Complexity For Modules Over The Classical Lie Superalgebra Gl(M Vertical Bar N)

Let g = g((0) over bar) circle plus g((1) over bar) be a classical Lie superalgebra and let F be the category of finite-dimensional g-supermodules which are completely reducible over the reductive Lie algebra g((0) over bar). 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 g((1) over bar). In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra gl(m vertical bar n). In both cases we show that the complexity is related to the atypicality of the block containing the module.