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.