A Hodge decomposition for the complex of injective words

非错排置换的单射字复形是可缩的

The complex of injective words of permutations which are not derangements is contractible

本文证明了对称群中非错排生成的单射字复形是可缩的，为全置换生成的复形同伦等价于特定球面楔和的事实提供了概念性解释。

Complexes of Injective Words and Their Commutation Classes

Let $S$ be a finite alphabet. An injective word over $S$ is a word over $S$ such that each letter in $S$ appears at most once in the word. We study Boolean cell complexes of injective words over $S$ and their commutation classes. This generalizes work by Farmer and by Björner and Wachs on the complex of all injective words.

The symmetric group action on rank-selected posets of injective words

The symmetric group $\mathfrak{S}_n$ acts naturally on the poset of injective words over the alphabet $\{1, 2,\dots,n\}$. The induced representation on the homology of this poset has been computed by Reiner and Webb. We generalize their result by computing the representation of $\mathfrak{S}_n$ on the homology of all rank-selected subposets, in the sense of Stanley. A further generalization to the poset of $r$-colored injective words is given.

欧拉量级同调：对角性、单射词与正则路径同调

Eulerian magnitude homology: diagonality, injective words, and regular path homology

本文探讨欧拉量级同调的代数结构与组合性质，分析其对角性条件刻画完全图，构造正则量级-路径谱序列并证明其收敛到单射词复形，还得到对角有向图正则路径同调的刻画结果。

A poset fiber theorem for doubly Cohen-Macaulay posets and its applications to non-crossing partitions and injective words

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen-Macaulayness of a poset. Applications to complexes of injective words are also included.

Combinatorics of injective words for Temperley-Lieb algebras

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…