论文标题
temperley-lieb代数的注射词的组合
Combinatorics of injective words for Temperley-Lieb algebras
论文作者
论文摘要
本文研究了“平面注射词的复合物”的组合特性，这是我们在同源稳定性的工作中出现的temendley-lieb代数模块的链复合体。尽管是线性而不是离散对象，但我们的链复合物仍然表现出有趣的组合特性。我们表明，该复合物的欧拉（Euler）特征是第n个罚款数字。我们获得了由其顶级同源模块给出的表示形式的交替总和公式，并且在接地环的进一步限制下，我们根据某些标准的Young Tableaux分解了该模块。这三个结果的灵感来自雷手和Webb的启发，以启发为注射词的复合体 - 可以将其视为对n个字母幻觉数量的“平面”或“ dyck路径”类似物的n -th -Fine数字的解释。这种解释在文献中具有前体，但在这里自然而然地来自同源稳定性的考虑。我们的最终结果表明，我们的复合物的边界图与雅各布斯特数字之间存在令人惊讶的联系。
This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableaux. This trio of results - inspired by results of Reiner and Webb for the complex of injective words - can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number of derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.
