Leonard triples, the Racah algebra, and some distance-regular graphs of Racah type

By a Leonard triple, we mean a triple of diagonalizable operators on a finite-dimensional vector space such that for each operator, there is an ordering of an eigenbasis for the selected operator with respect to which the other two operators are irreducible tridiagonal.Let CC denote the field of complex numbers and let D   denote an integer at least 3. Let 12H″(2D+1,2) denote the halved graph of the (2D+1)(2D+1)-cube with respect to the original P  -polynomial structure R0,R1,…,RD and another Q  -polynomial structure E0,E2,E4,…,E3,E1 in terms of the original ones. Let 12H¯(4D,2) denote the folded halved graph of the 4D  -cube and let 12H¯(4D+2,2) denote the folded halved graph of the (4D+2)(4D+2)-cube. Note that they are all distance-regular graphs of Racah type.In this paper we consider the relations between the above three graphs and the Leonard triples or the Racah algebra over CC. Our results are described as follows.1. Fix a vertex of 12H″(2D+1,2) and let T1T1 denote the corresponding Terwilliger algebra with respect to this vertex. We first construct three elements U1U1, U1⁎ and U1ε of T1T1. Then we show that the triple U1,U1⁎,U1ε acts on each irreducible T1T1-module as a Leonard triple. Moreover, let K1K1 be a Racah algebra with its generators and real parameters satisfying certain conditions. We display a CC-algebra homomorphism from K1K1 to T1T1.2. Fix a vertex of 12H¯(4D,2) and let T2T2 denote the Terwilliger algebra of 12H¯(4D,2) with respect to this vertex. We construct three elements U2U2, U2⁎, U2ε of T2T2 and show that the triple U2,U2⁎,U2ε not only acts on each irreducible T2T2-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W   denote an irreducible T2T2-module with type ψ   and let KψKψ be a Racah algebra with respect to ψ  . Then there exists a KψKψ-module structure on W.3. Fix a vertex of 12H¯(4D+2,2) and let T3T3 denote the Terwilliger algebra of 12H¯(4D+2,2) with respect to this vertex. We construct three elements U3,U3⁎,U3ε of T3T3 and show that the triple U3,U3⁎,U3ε not only acts on each irreducible T3T3-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W   denote an irreducible T3T3-module with auxiliary parameter e   and let KeKe be a Racah algebra with respect to e  . Then there exists a KeKe-module structure on W.