AI 中文总结
该研究将考斯特上同调相关结果从对称群不可约表示的 $d=2$ 情形推广至所有 $d$，证明了对应考斯特上同调的欧拉特征为有界次数多项式，依赖组合论证解决代数拓扑驱动的问题。
AI 中文摘要
本研究属于通过考斯特上同调计算霍奇同调群 $\text{HH}_*(\text{ℂ}[x_1,...,x_d]/(x_1,...,x_d)^3; \text{ℂ})$ 的研究计划，该计划利用了Larsen和Lindenstrauss给出的同构关系：$\text{HH}_i(\text{ℂ}[x_1,...,x_d]/(x_1,...,x_d)^3; \text{ℂ}) \ncong \n\bigoplus_{0 \nj \ni} H_C^j\bigg( S_{i+j}, V^{\n\bigotimes(i+j)} \bigg)$，其中 $H^*_C$ 表示考斯特上同调，$S_{i+j}$ 是 $i+j$ 个字母的对称群，$V$ 是 $\text{GL}_d(\text{ℂ})$ 在 $\text{ℂ}^d$ 上的标准表示。此前研究仅聚焦于 $d=2$ 的情况，本研究将部分结果推广至所有 $d$ 的取值，特别证明了当张量表示替换为不可约表示时，对应考斯特上同调的欧拉特征由次数有界的多项式给出。尽管该问题由代数与拓扑驱动，但解决方案主要依赖组合论证。
英文摘要
This work is part of a research program to compute the Hochschild homology groups $\text{HH}_*(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C})$ via Coxeter cohomology, which utilizes the isomorphism \[ \text{HH}_i(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C}) \cong \sum_{0 \leq j \leq i} H_C^j\bigg( S_{i+j}, V^{\otimes(i+j)} \bigg)\] provided by Larsen and Lindenstrauss. Here, $H^*_C$ denotes Coxeter cohomology, $S_{i+j}$ is the symmetric group on $i+j$ letters, and $V$ is the standard representation of $\text{GL}_d(\mathbb{C})$ on $\mathbb{C}^d$. While previous work has focused exclusively on the case $d=2$, we extend some results to all values of $d$. Notably, we show that when the tensor representation is replaced by an irreducible representation, the Euler characteristic of the corresponding Coxeter cohomology is given by a polynomial with bounded degree. Although the problem is motivated by algebra and topology, the solution relies primarily on combinatorial arguments.
CommentsTo be published in the proceedings of the HUIC 2026 STEM/STEAM and Education Conference
