论文标题
三角对，交替元素和距离规则图
Tridiagonal pairs, alternating elements, and distance-regular graphs
论文作者
论文摘要
$ u_q（\ hat {\ Mathfrak {sl}} _ 2）$ u_q的正零件$ u^+_ q $ $ w_0 $，$ w_1 $和两个称为$ q $ -serre关系的关系。代数$ u^+_ q $包含一些元素，据说是交替的。有四种交替元素，称为$ \ lbrace w _ { - k} \ rbrace_ {k \ in \ mathbb n} $，$ \ lbrace w_ {k+1} \ rbrace_ { \ mathbb n} $，$ \ lbrace {\ tilde g} _ {k+1} \ rbrace_ {k \ in \ mathbb n} $。每种相互通勤的交替元素。 Tridiagonal对是一对有序的可对角线线性地图$ A，在非零的有限维矢量空间$ V $上，每个人都以（块）三角形方式对另一个人的特征型进行。令$ a $，$ a^*$表示$ v $上的三角对。与这对相关的是$ V $的六个众所周知的直接总和分解；这些是$ a $和$ a^*$的特征空间分解，以及四个通常称为拆分的分解。 在我们的主要结果中，我们假设$ a $，$ a^*$具有$ q $ -serre类型。在此假设下，$ a $，$ a^*$满足$ q $ -serre关系，$ v $变成了不可约$ u^+_ q $ -module，$ w_0 = a $ and $ w_1 = a^*$。我们描述了$ u^+_ q $的交替元素在上述六个分解$ v $上的作用。我们表明，对于每个分解，每个交替的元素都以（块）对角线，（块）上比尼角，（块）下Bidiagonal或（块）Tridiagonal时尚。我们详细研究了两种特殊情况。在第一种情况下，$ a $和$ a^*$的特征空间都有一个尺寸。在第二种情况下，$ a $和$ a^*$是通过调整具有经典参数且正式自我划分的距离定型图的邻接矩阵和双邻接矩阵来获得的。
The positive part $U^+_q$ of $U_q(\hat{\mathfrak{sl}}_2)$ has a presentation with two generators $W_0$, $W_1$ and two relations called the $q$-Serre relations. The algebra $U^+_q$ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted $\lbrace W_{-k}\rbrace_{k\in \mathbb N}$, $\lbrace W_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace G_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace {\tilde G}_{k+1}\rbrace_{k \in \mathbb N}$. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps $A, A^*$ on a nonzero, finite-dimensional vector space $V$, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let $A$, $A^*$ denote a tridiagonal pair on $V$. Associated with this pair are six well-known direct sum decompositions of $V$; these are the eigenspace decompositions of $A$ and $A^*$, along with four decompositions of $V$ that are often called split. In our main results, we assume that $A$, $A^*$ has $q$-Serre type. Under this assumption $A$, $A^*$ satisfy the $q$-Serre relations, and $V$ becomes an irreducible $U^+_q$-module on which $W_0=A$ and $W_1=A^*$. We describe how the alternating elements of $U^+_q$ act on the above six decompositions of $V$. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of $A$ and $A^*$ all have dimension one. In the second case $A$ and $A^*$ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.
