Invariant theory is a branch of abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect on functions. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed

Invariant theory of finite groups has intimate connections with Galois theory. In Mathematics, a finite group is a group which has finitely many elements In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group S_n acting on the polynomial ring R[x₁, …, x_n] by permutations of the variables. In Mathematics, the term "symmetric function" can mean two different things In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In several fields of Mathematics the term permutation is used with different but closely related meanings More generally, Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. In mathematics the Chevalley–Shephard–Todd theorem in Invariant theory of Finite groups states that the ring of invariants of a finite group acting on a complex 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. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's Modular representation theory is a branch of Mathematics, and is that part of Representation theory which studies Linear representations of Finite group Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Linear algebra is the branch of Mathematics concerned with In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. One of the highlights of this relationship is the symbolic method. In Mathematics, the symbolic method in Invariant theory is a highly formal Algorithm developed in the 19th century for computing form invariants Representation theory of semisimple Lie groups has its roots in invariant theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i

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. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into In Mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in Algebraic geometry, used to construct moduli 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. In Mathematics, a linear algebraic group is a Subgroup of the group of invertible n × n matrices (under Matrix multiplication This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety 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. Gian-Carlo Rota ( April 27, 1932 &ndash April 18, 1999, known as Juan Carlos Rota A prominent example of this circle of ideas is given by the theory of standard monomials.

Contents 1 The nineteenth century origins 2 Geometric invariant theory 3 See also 4 References 5 External links

The nineteenth century origins

Classically, the term "invariant theory" refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same In Mathematics, a symmetric tensor is a Tensor that is invariant under a Permutation of its vector arguments In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This was a major field of study in the latter part of the nineteenth century, when it appeared that progress in this particular field (out of any number of possible mathematical formulations of invariance with respect to symmetry) was the key algorithmic discipline. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Despite some heroic efforts that promise was not fulfilled but many spin-off advances are connected to this research. Current theories relating to the symmetric group and symmetric functions, commutative algebra, moduli spaces and the representations of Lie groups are rooted in this area. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, the term "symmetric function" can mean two different things Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of In Mathematics and Theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous Symmetry

In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebra S(S^r(V)) of the polynomials of degree r over V, and the action on it of GL(V). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field It is actually more accurate to consider the projective representation of GL(V), or representations of SL(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. In the mathematical field of Representation theory, a projective representation of a group G on a Vector space V over a The point is then to define the subalgebra of invariants I(S^r(V)) for the (projective) action. We are, in classical language, looking at invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of SL(V) on S(V); this is an uninteresting problem as the only such invariants are constants. )

It is customary to say that the work of David Hilbert, proving abstractly that I(V) was finitely presented, put an end to classical invariant theory. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. Alfred Young ( 16 April 1873 – 15 December 1940) was a Mathematician. Explicit calculations for particular purposes have been known in modern times (for example Shioda, with the binary octavics).

Geometric invariant theory

The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. In Mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in Algebraic geometry, used to construct moduli David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into 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 theory, an apparently heuristic combinatorial notation, has been rehabilitated. In Mathematics, the symbolic method in Invariant theory is a highly formal Algorithm developed in the 19th century for computing form invariants

See also

• invariant theory of finite groups
• Molien series
• invariant (mathematics)

References

• Grace, J. In Mathematics, a Molien series is a Generating function attached to a Linear representation &rho of a group G on a Finite-dimensional In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. H. ; and Young, Alfred (1903). The algebra of invariants. Cambridge: Cambridge University Press.   A classic monograph.

• Grosshans, Frank D. (1997). Algebraic homogeneous spaces and invariant theory. New York: Springer. ISBN 3-540-63628-5.  

• Neusel, Mara D. ; and Smith, Larry (2002). Invariant Theory of Finite Groups. Providence, RI: American Mathematical Society. ISBN 0-8218-2916-5.   A recent resource for learning about modular invariants of finite groups.

• Olver, Peter J. (1999). Classical invariant theory. Cambridge: Cambridge University Press. ISBN 0-521-55821-2.   An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omega process starting at page 87. In Mathematics, the symbolic method in Invariant theory is a highly formal Algorithm developed in the 19th century for computing form invariants

• Popov, V. L. (2001), “Invariants, theory of”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

• Springer, T. The Encyclopaedia of Mathematics is a large reference work in Mathematics. A. (1977). Invariant Theory. New York: Springer. ISBN 0-387-08242-5.   An older but still useful survey.

• Sturmfels, Bernd (1993). Bernd Sturmfels (b March 28, 1962, Kassel, Germany) is a Professor of Mathematics and Computer Science at the University Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6.   A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.

External links

H. Kraft, C. Procesi, Classical Invariant Theory, a Primer

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