On the Coxeter Cohomology of Irreducible Representations of Symmetric Groups

Abstract

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.