## Graduate Research Theses & Dissertations

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  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