Terwilliger algebras of cyclotomic schemes and Jacobi sums
type: Item
instance of: scholarly article
title: Terwilliger algebras of cyclotomic schemes and Jacobi sums
zbMATH Open document ID: 0944.05101
author: Haruo Ishibashi
author: Tatsuro Ito
author: Mieko Yamada
published in: European Journal of Combinatorics
publication date: 2000-09-15T00:00:00Z
review text: Terwilliger algebras introduced in [\textit{P. Terwilliger}, J. Algebr. Comb. 1, No. 4, 363-388 (1992; Zbl 0785.05089), ibid. 2, No. 1, 73-103 (1993; Zbl 0785.05090), ibid. 2, No. 2, 177-210 (1993; 785.05091)] under the name subconstituent algebras and their representations are becoming important in the study of association schemes and related combinatorial objects. Let us consider an association scheme \(\mathcal X\) whose automorphism group acts transitively on each associate class \(R_i\). Then the Terwilliger algebra \(T\) is a combinatorial object in the centralizer algebra \(S\) of the stabilizer of the base vector \(x\) in \(\text{ Aut}({\mathcal X})\) and it is known that \(T\) not always coincides with \(S\). In the paper under review the authors consider cyclotomic schemes to discuss whether \(T\) coincides with \(S\). It turns out that it is sufficient to see whether an irreducible \(S\)-module is still irreducible as a \(T\)-module. The authors reduce the problem to a number-theoretical problem which itself is interesting: If \(K\) is a finite field, \(H\) is a multiplicative subgroup of index \(e\) in \(K^{\times}\) and \(\eta\) is a generator of the character group of \(K^{\times}/H\), then find all multiplicative characters \(\chi\) of \(K^{\times}\) such that \(\chi|_H\not=1|_H\) and the Gauss sum \(g(\eta^i\chi)\) equals \(\varepsilon^ig(\chi)\) for all \(i\), where \(\varepsilon\) is an \(e\)th root of unity independent of \(i\). The authors pay special attention to the case \(e=2\) and give the complete answer when \(T=S\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 1339813
zbMATH Keywords: subconstituent algebra
zbMATH Keywords: Terwilliger algebra
zbMATH Keywords: association scheme
zbMATH Keywords: Gauss sum
MaRDI profile type: Publication
full work available at URL: eujc.1998.0276
OpenAlex ID: W2056381349
DOI: 10.1006/EUJC.1998.0276
description: scientific article; zbMATH DE number 1339813