The GL-module structure of the Hochschild homology of truncated tensor algebras.
Let \(V\) be a finite dimensional complex vector space and \(A\) be a truncated tensor algebra of \(V\). The natural action of \(\text{GL}(V)\) on \(A\) extends to an action on the Hochschild homology of \(A\), \(HH(A)\). The goal of this paper is to determine the \(\text{GL}(V)\)-module structure of \(HH(A)\). Using work of \textit{M. J. Bardzell} [J. Algebra 188, No. 1, 69-89 (1997; Zbl 0885.16011)], the authors first construct a projective resolution of \(A\) over the enveloping algebra of \(A\). By decomposing this into subcomplexes, they are able to decompose \(HH(A)\) into a sum of eigenspaces. Knowing from Schur duality that \(V^{\otimes n}\) is completely reducible, the authors are then able to obtain formulas for the multiplicities of simple \(\text{GL}(V)\)-modules in a decomposition of \(HH(A)\). The simple \(\text{GL}(V)\)-modules are parameterized by Young diagrams and the authors then proceed to make specific computations of multiplicities for some ``standard'' Young diagrams and diagrams of depth two. The latter results make use of a hook-like formula due to \textit{V. M. Zhuravlev} [Sb. Math. 187, No. 2, 215-236 (1996); translation from Mat. Sb. 187, No. 2, 59-80 (1996; Zbl 0873.17008)]. Finally, the authors present some computations for truncation at low heights and results of some computer computations.
scientific article; zbMATH DE number 2092903