Brauer Lifting in Algebraic K-Theory of Finite Fields

Abstract:

Quillen’s calculation of algebraic K-theory of finite fields incorporates a wide range of techniques from group cohomology to representation theory. This exposition focuses on the proof of Green’s theoerm: For any finite group G and any representation of G over \mathbb{F}_q (q is a power of some prime p), one can construct a character called the Brauer character which is a virtual complex character. We will also introduce its application in algebraic K-theory. The Brauer character of the n-dimensional standard representation of GL_n(\mathbb{F}_q) is a virtual complex character, thus it induces a map GL_n(\mathbb{F}_q) \to GL(\mathbb{C}), hence a map BGL\mathbb{F}_q^+\to BGL(\mathbb{C}) \cong BU. Taking the n-th homotopy group gives a map \theta: K_n(\mathbb{F}_q)\to \pi_nBU. This map is a key construction that allows one to identify K_n(\mathbb{F}_q) with the homotopy groups of a better understood space.

This writing is part of the course project for MATH 506 Group Representation Theory: