The Steinberg Tensor Product Theorem for GL(m|n)

The Steinberg Tensor Product Theorem is a fundamental result in the modular representation theory of algebraic groups. The purpose of the present article is to formulate and prove the analogous theorem for the supergroup GL(m|n). This result was first mentioned without proof in [2]. We emphasize that our approach closely parallels the analogous result for the supergroup Q(n) proven by Brundan and Kleshchev [1], which in turn follows the approach of Cline, Parshall, and Scott [3]. The preliminaries are outlined in section 2. They are an abbreviated form of what can be found in [2] and [7]. Sections 3 and 4 contain the new results of the present article with the main theorem being the following version of the Steinberg Tensor Product Theorem. Before stating the result, we require some notation. We direct the reader to section 2 for precise statements of definitions. Throughout, let k be a fixed ground field of characteristic p > 0 which is algebraically closed. All objects under discussion are defined over k. Let T be the maximal torus of GL(m|n) consisting of diagonal matrices. We identify the character group X(T ) = Hom(T,Gm) with the free abelian group on generators ε1, . . . , εm+n, where εi picks out the ith entry of a diagonal matrix. We call the set