arXiv subjects

Eric M. Friedlander

Publications and source records attributed to Eric M. Friedlander.

24 records · Page 2Linked to original sources

arXiv2011

Representations of elementary abelian p-groups and bundles on Grassmannians

We initiate the study of representations of elementary abelian $p$-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by $r$ nilpotent elements, $k[t_1,..., t_r]/(t^p_1,..., t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.

math.RT↗

arXiv2010

Constructions for infinitesimal group schemes

Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $Θ_G: k[G] \to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter subgroups of G. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $\bP(G)$.

math.RT↗

arXiv2008

Cohomology of bifunctors

We initiate the study of the cohomology of (strict polynomial) bifunctors by introducing the foundational formalism, establishing numerous properties in analogy with the cohomology of functors, and providing computational techniques. Since one of the initial motivations for the study of functor cohomology was the determination of the cohomology of GL(k) with coefficients in a tensor product of a symmetric and an exterior power of the adjoint representation, we keep this challenging example in mind as we achieve numerous computations which illustrate our methods.

math.KT↗

arXiv2007

Modules of constant Jordan type

We introduce the class of modules of constant Jordan type for a finite group scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.

math.RT↗

arXiv1999

General linear and functor cohomology over finite fields

In recent years, there has been considerable success in computing Ext-groups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p.

math.RT↗

arXiv1992

A theory of algebraic cocycles

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a ``cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to $L$-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational $(p,p)$-cohomology class.

math.AG↗