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 ${\text{HH}}_{\ast }(\mathbb{C}[x_1,...,x_d]/(x_1,...,x_d{)}^3;\mathbb{C})$ 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, $H_C^{\ast }$ 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.

To be published in the proceedings of the HUIC 2026 STEM/STEAM and Education Conference