坦珀利-利布代数的中心
The Center of the Temperley-Lieb Algebra
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中文总结 AI 辅助
该研究计算特征零域上坦珀利-利布代数$\operatorname{TL_n}(\delta)$中心维度，利用胞腔滤过、表示理论等，通过扩张压缩映射及变形论证得出维度公式，还证明中心元素在特定自同构下不变，给出平凡根基情形同余准则及Gram矩阵计算。
中文摘要 AI 辅助
我们计算了特征为零的域上，参数δ取非零值时，坦珀利-利布代数$\operatorname{TL_n}(\delta)$的中心维度。证明利用了按杯数的胞腔滤过，以及坦珀利-利布代数表示理论的已知事实，特别是其标准模及其根基的结构。通过扩张和压缩映射比较中心在n和n - 2级的诱导分次部分，得到每个此类部分的上界为1。通过变形论证得到匹配的下界，从而得出$\dim Z(\operatorname{TL}_n(\delta))=1+\Bigl\lfloor \frac{n}{2}\Bigr\rfloor$。我们还证明了每个中心元素在典范反自同构和自然图反射自同构下是不变的。最后，我们给出了平凡根基情形的同余准则，并记录了首项的Gram矩阵计算。
英文摘要
We compute the dimension of the center of the Temperley--Lieb algebra $\operatorname{TL_n}(δ)$ over a field of characteristic zero for every nonzero value of the parameter $δ$. The proof uses the cellular filtration by cup number, together with known facts about the representation theory of the Temperley--Lieb algebra, especially the structure of its standard modules and their radicals. Dilation and compression maps compare the induced graded pieces of the center at levels $n$ and $n-2$, giving an upper bound of one for each such piece. A deformation argument gives the matching lower bound, and hence $ \dim Z(\operatorname{TL}_n(δ))=1+\Bigl\lfloor \frac{n}{2}\Bigr\rfloor$. We also prove that every central element is fixed by the canonical anti-automorphism and by the natural diagram-reflection automorphism. Finally, we give a congruence criterion for the trivial-radical case and record a Gram-matrix computation for leading terms.
