On standard bases of irreducible modules of Terwilliger algebras of Doob schemes.

For integers n ≥ 1 and m ≥ 0 , let D = D (n , m) denote the Doob scheme which is the direct product of n copies of Shrikhande graph and m copies of complete graph on four vertices. Let V denote the standard module of D with an inner product ⟨ u , v ⟩ = u t v ¯ where t and - denote transpose and complex conjugate, respectively. Fix a vertex x of D, and let T = T (x) denote the Terwilliger algebra of D with respect to x. We view T as a Lie algebra with respect to the usual commutator. Using Tanabe's results (JAC 6: 173–195, 1997) on characterization of irreducible T-modules, it was shown in (JAC 54: 979–998, 2021) that there exists a homomorphism π from the special orthogonal algebra so 4 to T and that each irreducible T-module is an irreducible π (so 4) -module. Let W denote an irreducible T-module. In this paper, we consider two Cartan subalgebras h and h ∗ of so 4 and obtain weight space decompositions W = ∑ r = 0 d ∑ s = 0 p W rs = ∑ k = 0 d ∑ l = 0 p W kl ∗ where d, p are uniquely determined by the parameters of W. We show that each W rs (resp. W kl ∗ ) is one-dimensional. Moreover, we describe how ⟨ W rs ∗ , W kl ⟩ is connected to Krawtchouk polynomials and we prove the relations π (h) W rs ∗ ⊆ W r - 1 , s ∗ + W r , s - 1 ∗ + W rs ∗ + W r + 1 , s ∗ + W r , s + 1 ∗ π (h ∗) W kl ⊆ W k - 1 , l + W k , l - 1 + W kl + W k + 1 , l + W k , l + 1 where W ij : = 0 and W ij ∗ : = 0 if i < 0 , j < 0 , i > d , or j > p . Additionally, we establish some connections between T and the tetrahedron algebra ⊠ . [ABSTRACT FROM AUTHOR]