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Hermite Reciprocity
In mathematics, Hermite's law of reciprocity, introduced by , states that the degree ''m'' covariants of a binary form of degree ''n'' correspond to the degree ''n'' covariants of a binary form of degree ''m''. In terms of 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. In essen ... it states that the representations ''S''''m'' ''S''''n'' C2 and ''S''''n'' ''S''''m'' C2 of ''GL''2 are isomorphic. References *{{Citation , last1=Hermite , first1=Charles , title=Sur la theorie des fonctions homogenes à deux indéterminées , authorlink=Charles Hermite , url=http://resolver.sub.uni-goettingen.de/purl?PPN600493962_0009 , year=1854 , journal=Cambridge and Dublin Mathematical Journal , volume=9 , pages=172–217 Invariant theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 equivariant map, SL_2(\mathbb)-equivariant. See also *module of covariants *Invariant of a binary form#Terminology *Transvectant - method/process of constructing covariants References * * {{algebra-stub category:Invariant theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
