A rank two Leonard pair in Terwilliger algebras of Doob graphs

Abstract

Let $\Gamma =\Gamma (n,m)$ denote the Doob graph formed by the Cartesian product of the nth Cartesian power of the Shrikhande graph and the mth Cartesian power of the complete graph on four vertices. Let $T=T\left(x\right)$ denote the Terwilliger algebra of Γ with respect to a fixed vertex x of Γ and let W denote an arbitrary non-thin irreducible T-module in the standard module of Γ. In (Morales and Palma, 2021 [25]), it was shown that there exists a Lie algebra embedding π from the special orthogonal algebra ${\mathrm{so}}_4$ into T and that W is an irreducible $\pi \left({\mathrm{so}}_4\right)$-module. In this paper, we consider two Cartan subalgebras $h,\tilde{h}$ of ${\mathrm{so}}_4$ such that $h,\tilde{h}$ generate ${\mathrm{so}}_4$. Using the embedding $\pi :{\mathrm{so}}_4\to T$, we show that $\pi \left(h\right)$ and $\pi \left(\tilde{h}\right)$ act on W as a rank two Leonard pair. We also obtain several direct sum decompositions of W akin to how split decompositions are obtained from Leonard pairs of rank one.