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Let ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

Let G be a connected reductive algebraic group with a Frobenius morphism F: G → G defined over a finite field □ pr . The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel G r = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.

Let Hq (d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive lth root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l 4 the multiplicities in a Specht or dual Specht module filtration of an Hq (d)-module are well defined. A cohomological criterion is given for when an Hq (d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p 5. These results are false in general for p = 2 or 3.