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

This work is part of a research program to compute the Hochschild homology groups HH \(_*({\mathbb {C}}[x_1,\ldots ,x_d]/(x_1,\ldots ,x_d)^3;{\mathbb {C}})\) ${}_{\ast }(\mathbb{C}[x_1,\dots ,x_d]/{(x_1,\dots ,x_d)}^3;\mathbb{C})$ in the case \(d = 2\) $d=2$ through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism \(\begin{aligned}\text {HH}_i({\mathbb {C}}[x_1,\ldots ,x_d]/(x_1,\ldots ,x_d)^3;{\mathbb {C}}) \cong \sum _{0\le j \le i} H^j_C \left( S_{i+j}, V^{\otimes (i+j)}\right) \end{aligned}\) $$\begin{array}{c}\hfill {\text{HH}}_i(\mathbb{C}[x_1,\dots ,x_d]/{(x_1,\dots ,x_d)}^3;\mathbb{C})\cong \sum _{0\le j\le i}{H}_C^j\left(S_{i+j},,,V^{\otimes (i+j)}\right)\end{array}$$ provided by Larsen and Lindenstrauss. Here, \(H_C^*\) $H_C^{\ast }$ denotes Coxeter cohomology, \(S_{i+j}\) $S_{i+j}$ denotes the symmetric group on \(i+j\) $i+j$ letters, and V is the standard representation of \(\textrm{GL}_d({\mathbb {C}})\) ${\text{GL}}_d(\mathbb{C})$ on \({\mathbb {C}}^d\) ${\mathbb{C}}^d$ . We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of \(S_n\) $S_n$ . In particular, the aforementioned tensor representation, and also several classes of irreducible representations of \(S_n\) $S_n$ . Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.