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The subconstituent algebra of an association scheme. I. (English) Zbl 0785.05089
From the author’s abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the $$P$$- and $$Q$$-polynomial schemes. Let $$Y$$ denote any commutative association scheme, and fix any vertex $$x$$ of $$Y$$. We introduce a non-commutative, associative, semi-simple $$\mathbb{C}$$-algebra $$T=T(x)$$ whose structure reflects the combinatorial structure of $$Y$$. We call $$T$$ the subconstituent algebra of $$Y$$ with respect to $$x$$. Roughly speaking, $$T$$ is a combinatorial analog of the centralizer algebra of the stabilizer of $$x$$ in the automorphism group of $$Y$$.
In general, the struture of $$T$$ is not determined by the intersection numbers of $$Y$$, but these parameters do give some information. Indeed, we find a relation among the generators of $$T$$ for each vanishing intersection number or Krein parameter.
We identify a class of irreducible $$T$$-modules whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say $$Y$$ is thin if every irreducible $$T(y)$$-module is thin for every vertex $$y$$ of $$Y$$. We compute the possible thin, irreducible $$Y$$-modules when $$Y$$ is $$P$$- and $$Q$$-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If $$Y$$ is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”.
We give a combinatorial characterization of the thin $$P$$- and $$Q$$- polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible $$T$$-modules actually occur.
We close with some conjectures and open problems.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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