A rank two Leonard pair in Terwilliger algebras of Doob graphs

Let Gamma = Gamma(n, m ) denote the Doob graph formed by the Cartesian product of the n th Cartesian power of the Shrikhande graph and the mth Cartesian power of the complete graph on four vertices. Let T = T (x) denote the Terwilliger algebra of Gamma with respect to a fixed vertex x of Gamma and let W denote an arbitrary non-thin irreducible T- module in the standard module of Gamma. In (Morales and Palma, 2021 [25]), it was shown that there exists a Lie algebra embedding pi from the special orthogonal algebra so(4) into T and that W is an irreducible pi ( so 4 )-module. In this paper, we consider two Cartan subalgebras h, such that h, of so(4) h,h generate so(4). Using the embedding pi : so(4) -> T, we show that pi (h)and pi (h) and pi ( h) 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.