  Let ${\mathcal H}_{q}(d)$ be the Iwahori-Hecke algebra for the symmetric
group, where $q$ is a primitive $l$th root of unity. In this paper we develop a
theory of support varieties which detects natural homological properties such
as the complexity of modules. The theory the authors develop has a canonical
description in an affine space where computations are tractable. The ideas
involve the interplay with the computation of the cohomology ring due to
Benson, Erdmann and Mikaelian, the theory of vertices due to Dipper and Du, and
branching results for cohomology by Hemmer and Nakano. Calculations of support
varieties and vertices are presented for permutation, Young and classes of
Specht modules. Furthermore, a discussion of how the authors' results can be
extended to other Hecke algebras for other classical groups is presented at the
end of the paper.
