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

Abstract

For integers $n \ge 1$ and $m \ge 0$, 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 $\langle u, v \rangle = u^t\bar{v}$ 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 $\pi$ from the special orthogonal algebra $\mathfrak {so}_4$ to T and that each irreducible T-module is an irreducible $\pi (\mathfrak {so}_4)$-module. Let W denote an irreducible T-module. In this paper, we consider two Cartan subalgebras $\mathfrak {h}$ and $\mathfrak {h}^*$ of $\mathfrak {so}_4$ and obtain weight space decompositions

$$\begin{aligned} W = \sum _{r=0}^d \sum _{s=0}^p W_{rs} = \sum _{k=0}^d \sum _{l=0}^p W_{kl}^* \end{aligned}$$

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 $\langle W_{rs}^*, W_{kl} \rangle$ is connected to Krawtchouk polynomials and we prove the relations

$$\begin{aligned} \pi (\mathfrak {h})W_{rs}^*&\subseteq W_{r-1,s}^*+W_{r,s-1}^*+W_{rs}^*+W_{r+1,s}^*+W_{r,s+1}^*\\ \pi (\mathfrak {h}^*)W_{kl}&\subseteq W_{k-1,l}+W_{k,l-1}+W_{kl}+W_{k+1,l}+W_{k,l+1} \end{aligned}$$

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 $\boxtimes$.