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In greater detail, given a finite-dimensional vector space V we can consider the symmetric algebra S(V), and the action on it of GL(V). It is actually more accurate to consider the projective representation of GL(V), if we are going to speak of invariants: that's because a scalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar. The point is then to define the subalgebra of invariants I(V) for the (projective) action. We are, in classical language, looking at n-ary r-ics, where n is the dimension of V.
These days it might be more natural to look to decompose the degree r part of S(V) into irreducible representations of GL(V): the formulation just given is the same as saying we are concerned only with the occurrence of one-dimensional representations. The representation theory required came later, though, with Issai SchurIssai Schur ( January 10, 1875 in Mogilyov January 10, 1941 in Tel Aviv) was a mathematician who worked in Germany for most of his life. He studied at Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at Bonn, professor.
To give the broader picture: what was actually studied in the classical phase of invariant theory related in fact to
where V* is the dual vector space to V. That is, the invariants as polynomials involved a contragredient set of coordinates, transforming in a dual fashion.
It is customary to say that the work of David HilbertDavid Hilbert ( January 23, 1862 February 14, 1943) was a German mathematician born in Wehlau, near Konigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th cen, proving abstractly that I(V) was finitely presented, put an end to classical invariant theory. That is far from being true: the classical epoch in the subject may have continued to the final publications of Alfred Young , more than 50 years later. Explicit calculations for particular purposes have been known in modern times (for example Shioda, with the binary octavics).
The modern formulation of geometric invariant theory is due to David MumfordDavid Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He is currently a professor in the Division of Applied Mathematics at Brown Uni, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtle theory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In a separate development the symbolic method of invariant theoryIn mathematics, the symbolic method in invariant theory is a highly formal algorithm developed in the 19th century for computing form invariants — invariants of algebraic forms. It is based on repeated applications of the " Cayley Omega process" (which in, an apparently heuristic combinatorial notation, has been rehabilitated.
For the invariant theory of finite groupGroup theory In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable grous, see Molien series . See also: invariantInvariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry..
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