On Inverting the Koszul

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 ∧ i 1 V ⊗ · · · ⊗ ∧ ip 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 ∧(V *) (k, k) = S t (V). Let k be a field, and let V be an n-dimensional vector space over k with n ≥ 2. We view the alternating and symmetric powers ∧ i V and S t V of V as representations of G = GL(V) ∼ = GL(n, k); we allow any i, t ∈ Z, with the understanding that ∧ i V and S t V are zero unless 0 ≤ i ≤ n or t ≥ 0. Working in the representation ring of G (i.e., the Grothendieck group of algebraic representations of G), we can write S t V as a polynomial in the fundamental representations ∧ 1 V = V, ∧ 2 V,. .. , ∧ n V ; as we shall see below, the terms of this polynomial can be ordered in a natural way with alternating signs. This suggests that there should exist an exact sequence of representations of G that concretely realizes this polynomial expression for S t V. This sequence, given in equation (3) below, has appeared in Sections 2 and 4 of [Tot97], as a modification of a construction of [Aki89], via the normalized bar resolution of k as a module for the exterior algebra ∧(V *). It is known that there is a natural isomorphism (1) Ext t ∧(V *) (k, k) = S t V, and computing the Ext group using the normalized bar resolution as in [Tot97] produces the exact sequence (3). It is however not immediate to write down the differentials explicitly, as this involves chasing through various dualizations and natural isomorphisms. Moreover, one needs to have prior knowledge of (1) to use this approach. In this note, we give a …