arXiv subjects

Michael Crumley

Publications and source records attributed to Michael Crumley.

4 recordsLinked to original sources

arXiv2011

Generic Representation Theory of the Additive and Heisenberg Groups

In this paper we give an intimate connection between the characteristic zero representation theories of the Additive and Heisenberg groups, and their characteristic p >0 theories when p is much larger than the dimension a representation. In particular, if p >> dimension, then all characteristic p representations for these groups can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of the group, one for each of the representation's Frobenius layers. In this sense, for a fixed dimension and large enough p, all representations for these groups look generically like representations for direct powers of themselves over a field of characteristic zero.

math.RT↗

arXiv2011

Generic Representation Theory of the Heisenberg Group

In this paper we extend a result for representations of the Additive group $G_a$ given in [3] to the Heisenberg group $H_1$. Namely, if $p$ is greater than 2d then all $d$-dimensional characteristic $p$ representations for $H_1$ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of $H_1$, one for each of the the representation's Frobenius layers. In this sense, for a fixed dimension and large enough $p$, all representations for $H_1$ look generically like representations for direct powers of it over a field of characteristic zero. The reader may consult chapter 13 of [1] for a fuller account of what follows.

math.RT↗

arXiv2011

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this paper we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups $U_n$), and under a certain hypothesis relating the characteristic $p$ to both $n$ and the dimension $d$ of a representation (specifically, $p \geq \text{max}(n,2d)$), Lie theory is completely sufficient to determine the representation theory of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their `generic' representation theory in characteristic $p$.

math.RT↗

arXiv2010

Ultraproducts of Tannakian Categories and Generic Representation Theory of Unipotent Algebraic Groups

The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and field k, and conversely. Originally motivated by an attempt to find a first-order explanation for generic cohomology of algebraic groups, we study neutral tannakian categories as abstract first-order structures and, in particular, ultraproducts of them. One of the main theorems of this dissertation is that certain naturally definable subcategories of these ultraproducts are themselves neutral tannakian categories, hence tensorially equivalent to Comod_A for some Hopf algebra A over a field k. We are able to give a fairly tidy description of the representing Hopf algebras of these categories, and explicitly compute them in several examples. For the second half of this dissertation we turn our attention to the representation theories of certain unipotent algebraic groups, namely the additive group G_a and the Heisenberg group H_1. The results we obtain for these groups in characteristic zero are not at all new or surprising, but in positive characteristic they perhaps are. In both cases we obtain that, for a given dimension n, if p is large enough with respect to n, all n-dimensional modules for these groups in characteristic p are given by commuting products of representations, with the constituent factors resembling representations of the same group in characteristic zero. We later use these results to extrapolate some generic cohomology results for these particular unipotent groups.

math.RT↗