Representations of elementary Abelian \(p\)-groups and bundles on Grassmannians. (Q409635)

Let \(E\) denote an elementary Abelian \(p\)-group of rank \(n\), and let \(k\) denote an algebraically closed field of characteristic \(p>0\). The representation theory of elementary Abelian \(p\)-groups has played a key role in understanding the modular representations of more general finite groups, particularly through the use of cohomological support varieties. In the case of elementary Abelian groups, the Avrunin-Scott Theorem identifies cohomological support varieties with rank varieties which are defined via cyclic shifted subgroups of \(kE\). A cyclic shifted subgroup of \(kE\) is actually a subalgebra of \(kE\) and isomorphic to \(k[t]/(t^p)\). Subalgebras of the form \(k[t]/(t^p)\) have played a key role over the past thirty years in the development of the modular representation theory of various structures. This culminated in a sense with Friedlander and Pevtsova's introduction of the notion of \(\pi\)-points (appropriate flat maps from \(k[t]/(t^p)\) to \(kG\)) to develop a general theory for an arbitrary finite group scheme \(G\).NEWLINENEWLINE This work takes a renewed (and quite fruitful) look at the representation theory of elementary Abelian groups by considering the more general notion of a rank \(r\) shifted subgroup \(k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) for \(1\leq r\leq n\). Let \(V\subset\text{Rad}(kE)\) be an \(n\)-dimensional subspace chosen so that the composite \(V\to\text{Rad}(kE)\to\text{Rad}(kE)/\text{Rad}^2(kE)\) is an isomorphism and consider the variety of Grassmannians \(\text{Grass}(r,V)\) of \(r\)-planes in \(V\). Associated to any \(U\in\text{Grass}(r,V)\) is an algebra \(C(U)\simeq k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) and an associated flat map \(C(U)\to kE\). For a finite dimensional \(kE\)-module \(M\), the \(r\)-rank variety of \(M\) is defined to be the subset \(\text{Grass}(r,V)_M\) of \(\text{Grass}(r,V)\) consisting of those \(U\) for which the restriction of \(M\) to \(C(U)\) is not free. The variety \(\text{Grass}(r,V)_M\) is shown to be closed in \(\text{Grass}(r,V)\) and essentially dependent only on \(M\) not the choice of \(V\). Unlike the \(r=1\) case, when \(r=2\), every closed subvariety of \(\text{Grass}(2,V)\) cannot be realized as some \(\text{Grass}(2,V)_M\).NEWLINENEWLINE For a given \(U\in\text{Grass}(r,V)\) and \(kE\)-module \(M\), one can consider the radical (or socle) of \(M\) upon restriction to \(C(U)\). More generally, one can consider higher radicals (or socles). This can be used to extend Friedlander and Pevtsova's notion of generalized support varieties (for \(r=1\)) to higher \(r\). Specifically, the authors define non-maximal \(r\)-radical (and \(r\)-socle) support varieties. Some computations of such are given including an appendix by the first author that involves some computer calculations.NEWLINENEWLINE Further, the authors introduce an analogue of modules of constant Jordan type: modules of constant \(r\)-radical (or \(r\)-socle) type. Again, this is done in full generality for higher radicals (or socles). Having constant radical (or socle) type means that the dimension of the radical (or socle) of \(M\) restricted to \(C(U)\) is independent of the choice of \(U\in\text{Grass}(r,V)\). The authors present a number of examples of such modules and also investigate when the Carlson \(L_\zeta\) modules have such type.NEWLINENEWLINE In the second part of the paper, the authors use modules of constant \(r\)-radical or \(r\)-socle type to construct algebraic vector bundles on \(\text{Grass}(r,V)\). Again, numerous examples are given as well as examples of how various standard bundles can be realized by these constructions.