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Representations On Coordinate Rings
In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties. Let ''X'' be an affine algebraic variety over an algebraically closed field ''k'' of characteristic zero with the action of a reductive algebraic group ''G''. ''G'' then acts on the coordinate ring k /math> of ''X'' as a left regular representation: (g \cdot f)(x) = f(g^ x). This is a representation of ''G'' on the coordinate ring of ''X''. The most basic case is when ''X'' is an affine space (that is, ''X'' is a finite-dimensional representation of ''G'') and the coordinate ring is a polynomial ring. The most important case is when ''X'' is a symmetric variety; i.e., the quotient of ''G'' by a fixed-point subgroup of an involution. Isotypic decomposition Let k be the sum of all ''G''-submodules of k /math> that are isomorphic to the simple module V^; it is called the \lambda- isotypic component of k /math>. Then there is a direct sum decomposition: :k = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Of A Group
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Algebraic Variety
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Algebraic Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left 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. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. Finite groups For a finite group ''G'', the left regular representation λ (over a field ''K'') is a linear representation on the ''K''-vector space ''V'' freely generated by the elements of ''G'', i. e. they can be identified with a basis of ''V''. Given ''g'' ∈ ''G'', λ''g'' is the linear map determined by its action on the basis by left translation by ''g'', i.e. :\lambda_:h\mapsto gh,\texth\in G. For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g'' ∈ ''G'', ρ''g'' is the linear map on ''V'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Variety
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Subgroup
In algebra, the fixed-point subgroup G^f of an automorphism ''f'' of a group ''G'' is the subgroup of ''G'': :G^f = \. More generally, if ''S'' is a set of automorphisms of ''G'' (i.e., a subset of the automorphism group of ''G''), then the set of the elements of ''G'' that are left fixed by every automorphism in ''S'' is a subgroup of ''G'', denoted by ''G''''S''. For example, take ''G'' to be the group of invertible ''n''-by-''n'' real matrices and f(g)=(g^T)^ (called the Cartan involution). Then G^f is the group O(n) of ''n''-by-''n'' orthogonal matrices. To give an abstract example, let ''S'' be a subset of a group ''G''. Then each element ''s'' of ''S'' can be associated with the automorphism g \mapsto sgs^, i.e. conjugation by ''s''. Then :G^S = \; that is, the centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotypic Component
The isotypic component of weight \lambda of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight \lambda. Definition * A finite-dimensional module V of a reductive Lie algebra \mathfrak (or of the corresponding Lie group) can be decomposed into irreducible submodules : V = \oplus_^N V_i . * Each finite-dimensional irreducible representation of \mathfrak is uniquely identified (up to isomorphism) by its highest weight :\forall i \in \ \exists \lambda \in P(\mathfrak) : V_i \simeq M_\lambda, where M_\lambda denotes the highest weight module with highest weight \lambda. * In the decomposition of V , a certain isomorphism class might appear more than once, hence : V \simeq \oplus_ ( \oplus_^ M_) . This defines the isotypic component of weight \lambda of V: \lambda(V) := \oplus_^ V_i \simeq \mathbb^ \otimes M_ where d_\lambda is maximal. See also * Lie algebra representation * Weight (representation theory) In the mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. Examples Linear complex structure One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as \mathbb = \mathbb (x^2+1), which corresponds to . Then a representation of C is a real vector space ''V'', together with an action of C on ''V'' (a map \mathbb \to \mathrm(V)). Concretely, this is just an action of , as this generates the algebra, and the operator representing (the image of in End(''V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Variety
In algebraic geometry, given a reductive algebraic group ''G'' and a Borel subgroup ''B'', a spherical variety is a ''G''-variety with an open dense ''B''-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be no .... There is also a notion of real spherical varieties. A projective spherical variety is a Mori dream space. Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory. In his seminal paper, developed a framework to classify complex spherical subgroups of reductive groups; he reduced the classification of spherical subgroups to wonderful subgroups. He further w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Coefficient
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group ''G'' obtained by composing a representation of ''G'' on a vector space ''V'' with a linear map from the endomorphisms of ''V'' into ''V'' 's underlying field. It is also called a representative function. They arise naturally from finite-dimensional representations of ''G'' as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on ''G'' are dense in the Hilbert space of square-integrable functions on ''G''. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group ''G'' . The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur. Let ''G'' be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of ''G'' are dense in the space ''C''(''G'') of continuous complex-valued functions on ''G'', and thus also in the space ''L''2(''G'') of square-integrable functions. The second part asserts the complete reducibility of unitary representations of ''G''. The third part then asserts that the regular representation of ''G'' on ''L''2(''G'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. Examples Linear complex structure One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as \mathbb = \mathbb (x^2+1), which corresponds to . Then a representation of C is a real vector space ''V'', together with an action of C on ''V'' (a map \mathbb \to \mathrm(V)). Concretely, this is just an action of , as this generates the algebra, and the operator representing (the image of in End(''V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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