# 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: