Tridiagonal pairs of shape (1,2,1)

Let $\mathbb{F}$ denote a field and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfies the following conditions: (i) each of $A,A^{\ast }$ is diagonalizable; (ii) there exists an ordering $\{V_i{\}}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1}+V_i+V_{i+1}$for$0 \leq i \leq d$,where$V_{-1} = 0$and$V_{d+1} = 0$;(iii)thereexistsanordering$\lbrace V^i \rbrace{i=0}^{\delta}$oftheeigenspacesof$A^$suchthat$AV^_i \subseteq V^{i-1}+V^_i+V^{i+1}$for$0 \leq i \leq \delta $,where$V^_{-1} = 0$and$V^{\delta+1} = 0$;(iv)thereisnosubspace$W$of$V$suchthat$AW\subseteq W$,$A^*W\subseteq W$,$W \neq 0, W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=\delta$ and that for $0 \leq i \leq d$thedimensionsof$V_i, V{d-i}, V^_i, V^{d-i}$ coincide; we denote this common value by ${\rho }_i$. The sequence $\lbrace \rho_i\rbrace{i=0}^d$ is called the {\it shape} of the pair. In this paper we assume the shape is $(1,2,1)$ and obtain the following results. We describe six bases for $V$; one diagonalizes $A$, another diagonalizes $A^{\ast }$, and the other four underlie the split decompositions for $A,A^{\ast }$. We give the action of $A$ and $A^{\ast }$ on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape $(1,2,1)$ in terms of a sequence of scalars called the parameter array.