  In this paper we discuss the dimensions of the (higher) secant varieties to
the Grassmann varieties, embedded via the Plucker embeddings. We use
Terracini's Lemma and the duality in the exterior algebra of a finite
dimensional vector space to translate the problem into that of finding the
dimension of a graded piece of a " fat" ideal in the exterior algebra. Among
other things, our methods prove that (apart from the chordal varieties to the
Grassmannians of lines) all the chordal varieties of Grassmannians have the
expected dimension.
  We also discuss some Grassmannians with deficient secant varieties, giving
new proofs that G(3,7) and G(3,9) are of this type. We also find a new
Grassmannian, namely G(4,8) with deficient secant varieties.
