相对于LR三元组的三个分解为三对角的线性变换
Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple
摘要：
固定整数$d \geq 0$、字段$\mathbb{F}$和$\mathbb{F}$上具有维度$d+1$的向量空间$V$。通过$V$的分解，我们表示$V$的$1$维子空间的序列$\{V_i\}_{i=0}^d$，其和是$V$。对于从$V$到$V$的线性变换$A$，我们说$A$降低$\{V_i\}_{i=0}^d$，只要$A V_i = V_{i-1}$为$0 \leq i \leq d$，其中$V_{-1}=0$。当$A V_i = V_{i+1}$为$0 \leq i \leq d$时，我们说$A$引发$\{V_i\}_{i=0}^d$，其中$V_{d+1}=0$。从$V$到$V$的有序线性变换$A,B$对称为LR，只要存在$V$的分解$\{V_i\}_{i=0}^d$，该分解由$A$降低并由$B$升高。在这种情况下，分解$\{V_i\}_{i=0}^d$由$A,B$唯一确定;我们称之为$V$的$(A,B)$ -分解。考虑从$V$到$V$的线性变换$A$、$B$、$C$的$3$元组，使得$A$、$B$、$C$中的任何两个在$V$上形成LR对。这样的$3$ -元组在$V$上称为LR三元组。令$α$、$β$、$γ$为$\mathbb{F}$中的非零标量。三元组$αA, βB, γC$是$V$上的LR三元组，据说与$A,B,C$相关。令$\{V_i\}_{i=0}^d$是$V$的分解，令$X$是从$V$到$V$的线性变换。当$X V_i \subseteq V_{i-1} + V_i + V_{i+1}$对应于$0 \leq i \leq d$时，我们说$X$相对于$\{V_i\}_{i=0}^d$是三对角的。令$\cal X$是$\mathbb{F}$上的向量空间，由从$V$到$V$的线性变换组成，这些线性变换相对于$V$的$(A,B)$和$(B,C)$以及$(C,A)$分解是三对角的。有一类特殊的LR三元组，称为$q$-Weyl型.在本文中，我们找到了一个基础的$\cal X$的每一个LR三元组，是不相关的LR三元组的$q$-Weyl型。
英文摘要：
Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional subspaces of $V$ whose sum is $V$. For a linear transformation $A$ from $V$ to $V$, we say $A$ lowers $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i-1}$ for $0 \leq i \leq d$, where $V_{-1}=0$. We say $A$ raises $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{d+1}=0$. An ordered pair of linear transformations $A,B$ from $V$ to $V$ is called LR whenever there exists a decomposition $\{V_i\}_{i=0}^d$ of $V$ that is lowered by $A$ and raised by $B$. In this case the decomposition $\{V_i\}_{i=0}^d$ is uniquely determined by $A,B$; we call it the $(A,B)$-decomposition of $V$. Consider a $3$-tuple of linear transformations $A$, $B$, $C$ from $V$ to $V$ such that any two of $A$, $B$, $C$ form an LR pair on $V$. Such a $3$-tuple is called an LR triple on $V$. Let $α$, $β$, $γ$ be nonzero scalars in $\mathbb{F}$. The triple $αA, βB, γC$ is an LR triple on $V$, said to be associated to $A,B,C$. Let $\{V_i\}_{i=0}^d$ be a decomposition of $V$ and let $X$ be a linear transformation from $V$ to $V$. We say $X$ is tridiagonal with respect to $\{V_i\}_{i=0}^d$ whenever $X V_i \subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$. Let $\cal X$ be the vector space over $\mathbb{F}$ consisting of the linear transformations from $V$ to $V$ that are tridiagonal with respect to the $(A,B)$ and $(B,C)$ and $(C,A)$ decompositions of $V$. There is a special class of LR triples, called $q$-Weyl type. In the present paper, we find a basis of $\cal X$ for each LR triple that is not associated to an LR triple of $q$-Weyl type.
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