The Irreducible Modules of the Terwilliger Algebras of Doob Schemes
Kenichiro Tanabe
Kyushu University Graduate School of Mathematics 6-10-1, Hakozaki, Higashi-ku 812 Fukuoka-shi Japan
DOI: 10.1023/A:1008647205853
Abstract
Let Y be any commutative association scheme and we fix any vertex x of Y. Terwilleger introduced a non-commutative, associative, and semi-simple C-algebra T=T(x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra of Y with respect to x.
Let W( Ì C | X | ) W( \subset C^{|X|} ) be an irreducible T(x)-module. W is said to be thin if W satisfies a certain simple condition. Y is said to be thin with respect to x if each irreducible T(x) -module is thin. Y is said to be thin if Y is thin with respect to each vertex in X.The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphs K 4 . Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible T(x)-modules of Doob schemes.
Pages: 173–195
Full Text: PDF
References
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58. Menlo Park, 1984.
2. A. Brouwer, A. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer Verlag, New York, 1989.
3. Y. Egawa, “Characterization of H (n, q) by the parameters,” Journal of Combinatorial Theory, Series A 31 (1981), 108-125.
4. P. Terwilliger, “The subconstituent algebra of an association scheme,” (Part I): Journal of Algebraic Combinatorics 1 (1992), 363-388; (Part II): Journal of Algebraic Combinatorics 2 (1993), 73-103; (Part III): Journal of Algebraic Combinatorics 2 (1993), 177-210.
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