welcome. The ranks of elements of the adjacency algebra over an algebraically closed field \(K\) of characteristic \(p\) have proven to be successful at this in some cases. To that end, this paper focuses on understanding the adjacency algebra of some cyclotomic association schemes over \(K\). Let \(N\) be the additive group of the finite field \(\mathbb{F}_{p^{a}}\), and let \(H\) be the cyclic group of order \(p^{a} - 1\) generated by a primitive element of \(\mathbb{F}_{p^{a}}\). If \(d | p^{a}-1\), then there is a unique subgroup \(H_{d} \subseteq H\) of index \(d\), and \(H_{d}\) acts on \(N\) in a natural way. The cyclotomic scheme, denoted \(\operatorname{Cyc}(p^{a},d)\), is the association scheme built on the data of this action. The paper focuses on the algebraic data associated to this scheme, and describes the adjacency algebra of the cyclotomic scheme. Explicit descriptions of this algebra as the quotient of a polynomial ring are given in the cases where \(p \equiv \pm 1 \pmod d\), or where \(d=2,3,4,5\), or 6. Moreover, the authors give the precise structure of the standard module, and the \(p\)-ranks of some certain elements of the adjacency algebra, in the cases \(p \equiv 1 \pmod d\) and \(d=3\).
scientific article; zbMATH DE number 6656213