设$K$表示域，$V$表示$K$上具有有限正维数的向量空间。令$End(V)$表示由从$V$到$V$的所有$K$ -线性变换组成的$K$ -代数.我们考虑满足以下（i）-（iv）的一对$A,A^* \in End(V)$：（i）每个$A,A^*$是可对角化的。(ii)存在$A$的特征空间的排序$\{V_i\}_{i=0}^d$，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$A^*$的本征空间的序$\{V^*_i\}_{i=0}^δ$，使得对于$0 \leq i \leq δ$有$A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$，其中$V^*_{-1}=0$和$V^*_{δ+1}=0$。(iv)不存在$V$的子空间$W$使得$AW \subseteq W$，$A^* W \subseteq W$，$W \neq 0$，$W \neq V$ .我们在$V$上将这样的对称为{ \em三对角对}。令$E^*_0$表示$End(V)$的元素，使得$(E^*_0-I)V^*_0=0$和$E^*_0V^*_i=0$用于$1 \leq i \leq d$。令$D$（resp. $D^*$）表示由$A$（resp. $A^*$）。本文证明了$E^*_0 D D^*DE^*_0$的跨度等于$E^*_0D E^*_0DE^*_0$的跨度，并且$E^*_0 D E^*_0$的元素相互可换。我们将这些结果与伊藤龙郎和第二作者的一些成果相关联，这些成果有望在三对角对的分类中发挥作用。
英文摘要：
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \in End(V)$ that satisfy (i)--(iv) below:
(i) Each of $A,A^*$ 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) There exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ 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 {\em tridiagonal pair} on $V$. Let $E^*_0$ denote the element of $End(V)$ such that $(E^*_0-I)V^*_0=0$ and $E^*_0V^*_i=0$ for $1 \leq i \leq d$. Let $D$ (resp. $D^*$) denote the $K$-subalgebra of $End(V)$ generated by $A$ (resp. $A^*$). In this paper we prove that the span of $E^*_0 D D^*DE^*_0$ equals the span of $E^*_0D E^*_0DE^*_0$, and that the elements of $E^*_0 D E^*_0$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.
Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy 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 $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$…
A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair $(A, A^*)$ of $q$-Serre type; for such a pair the maps $A$ and $A^*$ satisfy the $q$-Serre relations. There is a linear map $K$ in the literature that is used to describe how $A$ and $A^*$ are related. We investigate a pair of linear maps $B=A$ and $B^* = tA^* + (1-t)K$, where $t$ is any scalar. Our goal is to find a necessary and sufficient condition on $t$ for the pair $(B…
Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy 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 $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1…
Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ 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) there exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy 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 $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$…
Transition maps between the 24 bases for a Leonard pair
设$V$表示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$A^* : V \to V$满足以下（i）和（ii）：（i）存在一个基$V$，关于该基，表示$A$的矩阵是不可约三对角的，表示$A^*$的矩阵是对角的. (ii)存在$V$的基，关于该基，表示$A^*$的矩阵是不可约三对角的，并且表示$A$的矩阵是对角的。我们称这样的对为$V$上的Leonard对。在以前的论文中，我们描述了24个特殊基地$V$。这些基的一个特征是，对于它们中的每一个，表示$A$和$A^*$的矩阵是（i）对角和不可约三对角的或（ii）不可约三对角和对角的或（iii）下双对角和上双对角的或（iv）上双对角和下双对角的。对于24个有序基对中的每一个，都存在从$V$到$V$的唯一线性变换，它将第一个基发送到第二个基;我们称之为转换映射。在本文中，我们发现每个转换映射明确作为一个多项式在$A,A^*$。
Let $\mathbb{K}$ denote an algebraically closed field and let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$. We assume that $A,A^*$ belongs to a family of tridiagonal pairs said to have $q$-Racah type. Let $\{U_i\}_{i=0}^d$ and $\{U_i^\Downarrow\}_{i=0}^{d}$ denote the first and second split decompositions of $V$. In an earlier paper we introduced a double lowering operator $ψ:V\to V$ with the notable feature that both $ψU_i\subseteq U_{i-1}$ and $ψU_i^\Downarrow\subseteq U_{i-1}^\Downarrow$ for $0\leq i\leq d$, where $…
Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*:V\to V$ such that (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 $A V^*_i \subseteq V^*_{i-1} + V^*_i+ V^*_{i+1} $ for…
Let \K denote an algebraically closed field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V,A*:V \to V that satisfy the following conditions:(i)Each of A,A* 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)there exists an ordering {V*_i}_{i=0}^δof the eigenspaces of A* such that A V*_i\subseteq V*_{i-1}+V*_i+V*_{i+1} for 0\leq i\leqδ, where V*_{-1}=0 and V*_{δ+1}=0;(iv)there d…
