nLab representation theory

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

Idea

Representation theory is concerned with the study of algebraic structures via their representations. This concerns notably groups, directly or in their incarnation as group algebras, Hopf algebras or Lie algebras, and usually concerns linear representations, hence modules of these structures. But more generally representation theory also studies representations/modules/actions of generalizations of such structures, such as coalgebras via their comodules etc.

In homotopy type theory

The fundamental concepts of representation theory have a particular natural formulation in homotopy theory and in fact in homotopy type theory, which also refines it from the study of representations of groups to that of ∞-representations of ∞-groups. This includes both discrete ∞-groups as well as geometric homotopy types such as smooth ∞-groups, the higher analog of Lie groups.

The key observation to this translation is that

1. an ∞-group $G$ is equivalently given by its delooping $\mathbf{B}G$ regarded with its canonical point (see at looping and delooping), hence the universal $G$-principal ∞-bundle

$\array{ G &\longrightarrow& \ast \\ && \downarrow \\ && \mathbf{B}G }$
2. an ∞-action $\rho$ of $G$ on any geometric homotopy type $V$ is equivalently given by a homotopy fiber sequence of the form

$\array{ V &\stackrel{}{\longrightarrow}& V//_\rho G \\ && \downarrow \\ && \mathbf{B}G } \,,$

hence by a $V$-fiber ∞-bundle over $\mathbf{B}G$ which is the $\rho$-associated ∞-bundle to the universal $G$-principal ∞-bundle (see at ∞-action for more on this).

Under this identification, the representation theory of $G$ is equivalently

More in detail, this yields the following identifications:

homotopy type theoryrepresentation theory
pointed connected context $\mathbf{B}G$∞-group $G$
dependent type on $\mathbf{B}G$$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$trivial representation
dependent product along $\mathbf{B}G \to \ast$homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$coinduced representation
spectrum object in context $\mathbf{B}G$spectrum with G-action (naive G-spectrum)

Lecture notes include

Textbooks include

• Charles Curtis, Irving Reiner, Representation theory of finite groups and associative algebras, AMS 1962

• William Fulton, Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991 (pdf)

• Klaus Lux, Herbert Pahlings, Representations of groups – A computational approach, Cambridge University Press 2010 (author page, publisher page)

A list of texts on representation theory is maintained at

The relation to number theory and the Langlands program is discussed in