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Covariant (invariant Theory)
In invariant theory, a branch of algebra, given a group ''G'', a covariant is a ''G''-equivariant polynomial map V \to W between linear representations ''V'', ''W'' of ''G''. It is a generalization of a classical convariant, which is a homogeneous polynomial map from the space of binary ''m''-forms to the space of binary ''p''-forms (over the complex numbers) that is SL_2(\mathbb)-equivariant. See also *module of covariants * Invariant of a binary form#Terminology *Transvectant In mathematical invariant theory, a transvectant is an invariant formed from ''n'' invariants in ''n'' variables using Cayley's Ω process. Definition If ''Q''1,...,''Q'n'' are functions of ''n'' variables x = (''x''1,...,''x'n'' ... - method/process of constructing covariants References * * {{algebra-stub category:Invariant theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Map
In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as :P(v) = \sum_ \lambda_(v) \cdots \lambda_(v) w_ where the \lambda_: V \to k are linear functionals and the w_ are vectors in ''W''. For example, if W = k^m, then a polynomial mapping can be expressed as P(v) = (P_1(v), \dots, P_m(v)) where the P_i are (scalar-valued) polynomial functions on ''V''. (The abstract definition has an advantage that the map is manifestly free of a choice of basis.) When ''V'', ''W'' are finite-dimensional vector spaces and are viewed as algebraic varieties, then a polynomial mapping is precisely a morphism of algebraic varieties. One fundamental outstanding question regarding polynomial mappings is the Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible. See also *Polynomial functor References *Claudio Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Representation
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. 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. 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 in such a way that the group operation is m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Map
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Of Covariants
In algebra, given an algebraic group ''G'', a ''G''-module ''M'' and a ''G''-algebra ''A'', all over a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ... ''k'', the module of covariants of type ''M'' is the A^G-module : (M \otimes_k A)^G. where -^G refers to taking the elements fixed by the action of ''G''; thus, A^G is the ring of invariants of ''A''. See also * Local cohomology References * M. Brion, ''Sur les modules de covariants'', Ann. Sci. École Norm. Sup. (4) 26 (1993), 1 21. * M. Van den Bergh, ''Modules of covariants'', Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhauser, Basel, pp. 352–362, 1995. Module theory {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Of A Binary Form
In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables ''x'' and ''y'' that remains invariant under the special linear group acting on the variables ''x'' and ''y''. Terminology A binary form (of degree ''n'') is a homogeneous polynomial Σ ()''a''''n''−''i''''x''''n''−''i''''y''''i'' = ''a''''n''''x''''n'' + ()''a''''n''−1''x''''n''−1''y'' + ... + ''a''0''y''''n''. The group ''SL''2(C) acts on these forms by taking ''x'' to ''ax'' + ''by'' and ''y'' to ''cx'' + ''dy''. This induces an action on the space spanned by ''a''0, ..., ''a''''n'' and on the polynomials in these variables. An invariant is a polynomial in these ''n'' + 1 variables ''a''0, ..., ''a''''n'' that is invariant under this action. More generally a covariant is a polynomial in ''a''0, ..., ''a''''n'', ''x'', ''y'' that is invariant, so an invariant is a special case of a cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transvectant
In mathematical invariant theory, a transvectant is an invariant formed from ''n'' invariants in ''n'' variables using Cayley's Ω process. Definition If ''Q''1,...,''Q''''n'' are functions of ''n'' variables x = (''x''1,...,''x''''n'') and ''r'' ≥ 0 is an integer then the ''r''th transvectant of these functions is a function of ''n'' variables given by : tr \Omega^r(Q_1\otimes\cdots \otimes Q_n) where Ω is Cayley's Ω process, the tensor product means take a product of functions with different variables x1,..., x''n'', and tr means set all the vectors x''k'' equal. Examples The zeroth transvectant is the product of the ''n'' functions. The first transvectant is the Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ... of the ''n'' functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
