扭曲Grassmann图的Terwilliger代数：薄情形
The Terwilliger algebra of the twisted Grassmann graph: the thin case
摘要：
有限连通简单图$Γ$关于顶点$x$的Terwilliger代数$T(x)$是由$Γ$的邻接矩阵$A$和对角矩阵$E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$生成的复半单矩阵代数，其中$v_i$表示距离$x$ $i$的顶点集的特征向量。Van Dam和Koolen在2005年发现的扭曲Grassmann图$\tilde{J}_q(2D+1,D)$在其顶点集上有两个自同构群的轨道，已知其中一个轨道具有$T(x)$是瘦的性质，即，只要从中选择$x$，每个不可约$T(x)$ -模块$W$对所有$i$都满足$\dim E_i^*(x)W\leqslant 1$。在本文中，我们确定了所有的不可约$T(x)$ -模的$\tilde{J}_q(2D+1,D)$的“薄”的情况下。
英文摘要：
The Terwilliger algebra $T(x)$ of a finite connected simple graph $Γ$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $Γ$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
