  Let d1,...,dn be a strictly increasing sequence of integers. Boij and
S\"oderberg [arXiv:math/0611081] have conjectured the existence of a graded
module M of finite length over any polynomial ring K[x_1,..., x_n], whose
minimal free resolution is pure of type (d1,...,dn), in the sense that its i-th
syzygies are generated in degree di.
  In this paper we prove a stronger statement, in characteristic zero: Such
modules not only exist, but can be taken to be GL(n)-equivariant. In fact, we
give two different equivariant constructions, and we construct pure resolutions
over exterior algebras and Z/2-graded algebras as well.
  The constructions use the combinatorics of Schur functors and Bott's Theorem
on the direct images of equivariant vector bundles on Grassmann varieties.
