The Terwilliger algebra of the Odd graph revisited from the viewpoint of group action

2022/07/04 by Lihang, Hou, Suogang, Gao, Na, Kang +1
#05C50 #05E15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.01265

Abstract

Let O_m+1 denote the Odd graph on a set of cardinality 2m+1, where m is a positive integer. Denote by X its vertex set and by T:=T(x₀) its Terwilliger algebra with respect to any fixed vertex x₀∈ X. In this paper, we first prove that T coincides with the centralizer algebra of the stabilizer of x₀ in the automorphism group of O_m+1 by considering the action of this automorphism group on X× X× X. Then we give the decomposition of T for m≥ 3 by using all the homogeneous components of V:=ℂ^X, each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. Finally, we display an orthogonal basis for every homogeneous component of V.

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