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The irreducible modules of the Terwilliger algebras of Doob schemes. (English) Zbl 0868.05056

Let \(Y = (X,(R_i)_{0 \leq i \leq D})\) be a commutative association scheme. For each \(x \in X\) and \(0 \leq i \leq D\) one forms the diagonal matrix \(E_i^*(x)\) with rows and columns indexed by \(X\) whose entry in position \(yy\) is \(1\) if \((x,y) \in R_i\) and \(0\) otherwise. Then one considers the algebra \(T(x)\) that is the matrix algebra generated by \(E_i^*(x)\) and \(A_i\), \(0 \leq i \leq D\), where \(A_i\) is the \(i\)th associate matrix of \(Y\). The algebra \(T(x)\) is called the Terwilliger algebra of \(Y\) with respect to \(x\). The main result of the paper is the classification of all irreducible \(T(x)\)-modules for Doob schemes \(Y\). The actual result is too technical to be stated here. A Doob scheme is the scheme associated to a direct product of some complete graphs and some Shrikhande graphs. Doob schemes have been considered by P. Terwilliger [J. Algebr. Comb. 1, No. 4, 363-388 (1992; Zbl 0785.05089)] who proved that Doob schemes are not thin if the diameter is greater than \(2\).
Reviewer: V.Welker (Essen)

MSC:

05E30 Association schemes, strongly regular graphs

Citations:

Zbl 0785.05089
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References:

[1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings Lecture Note 58. Menlo Park, 1984.
[2] A. Brouwer, A. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer Verlag, New York, 1989.
[3] Egawa, Y., Characterization of H(n,q) by the parameters, Journal of Combinatorial Theory, Series A, 31, 108-125, (1981) · Zbl 0472.05056
[4] Terwilliger, P., The subconstituent algebra of an association scheme, Journal of Algebraic Combinatorics, 1, 363-388, (1992) · Zbl 0785.05089
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