The Terwilliger algebra of the doubled Odd graph

Abstract

Let $2.O_{m+1}$ denote the doubled Odd graph with vertex set $X$ on a set of cardinality $2m+1$, where $m\geq 1$. Fix a vertex $x_0\in X$. Let $\mathcal{A}:=\mathcal{A}(x_0)$ denote the centralizer algebra of the stabilizer of $x_0$ in the automorphism group of $2.O_{m+1}$, and $T:=T(x_0)$ the Terwilliger algebra of $2.O_{m+1}$. In this paper, we first give a basis of $\mathcal{A}$ by considering the action of the stabilizer of $x_0$ on $X\times X$ and determine the dimension of $\mathcal{A}$. Furthermore, we give three subalgebras of $\mathcal{A}$ such that their direct sum is $\mathcal{A}$ as vector space. Next, for $m\geq 3$ we find all isomorphism classes of irreducible $T$-modules to display the decomposition of $T$ in a block-diagonalization form. Finally, we show that the two algebras $\mathcal{A}$ and $T$ coincide. This result tells us that the graph $2.O_{m+1}$ may be the first example of bipartite but not $Q$-polynomial distance-transitive graph for which the corresponding centralizer algebra and Terwilliger algebra are equal.