Modular adjacency algebras, standard representations, and \(p\)-ranks of cyclotomic association schemes
type: Item
instance of: scholarly article
title: Modular adjacency algebras, standard representations, and \(p\)-ranks of cyclotomic association schemes
zbMATH Open document ID: 1354.05147
author: Akihide Hanaki
published in: Journal of Algebraic Combinatorics
publication date: 2016-11-25T00:00:00Z
review text: 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 \(d=3\).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 6656213
zbMATH Keywords: association schemes
zbMATH Keywords: cyclotomic schemes
zbMATH Keywords: representations
zbMATH Keywords: \(p\)-rank
MaRDI profile type: Publication
full work available at URL: s10801-016-0681-y
OpenAlex ID: W2470941396
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DOI: 10.1007/S10801-016-0681-Y
description: scientific article; zbMATH DE number 6656213