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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.[1][2] In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

This article is about the theory of representations of algebraic structures by linear transformations and matrices. For representation theory in other disciplines, see Representation (disambiguation).

The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.[3][4]


Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood.[5] For instance, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups.[6][7] Furthermore, representation theory is important in physics because it can describe how the symmetry group of a physical system affects the solutions of equations describing that system.[8]


Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse.[9] In addition to its impact on algebra, representation theory


There are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.[13]


The success of representation theory has led to numerous generalizations. One of the most general is in category theory.[14] The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces.[4] This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

The set of all matrices is a group under matrix multiplication, and the representation theory of groups analyzes a group by describing ("representing") its elements in terms of invertible matrices.

invertible

Matrix addition and multiplication make the set of all matrices into an associative algebra, and hence there is a corresponding .

representation theory of associative algebras

If we replace matrix multiplication by the matrix , then the matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.

commutator

: in the category of projective spaces. These can be described as "linear representations up to scalar transformations".

projective representations

: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space.

affine representations

: in the category of complex vector spaces with morphisms being linear or antilinear transformations.

corepresentations of unitary and antiunitary groups

(1986), Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups, Cambridge University Press, ISBN 978-0-521-44926-7.

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Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", , 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129.

Annals of Mathematics

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Borel, Armand

Borel, Armand; Casselman, W. (1979), Automorphic Forms, Representations, and L-functions, American Mathematical Society,  978-0-8218-1435-2.

ISBN

; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0-470-18975-7.

Curtis, Charles W.

Folland, Gerald B. (1995), A Course in Abstract Harmonic Analysis, CRC Press,  978-0-8493-8490-5.

ISBN

; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..

Fulton, William

(1984), "An Elementary Introduction to the Langlands Program", Bulletin of the American Mathematical Society, 10 (2): 177–219, doi:10.1090/S0273-0979-1984-15237-6.

Gelbart, Stephen

Goodman, Roe; Wallach, Nolan R. (1998), Representations and Invariants of the Classical Groups, Cambridge University Press,  978-0-521-66348-9.

ISBN

Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,  978-3319134666

ISBN

Helgason, Sigurdur (1978), Differential Geometry, Lie groups and Symmetric Spaces, Academic Press,  978-0-12-338460-7

ISBN

Humphreys, James E. (1972a), , Birkhäuser, ISBN 978-0-387-90053-7.

Introduction to Lie Algebras and Representation Theory

Humphreys, James E. (1972b), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Berlin, New York: , ISBN 978-0-387-90108-4, MR 0396773

Springer-Verlag

James, Gordon; (1993), Representations and Characters of Groups, Cambridge: Cambridge University Press, ISBN 978-0-521-44590-0.

Liebeck, Martin

Jantzen, Jens Carsten (2003), Representations of Algebraic Groups, American Mathematical Society,  978-0-8218-3527-2.

ISBN

(1977), "Lie superalgebras", Advances in Mathematics, 26 (1): 8–96, doi:10.1016/0001-8708(77)90017-2.

Kac, Victor G.

Kac, Victor G. (1990), Infinite Dimensional Lie Algebras (3rd ed.), Cambridge University Press,  978-0-521-46693-6.

ISBN

Kim, Shoon Kyung (1999), Group Theoretical Methods and Applications to Molecules and Crystals: And Applications to Molecules and Crystals, Cambridge University Press,  978-0-521-64062-6.

ISBN

(2001), Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, ISBN 978-0-691-09089-4.

Knapp, Anthony W.

; Manin, Yuri I. (1997), Linear Algebra and Geometry, Taylor & Francis, ISBN 978-90-5699-049-7.

Kostrikin, A. I.

Lam, T. Y. (1998), "Representations of finite groups: a hundred years", Notices of the AMS, 45 (3, 4): , 465–474 (Part II).

361–372 (Part I)

Lyubich, Yurii I. (1988). Introduction to the Theory of Banach Representations of Groups. Operator Theory: Advances and Applications. Vol. 30. Basel: Birkhauser.  978-3-7643-2207-6.

ISBN

; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 0214602; MR0719371 (2nd ed.); MR1304906(3rd ed.)

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(1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 978-0-521-55821-1.

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Peter, F.; Weyl, Hermann (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", , 97 (1): 737–755, doi:10.1007/BF01447892, S2CID 120013521.

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Pontrjagin, Lev S.

; Vogan, David A. (1989), Representation Theory and Harmonic Analysis on Semisimple Lie Groups, American Mathematical Society, ISBN 978-0-8218-1526-7.

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(1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 978-0387901909.

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Sharpe, Richard W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer,  978-0-387-94732-7.

ISBN

Simson, Daniel; Skowronski, Andrzej; Assem, Ibrahim (2007), Elements of the Representation Theory of Associative Algebras, Cambridge University Press,  978-0-521-88218-7.

ISBN

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"Representation theory"

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Alexander Kirillov Jr.

Kevin Hartnett,

(2020), article on representation theory in Quanta magazine