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*(C[x(1),...,x(d)]/(x(1),...,x(d))(3);C) in the case d = 2 through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism HHi(C[x(1),...,x(d)]/(x(1),...,x(d))(3);C) congruent to to Sigma(0 <= j <=) H-C(j) (Si+j,V circle times(i+j)) provided by Larsen and Lindenstrauss. Here, H-C* denotes Coxeter cohomology, Si+j denotes the symmetric group on i + j letters, and V is the standard representation of GL(d) (C) on 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.