Our description of B depends on a fixed, but arbitrary, decomposition of A1 of the form kx1⊕V0, for some non-zero element x1 and some (d−1) dimensional subspace V0 of A1. Much information about B is already contained in the complex , which we call the skeleton of B. One striking feature of B is the fact that the skeleton of B is completely determined by the data (d,n); no other information about A is used in the construction of .
The skeleton is the mapping cone of zero:K→L, where L is a well known resolution of Buchsbaum and Eisenbud; K is the dual of L; and L and K are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of .
The differentials of B are explicitly given, in a polynomial manner, in terms of the coefficients of a Macaulay inverse system for A . In light of the properties of , the description of the differentials of B amounts to giving a minimal generating set of I , and, for the interior differentials, giving the coefficients of x1. As an application we observe that every non-zero element of A1 is a weak Lefschetz element for A.