The GL(V)-module structure of the Hochschild homology of truncated tensor algebras

G. AMES; L. CAGLIERO; P. TIRAO

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Año: 2004 vol. 193 p. 11 - 26

0022-4049

Let $V$ be a finite dimensional complex vector space and $A$ be a truncated tensor algebra of $V$. The natural action of $ ext{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 $ ext{GL}(V)$-module structure of $HH(A)$.par Using work of {it 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 $ ext{GL}(V)$-modules in a decomposition of $HH(A)$.par The simple $ ext{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 {it 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.