形状为（1，2，1）的三对角对

摘要：

设$\mathbb F$表示域，$V$表示$\mathbb F$上具有有限正维数的向量空间。我们考虑一对线性变换$A:V\to V$和$A^*:V\to V$，它们满足下列条件：（i）每个$A,A^*$都是可对角化的;（ii）存在$A$的特征空间的序$\lbrace V_i \rbrace_{i=0}^d$使得$A^* V_i \subseteq V_{i-1}+V_i+V_{i+1}$对于$0 \leq i \leq d$，其中$V_{-1} = 0$和$V_{d+1} = 0$ ;（iii）存在$A^*$的特征空间的序$\lbrace V^*_i \rbrace_{i=0}^δ$使得$AV^*_i \subseteq V^*_{i-1}+V^*_i+V^*_{i+1}$对于$0 \leq i \leq δ$，其中$V^*_{-1} = 0$和$V^*_{δ+1} = 0$ ;（iv）不存在$V$的子空间$W$使得$AW\subseteq W$，$A^*W\subseteq W$，$W \neq 0, W \neq V$ .我们称这样的对为$V$上的{ \it三对角对}。已知$d = δ$和$0 \leq i \leq d$的$V_i, V_{d-i}, V^*_i, V^*_{d-i}$的维数一致;我们用$ρ_i$表示这个公共值。序列$\lbrace ρ_i\rbrace_{i=0}^d$称为该对的{ \it形状}。在本文中，我们假设形状是$(1,2,1)$，并得到以下结果。我们描述了六个基地$V$ ;一个diagonizes $A$，另一个diagonizes $A^*$，和其他四个基础的分裂分解$A,A^*$。并在此基础上给出了$A$和$A^*$的作用。对于六个基中的每一个有序对，我们给出了转移矩阵。在最后，我们分类的三对角对形状$(1,2,1)$的标量序列称为参数数组。

英文摘要：

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 $A^*:V\to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ 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) there exists an ordering $\lbrace V^*_i \rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $AV^*_i \subseteq V^*_{i-1}+V^*_i+V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1} = 0$ and $V^*_{δ+1} = 0$; (iv) there is no subspace $W$ of $V$ such that $AW\subseteq W$, $A^*W\subseteq W$, $W \neq 0, W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d = δ$ and that for $0 \leq i \leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide; we denote this common value by $ρ_i$. The sequence $\lbrace ρ_i\rbrace_{i=0}^d$ 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 $A^*$, and the other four underlie the split decompositions for $A,A^*$. We give the action of $A$ and $A^*$ 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.

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