Group Representations and Complexity Theory: A Review

Group Representations and Character Theory

In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur’s Lemma. We then consider characters, class functions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogo...

Chaos/Complexity Theory and Education

Sciences exist to demonstrate the fundamental order underlying nature. Chaos/complexity theory is a novel and amazing field of scientific inquiry. Notions of our everyday experiences are somehow in connection to the laws of nature through chaos/complexity theory’s concerns with the relationships between simplicity and complexity, between orderliness and randomness (Retrieved from http://www.inc...

Group Quasi-representations and Index Theory

Let M be a closed connected manifold and let D be an elliptic operator on M . Let G be a discrete countable group and let M̃ → M be a principal G-bundle. Connes and Moscovici showed that this data defines an analytic index ind`1(G)(D) ∈ K0(`(G)). If B is a unital tracial C*-algebra, we give a formula for the trace of the image of ind`1(G)(D) in K0(B) under the map induced by a quasi-representati...

Deformation of Outer Representations of Galois Group

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

An Introduction to Wedderburn Theory & Group Representations