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:

 5e+30 Association schemes, strongly regular graphs

Zbl 0785.05089
Full Text:

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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