Brauer Lifting in Algebraic K-Theory of Finite Fields

Likun Xie (they/them) Algebra, Mathematics, Representation Theory May 21, 2022May 21, 2022

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.