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• Invariant_theory abstract "Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties 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. Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group Sn acting on the polynomial ring R[x1, …, xn] by permutations of the variables. More generally, the Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invariant theory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. The case of positive characteristic, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology.Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. One of the highlights of this relationship is the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creation of a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions in more constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is presently seen to encompass the theory of actions of linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rota and his school. A prominent example of this circle of ideas is given by the theory of standard monomials.".
• Invariant_theory wikiPageExternalLink algebraofinvaria00graciala.
• Invariant_theory wikiPageExternalLink books?isbn=0691057567.
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• Invariant_theory wikiPageExternalLink ?PPN=PPN235181684_0042&DMDID=DMDLOG_0034.
• Invariant_theory wikiPageExternalLink 1077491405.
• Invariant_theory wikiPageExternalLink S0273-0979-1984-15188-7.
• Invariant_theory wikiPageExternalLink KP-Primer.pdf.
• Invariant_theory wikiPageID "363762".
• Invariant_theory wikiPageRevisionID "599144359".
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• Invariant_theory first "V.L.".
• Invariant_theory hasPhotoCollection Invariant_theory.
• Invariant_theory id "i/i052350".
• Invariant_theory last "Popov".
• Invariant_theory quote "Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics.".
• Invariant_theory quote "The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head.".
• Invariant_theory title "Invariants, theory of".
• Invariant_theory width "33.0".
• Invariant_theory subject Category:Invariant_theory.
• Invariant_theory comment "Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties 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. Invariant theory of finite groups has intimate connections with Galois theory.".
• Invariant_theory label "Invariant theory".
• Invariant_theory label "Invariantentheorie".
• Invariant_theory label "Théorie des invariants".
• Invariant_theory label "不變量理論".
• Invariant_theory sameAs Teorie_invariantů.
• Invariant_theory sameAs Théorie_des_invariants.
• Invariant_theory sameAs Invariantentheorie.
• Invariant_theory sameAs m.01_r17.
• Invariant_theory sameAs Q1855669.
• Invariant_theory sameAs Q1855669.
• Invariant_theory wasDerivedFrom Invariant_theory?oldid=599144359.
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