In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0 \leq i \leq d$ let $θ_i$ (resp. $θ^*_i$) denote a standard ordering of the eigenvalues of A (resp. A^*). Fix a nonzero scalar q which is not a root of unity. T. Ito and P. Terwilliger have shown that when $θ_i = q^{2i-d}$ and $θ^*_i = q^{d-2i}$ there exists an irreducible $\boxtimes_q$-module structure on V such that the $\boxtimes_q$ generato…
令$V$表示具有有限正维度的向量空间。我们考虑一对线性变换$A : V \to V$和$A^* : V \to V$满足以下（i）和（ii）：（i）存在一个基$V$，关于该基，表示$A$的矩阵是不可约三对角的，表示$A^*$的矩阵是对角的. (ii)存在$V$的基，关于该基，表示$A^*$的矩阵是不可约三对角的，并且表示$A$的矩阵是对角的。我们在$V$上称这样的对为{ \em Leonard对}。令$v_0,v_1,...,v_d$（分别$w_0,w_1,...,w_d$）表示（i）中所述的$V$的基础（分别（ii））。我们发现，存在一个独特的线性变换$S: V \to V$发送$v_0$的$v_d$的标量倍数，固定$w_0$，并发送$w_i$的$w_i$的标量倍数为$1 \leq i \leq d$。我们称$S$为{ \it交换元素}。我们描述$S$从许多角度来看。
Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let $\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n \times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices in $\text{Mat}_{n}(\mathbb{F})$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{n}(\mathbb{F})$. In the present paper, we find all Leonard pairs $A,A^*$ in $\text{Mat}_{n}(\mathbb{F})$ such that each of $A$ and $A^*$ is irreducible tridiagonal wi…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In this paper, we characterize the Leonard pairs using the noti…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Arlene Pascasio recently obtained a characterization of the $Q$…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^{*}: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{*}$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{*}$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In the literature, there are some paramet…
Let \K denote a field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V and A*:V \to V that satisfy the following four conditions: (i) Each of A,A* 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) there exists an ordering {V*_i}_{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\leqδ, where V*_{-1}=0 and V*_{δ+1}=0; (iv) there does n…
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 decompositio…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Very roughly speaking, a Leonard pair is a line…
Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such that for each of these transformations there exists a basis of $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the L…
Let $\F$ denote a field and let $V$ denote a vector space over $\F$ with finite positive dimension. Consider a pair $A,A^*$ of diagonalizable $\F$-linear maps on $V$, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of $V$ that swaps $A$ and $A^*$. Such an automorphism is unique, and called the duality $A \leftrightarrow A^*$. In the present paper we give a comprehensive description of this duality. In particular, we …
In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r…
Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $(\{þ_i\}_{i=0}^d, \{þ^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d, \{ϕ_i\}_{i=1}^d)$, called its parameter array. The scalars $\{þ…
In this paper, we shall recall and arrange the relationship between hyperbolic elements and parabolic subalgebras at first. And then, we shall classify all the compatible parabolic subalgebras containing a $τ$-stable Borel subalgebra for complex symmetric pairs $(\mathfrak{g},\mathfrak{g}^τ)$, where $τ$ is a involutive automorphism of $\mathfrak{g}$.
The author classifies Klein four symmetric pairs of holomorphic type for non-compact Lie group $\mathrm{E}_{6(-14)}$, which gives a class of pairs $(G,G')$ of real reductive Lie group $G$ and its reductive subgroup $G'$ such that there exist irreducible unitary representations $π$ of $G$, which are admissible upon restriction to $G'$.
