Calculations of the Euler characteristic of the Coxeter cohomology of symmetric groups

Abstract

This work is part of a research program to compute the Hochschild homology groups HH$_*({\mathbb {C}}[x_1,\ldots ,x_d]/(x_1,\ldots ,x_d)^3;{\mathbb {C}})$ in the case $d = 2$ through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism

$$\begin{aligned}\text {HH}_i({\mathbb {C}}[x_1,\ldots ,x_d]/(x_1,\ldots ,x_d)^3;{\mathbb {C}}) \cong \sum _{0\le j \le i} H^j_C \left( S_{i+j}, V^{\otimes (i+j)}\right) \end{aligned}$$

provided by Larsen and Lindenstrauss. Here, $H_C^*$ denotes Coxeter cohomology, $S_{i+j}$ denotes the symmetric group on $i+j$ letters, and V is the standard representation of $\textrm{GL}_d({\mathbb {C}})$ on ${\mathbb {C}}^d$. We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of $S_n$. In particular, the aforementioned tensor representation, and also several classes of irreducible representations of $S_n$. Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.