The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph∗
نویسنده
چکیده
Let be a distance-regular graph of diameter D. Let X denote the vertex set of and let Y be a nonempty subset of X . We define an algebra T = T (Y ). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then T is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible T -modules. We define endpoints and thin condition on irreducible T -modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y , it is thin if and only if Y is a completely regular code of . By considering a suitable subset Y , every irreducible T (x)-module of endpoint i can be regarded as an irreducible T (Y )-module of endpoint 0.
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