# Geometric Properties of Representation Varieties For An Elementary Abelian Group of Rank 2 in Positive Characteristic

2021

## Document Type

Dissertation/Thesis

Geline, Michael

## Degree Name

Ph.D. (Doctor of Philosophy)

## Legacy Department

Department of Mathematical Sciences

## Abstract

This dissertation’s motivation is the exploration of the irreducible components of Repn(kG), the affine variety whose points are n-dimensional representations of a finite group G over a field k. We let G = Z/pZ×Z/pZ and assume k is algebraically closed with char(k) = p > 0. In this case there is an isomorphism of affine varieties φ : Repn (kG) → C nil 1 (n) where C nil 1 (n) = {(x, y) Mn(k)×Mn(k) | x p = y p = xy−yx = 0}. Hence, for an irreducible component X of Repn (kG), φ(X) is an irreducible component of C nil 1 (n) with the same dimension as X. When n ≤ p, Premet showed in [15] that C nil 1 (n) is irreducible. We further assume that char(k) = p > 2 and that n > p. It is known that if X is an irreducible component of C nil 1 (n), then X = GLn(k) · (e, V ) where e is a n×n nilpotent matrix and V is an irreducible component of the restricted nilpotent centralizer of e, z(e) ∩ N1(n). Main Result 1 proves that e must have at least one Jordan block of size greater than 2. We also show that V is the only irreducible component of z(e) ∩ N1(n) with X = GLn(k) · (e, V ), and that dim X = dim GLn(k) · e + dim V . Lastly, Main Result 2 proves that C nil 1 (n) has at least as many irreducible components as the affine variety z(ed) ∩ N1(n), where GLn(k) · ed is the GLn(k)-orbit that dominates all others in the dominance ordering.

99 pages

eng

## Publisher

Northern Illinois University

## Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

Text

COinS