nLab representation theory

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.

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 $G$ 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

• the homotopy theory in the slice (∞,1)-topos over $\mathbf{B}G$;

• the homotopy type theory in the context of/dependent on $\mathbf{B}G$.

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 theory	representation 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)

Examples

• representation theory of the symmetric group

• representation theory of the general linear group

• representation theory of the special unitary group

• Schur-Weyl duality

Related entries

• representation, action, module

  • group, group algebra, groupoid, algebraic group, Lie algebra

  • vector space, affine space, symplectic vector space

  • character

  • action, module, equivariant object

  • bimodule, Morita equivalence, induced representation

  • Hilbert space, Banach space, Fourier transform, functional analysis

  • weight (in representation theory)

  • orbit, coadjoint orbit, Killing form

  • geometric quantization, coherent state

  • socle, quiver

  • module algebra, comodule algebra, Hopf action, measuring

• representation ring

• irreducible representation

• Schur's lemma

• Young diagram

• Schur index

• McKay correspondence, ADE classification

• asymptotic representation theory

• geometric representation theory

  • Borel-Weil theorem, Beilinson-Bernstein localization

  • D-module, perverse sheaf, BBDG decomposition theorem

  • Kazhdan-Lusztig theory

  • Dirac induction

• Verma module, BGG resolution

• representation stability

• Grothendieck group, lambda-ring, symmetric function, formal group

• principal bundle, torsor, vector bundle, Atiyah Lie algebroid

• character sheaf?, Harish Chandra transform

• geometric function theory, groupoidification

• Eilenberg-Moore category, algebra over an operad, actegory, crossed module

• reconstruction theorems

References

Lecture notes:

• Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina:

  Introduction to representation theory, Student Mathematical Library 59, AMS (2011) [arXiv:0901.0827, ams:stml-59]

• Tammo tom Dieck, Representation theory (2009) [pdf, pdf]

• Constantin Teleman, Representation theory, lecture notes 2005 (pdf)

• Joel Robbin, Real, Complex and Quaternionic representations, 2006 (pdf, pdf)

• Igor R. Shafarevich, Alexey O. Remizov: §14 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]

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:

• Tammo tom Dieck, Theodor Bröcker, Representations of compact Lie groups, Springer (1985) [doi:10.1007/978-3-662-12918-0]

• William Fulton, Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991 (doi:10.1007/978-1-4612-0979-9)

In the context of quantum mechanics:

• Peter Woit, Quantum Theory, Groups and Representations: An Introduction, Springer 2017 [doi:10.1007/978-3-319-64612-1, ISBN:978-3-319-64610-7]

Discussion via string diagrams/Penrose notation:

• Jeffrey Ellis Mandula, Diagrammatic techniques in group theory, Southampton Univ. Phys. Dept. (1981) (cds:129911, pdf)

• Predrag Cvitanović, Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press July 2008 (PUP, birdtracks.eu, pdf)

  (aimed at Lie theory and gauge theory)

Further references:

• Mikhail Khovanov, Resources.

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

• Robert Langlands, Representation theory: Its rise and its role in number theory (web)