Mathematics – Representation Theory
Scientific paper
2007-02-28
Communications in Algebra 36 (2008), 1830-1837
Mathematics
Representation Theory
7 pages, uses xypic. Second revision, after referee's comments. Includes more details of the proof of Theorem 2, and some othe
Scientific paper
Let V be an n-dimensional vector space. We give a direct construction of an exact sequence that gives a GL(V)-equivariant "resolution" of each symmetric power S^t V in terms of direct sums of tensor products of the form \wedge^{i_1} V \otimes ... \otimes \wedge^{i_p} V. This exact sequence corresponds to inverting the relation in the representation ring of GL(V) that is described by the Koszul complex, and has appeared before in work by B. Totaro, analogously to a construction of K. Akin involving the normalized bar resolution. Our approach yields a concrete description of the differentials, and provides an alternate direct proof that Ext^t_{\wedge (V^*)}(k,k) = S^t V.
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