Terwilliger algebras of cyclotomic schemes and Jacobi sums
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 \(T=S\).
scientific article; zbMATH DE number 1339813