General representation theory

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 studiesmodules over these abstract algebraic structures...

• Linear representation
  • Unitary representation
    Unitary representation
    In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
    
    
• Trivial representation
  Trivial representation
  In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
  
  
• Irreducible representation
  • Semisimple
    Semisimple
    In mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way...
    
    
• Complex representation
  Complex representation
  The term complex representation has slightly different meanings in mathematics and physics.In mathematics, a complex representation is a group representationof a group on a complex vector space....
  
  
• Real representation
  Real representation
  In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...
  
  
• Quaternionic representation
• Pseudo-real representation
• Symplectic representation
  Symplectic representation
  In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant mapj\colon V\to V\,which satisfiesj^2=-1.\,...
  
  
• Schur's lemma
  Schur's lemma
  In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
  
  
• Restricted representation
  Restricted representation
  In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
  
  

Representation theory of groups

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

• Group representation
  Group representation
  In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
  
  
  • Group ring
    Group ring
    In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
    
    
• Maschke's theorem
  Maschke's theorem
  In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces...
  
  
• Regular representation
  Regular representation
  In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
  
  
• Character (mathematics)
  Character (mathematics)
  In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...
  
  
• Character theory
  Character theory
  In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
  
  
• Class function
  Class function
  In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...
  
  
• Representation theory of finite groups
  Representation theory of finite groups
  In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
  
  
  • Modular representation theory
    Modular representation theory
    Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
    
    
• Restricted representation
  Restricted representation
  In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
  
  
• Induced representation
  Induced representation
  In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
  
  
• Peter-Weyl theorem
• Young tableau
  Young tableau
  In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
  
  
• Spherical harmonic
  Spherical Harmonic
  Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
  
  
• Hecke operator
  Hecke operator
  In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
  
  
• Representation theory of the symmetric group
  Representation theory of the symmetric group
  In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...
  
  
• Representation theory of diffeomorphism groups
  Representation theory of diffeomorphism groups
  In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that that group acts transitively on M.-History:...
  
  
• Permutation representation
• Affine representation
  Affine representation
  An affine representation of a topological group G on an affine space A is a continuous group homomorphism from G to the automorphism group of A, the affine group Aff...
  
  
• Projective representation
  Projective representation
  In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
  
  
  • Central extension

Representation theory of Lie groups and Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

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• Representation of a Lie group
  Representation of a Lie group
  In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie...
  
  
• Lie algebra representation, Representation of a Lie superalgebra
  Representation of a Lie superalgebra
  In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2-graded vector space V, such that if A and B are any two pure elements of L and X and Y are any two pure elements of V, then[X]=c_1 A[X] + c_2 B[X]\,A[c_1 X + c_2...
  
  
• Universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
  
  
  • Casimir operator
  • Infinitesimal character
    Infinitesimal character
    In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation...
    
    
  • Harish-Chandra homomorphism
    Harish-Chandra homomorphism
    In mathematics, the Harish-Chandra isomorphism, introduced by ,is an isomorphism of commutative rings constructed in the theory of Lie algebras...
    
    
• Fundamental representation
  Fundamental representation
  In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
  
  
• Antifundamental representation
  Antifundamental representation
  In mathematics, an antifundamental representation is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each of them is a...
  
  
• Bifundamental representation
  Bifundamental representation
  In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations....
  
  
• Adjoint representation
  Adjoint representation
  In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
  
  
• Weight (representation theory)
  Weight (representation theory)
  In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
  
  
• Cartan's theorem
  Cartan's theorem
  In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....
  
  
• Spinor
  Spinor
  In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
  
  
• Wigner's classification
  Wigner's classification
  In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues...
  
  , Representation theory of the Poincaré group
  Representation theory of the Poincaré group
  In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics....
  
  
• Wigner-Eckart theorem
  Wigner-Eckart theorem
  The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum...
  
  
• Stone-von Neumann theorem
• Orbit method
  • Kirillov Character Formula
    Kirillov character formula
    In mathematics, for a Lie group G, the Kirillov orbit method gives a heuristic method in representation theory. It connects the Fourier transforms of coadjoint orbits, which lie in the dual space of the Lie algebra of G, to the infinitesimal characters of the irreducible representations...
    
    
• Weyl character formula
  Weyl character formula
  In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
  
  
• Discrete series representation
  Discrete series representation
  In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²...
  
  
• Principal series representation
  Principal series representation
  In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group. There, by analogy with spectral theory, one expects that the regular representation of G will decompose according to some kind of continuous spectrum, of...
  
  
• Borel–Weil theorem
  Borel–Weil theorem
  In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global...
  
  , Borel–Weil–Bott theorem

Representation theory of algebras

• Algebra representation
• Representation theory of Hopf algebras