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An $$A$$-invariant subspace for bipartite distance-regular graphs with exactly two irreducible $$T$$-modules with endpoint 2, both thin. (English) Zbl 1404.05042
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with vertex set $$X$$, diameter $$D \geq 4$$, and valency $$k \geq 3$$. Let $$\mathbb{C}^X$$ denote the vector space over $$\mathbb{C}$$ consisting of column vectors with entries in $$\mathbb{C}$$ and rows indexed by $$X$$. For $$z \in X$$, let $$\widehat{z}$$ denote the vector in $$\mathbb{C}^X$$ with a 1 in the $$z$$-coordinate, and 0 in all other coordinates. Fix a vertex $$x$$ of $$\Gamma$$ and let $$T = T(x)$$ denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible $$T$$-modules with endpoint 2, and they both are thin. Fix $$y \in X$$ such that $$\partial(x,y)=2$$, where $$\partial$$ denotes path-length distance. For $$0 \leq i$$, $$j \leq D$$ define $$w_{ij}=\sum \widehat{z}$$, where the sum is over all $$z \in X$$ such that $$\partial (x,z)=i$$ and $$\partial (y,z)=j$$. We define $$W=\operatorname{span}\{w_{ij} \mid 0 \leq i,j \leq D\}$$. In this paper we consider the space $$MW=\operatorname{span}\{mw \mid m \in M, w \in W\}$$, where $$M$$ is the Bose-Mesner algebra of $$\Gamma$$. We observe that $$MW$$ is the minimal $$A$$-invariant subspace of $$\mathbb{C}^X$$ which contains $$W$$, where $$A$$ is the adjacency matrix of $$\Gamma$$. We show that $$4D-6 \leq \operatorname{dim}(MW) \leq 4D-2$$. We display a basis for $$MW$$ for each of these five cases, and we give the action of $$A$$ on these bases.

MSC:
 05C12 Distance in graphs
Full Text:
References:
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