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List of representation theory topics

Index List of representation theory topics

This is a list of representation theory topics, by Wikipedia page. [1]

61 relations: Adjoint representation, Affine representation, Algebra representation, Antifundamental representation, Bifundamental representation, Borel–Weil–Bott theorem, Casimir element, Character (mathematics), Character theory, Class function (algebra), Closed-subgroup theorem, Complex representation, Discrete series representation, Frobenius reciprocity, Fundamental representation, Glossary of representation theory, Group representation, Group ring, Harish-Chandra homomorphism, Hecke operator, Induced representation, Infinitesimal character, Irreducible representation, Kirillov character formula, Lie algebra, Lie algebra representation, List of harmonic analysis topics, Maschke's theorem, Mathematical analysis, Modular representation theory, Orbit method, Permutation representation, Peter–Weyl theorem, Principal series representation, Projective representation, Quaternionic representation, Real representation, Regular representation, Representation of a Lie group, Representation of a Lie superalgebra, Representation theory, Representation theory of diffeomorphism groups, Representation theory of finite groups, Representation theory of Hopf algebras, Representation theory of the Poincaré group, Representation theory of the symmetric group, Restricted representation, Schur's lemma, Semi-simplicity, Spherical harmonics, ..., Spinor, Stone–von Neumann theorem, Symplectic representation, Trivial representation, Unitary representation, Universal enveloping algebra, Weight (representation theory), Weyl character formula, Wigner's classification, Wigner–Eckart theorem, Young tableau. Expand index (11 more) »

Adjoint representation

In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.

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Affine representation

An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A).

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Algebra representation

In abstract algebra, a representation of an associative algebra is a module for that algebra.

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Antifundamental representation

In mathematics, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental representation is a matter of convention.

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Bifundamental representation

In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations.

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Borel–Weil–Bott theorem

In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles.

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Casimir element

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra.

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Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).

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Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.

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Class function (algebra)

In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G.

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Closed-subgroup theorem

In mathematics, the closed-subgroup theorem (sometimes referred to Cartan's theorem) is a theorem in the theory of Lie groups.

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Complex representation

In mathematics, a complex representation is a group representation of a group (or that of Lie algebra) on a complex vector space.

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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(G).

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Frobenius reciprocity

In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting.

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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 or Lie algebra whose highest weight is a fundamental weight.

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Glossary of representation theory

This is a glossary of representation theory in mathematics.

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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.

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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.

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Harish-Chandra homomorphism

In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra.

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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.

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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 to a representation of the (whole) group itself.

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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.

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Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.

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Kirillov character formula

In mathematics, for a Lie group G, the Kirillov orbit method gives a heuristic method in representation theory.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.

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List of harmonic analysis topics

This is a list of harmonic analysis topics.

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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.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Modular representation theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.

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Orbit method

In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra.

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Permutation representation

In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices.

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Peter–Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian.

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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.

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Projective representation

In the field of representation theory in mathematics, 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 group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.

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Quaternionic 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 map which satisfies Together with the imaginary unit i and the antilinear map k.

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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 map which satisfies The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V.

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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.

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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.

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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 '''Z'''2-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 Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above.

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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.

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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 (for M connected) that group acts transitively on M.

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Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

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Representation theory of Hopf algebras

In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra.

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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.

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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.

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Restricted representation

In mathematics, restriction is a fundamental construction in representation theory of groups.

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Schur's lemma

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras.

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Semi-simplicity

In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry.

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Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.

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Spinor

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

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Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.

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Symplectic representation

In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form where F is the field of scalars.

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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. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector.

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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(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

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Universal enveloping algebra

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

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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, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

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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.

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Wigner's classification

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group which have sharp mass eigenvalues.

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Wigner–Eckart theorem

The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics.

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Young tableau

In mathematics, a Young tableau (plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.

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References

[1] https://en.wikipedia.org/wiki/List_of_representation_theory_topics

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