Categories of modules for elementary abelian p-groups and generalized Beilinson algebras

In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian p-groups E_r of rank r \geq 2 by exploiting a functor from the module category of a generalized Beilinson algebra B(n,r), n \leq p, to mod E_r. We define analogs of the above mentioned properties in mod B(n,r) and give a homological characterization of the resulting subcategories via a P^{r-1}-family of B(n,r)-modules of projective dimension one. This enables us to apply homological methods from Auslander-Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length two over E_2 with the case r > 2. Moreover, we give a generalization of the W-modules defined by Carlson, Friedlander and Suslin.Comment: 21 page

Mixed Tensors of the General Linear Supergroup

We describe the image of the canonical tensor functor from Deligne's interpolating category $Rep(GL_{m-n})$ to $Rep(GL(m|n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of $GL(m|n)$, covering the decomposition between any two irreducible $GL(m|1)$-representations. We also obtain character and dimension formulas. For $m>n$ we classify the mixed tensors with non-vanishing superdimension. For $m=n$ we characterize the maximally atypical mixed tensors and show some applications regarding tensor products.Comment: v3: Improved exposition, corrected minor mistakes v2: shortened and revised version. Comments welcom

Modular Centralizer Algebras Corresponding to p-Groups

We study the Loewy structure of the centralizer algebra kP^Q for P a p-group with subgroup Q and k a field of characteristic p. Here kP^Q is a special type of Hecke algebra. The main tool we employ is the decomposition of kP^Q as a split extension of a nilpotent ideal I by the group algebra kC_P(Q). We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of $P^Q. Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As a final application of the decomposition, we determine the representation type of kP^Q