nLab representation theory



Representation theory




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.

See also at geometric representation theory.

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

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

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

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

Under this identification, the representation theory of GG is equivalently

More in detail, this yields the following identifications:

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)



Lecture notes:

Textbook accounts

for finite groups:

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

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

  • Caroline Gruson, Vera Serganova, From Finite Groups to Quivers via Algebras – A Journey Through Representation Theory, Springer (2018) [doi:10.1007/978-3-319-98271-7]

and more generally for compact Lie groups:

In the context of quantum mechanics:

Discussion via string diagrams/Penrose notation:

Further references:

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

Last revised on June 10, 2024 at 20:07:45. See the history of this page for a list of all contributions to it.