论文标题
terwilliger代数加倍奇数图
The Terwilliger algebra of the doubled Odd graph
论文作者
论文摘要
令$ 2.O_ {m+1} $表示一组基数$ x $ $ x $的双重奇数图表示$ x $ $ 2M+1 $，其中$ m \ geq 1 $。修复x $中的顶点$ x_0 \。令$ \ Mathcal {a}：= \ Mathcal {a}（x_0）$表示$ 2.O_ {m+1} $ $ x_0 $ $ x_0 $的稳定器的中心化代数在本文中，我们首先通过考虑$ x_0 $在$ x \ times x $上的稳定剂的稳定剂的操作，并确定$ \ mathcal {a} $的尺寸，从而给出$ \ mathcal {a} $的基础。此外，我们给出了$ \ mathcal {a} $的三个子代理，使它们的直接总和为$ \ nathcal {a} $作为向量空间。接下来，对于$ m \ geq 3 $，我们找到了所有不可约$ t $模型的同构类别，以显示$ t $的分解形式。最后，我们证明了两个代数$ \ MATHCAL {a} $和$ t $重合。该结果告诉我们，图$ 2.O_ {M+1} $可能是两部分的第一个示例，而不是$ q $ -PolyNomial距离传播图，相应的中央式代数和Terwilliger代数相等。
Let $2.O_{m+1}$ denote the doubled Odd graph with vertex set $X$ on a set of cardinality $2m+1$, where $m\geq 1$. Fix a vertex $x_0\in X$. Let $\mathcal{A}:=\mathcal{A}(x_0)$ denote the centralizer algebra of the stabilizer of $x_0$ in the automorphism group of $2.O_{m+1}$, and $T:=T(x_0)$ the Terwilliger algebra of $2.O_{m+1}$. In this paper, we first give a basis of $\mathcal{A}$ by considering the action of the stabilizer of $x_0$ on $X\times X$ and determine the dimension of $\mathcal{A}$. Furthermore, we give three subalgebras of $\mathcal{A}$ such that their direct sum is $\mathcal{A}$ as vector space. Next, for $m\geq 3$ we find all isomorphism classes of irreducible $T$-modules to display the decomposition of $T$ in a block-diagonalization form. Finally, we show that the two algebras $\mathcal{A}$ and $T$ coincide. This result tells us that the graph $2.O_{m+1}$ may be the first example of bipartite but not $Q$-polynomial distance-transitive graph for which the corresponding centralizer algebra and Terwilliger algebra are equal.
