The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group \(\Sigma_n\), the Lie module \(\mathsf{Lie}(n)\) has attracted a great deal of interest in recent years. In this talk, I will show that the complexity of \(\mathsf{Lie}(n)\) in characteristic \(p\) is \(t\) where \(p^t\) is the largest power of \(p\) dividing \(n\), thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology \(\operatorname{H}^{\bullet}(\Sigma_n, \mathsf{Lie}(n))\) and earlier work of Hemmer and Nakano on complexity for modules over \(\Sigma_n\) that involves restriction to Young subgroups. This part is joint work with David Hemmer and Fred Cohen.

The aforementioned ideas will be employed in a second application where \(\mathcal{H}_{q}(d)\) is the Iwahori-Hecke algebra for the symmetric group and \(q\) is a primitive \(l\)th root of unity. In joint work with Ziqing Xiang, we developed a theory of support varieties which detects natural homological properties such as the complexity of modules. The theory has a canonical description in an affine space where computations are tractable. The ideas involved 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 will be 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 talk.