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Invariant theory is a branch of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
dealing with actions of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field k (which in classical invariant theory was usually assumed to be the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s). 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 V, then the group action of G on V produces an action on k /math> by the following formula: :(g \cdot f)(x) := f(g^ (x)) \qquad \forall x \in V, g \in G, f\in k With this action it is natural to consider the subspace of all polynomial functions which are invariant under this group action, in other words the set of polynomials such that g\cdot f = f for all g\in G. This space of invariant polynomials is denoted k G. First problem of invariant theory: Is k G a finitely generated algebra over k? For example, if G=SL_n and V=M_n the space of square matrices, and the action of G on V is given by left multiplication, then k G is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, k G is finitely generated over k. If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over k /math>. Invariant theory of
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
s 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 S_n acting on the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
R _1, \ldots, x_nby
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s 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 Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ha ...
, is an area of active study, with links to
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. Invariant theory of infinite groups is inextricably linked with the development of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, especially, the theories of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s and
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s. Another subject with strong mutual influence was
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, 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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
'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 seen to encompass the theory of actions of
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I ...
s 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 monomial In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of ...
s.


Examples

Simple examples of invariant theory come from computing the invariant
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s from a group action. For example, consider the \mathbb/2\mathbb-action on \mathbb ,y/math> sending : \begin x\mapsto -x && y \mapsto -y \end Then, since x^2,xy,y^2 are the lowest degree monomials which are invariant, we have that :\mathbb ,y \cong \mathbb ^2,xy,y^2\cong \frac This example forms the basis for doing many computations.


The nineteenth-century origins

Cayley first established invariant theory in his "On the Theory of Linear Transformations (1845)." In the opening of his paper, Cayley credits an 1841 paper of
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
, "investigations were suggested to me by a very elegant paper on the same subject... by Mr Boole." (Boole's paper was Exposition of a General Theory of Linear Transformations, Cambridge Mathematical Journal.) Classically, the term "invariant theory" refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenth century. Current theories relating to the symmetric group and symmetric functions,
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
,
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s and the representations of Lie groups are rooted in this area. In greater detail, given a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''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''). It is actually more accurate to consider the relative invariants of GL(''V''), or representations of SL(''V''), if we are going to speak of ''invariants'': that is 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''(''S''''r''(''V'')) for the 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 GL(''V'') on S(''V''); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied was invariants of binary forms where ''n'' = 2. Other work included that of Felix Klein in computing the invariant rings of finite group actions on \mathbf^2 (the binary polyhedral groups, classified by the
ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
); these are the coordinate rings of du Val singularities. The work of
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, proving that ''I''(''V'') was finitely presented in many cases, almost put an end to classical invariant theory for several decades, though the classical epoch in the subject 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).


Hilbert's theorems

proved that if ''V'' is a finite-dimensional representation of the complex algebraic group ''G'' = SL''n''(''C'') then the ring of invariants of ''G'' acting on the ring of polynomials ''R'' = ''S''(''V'') is finitely generated. His proof used the
Reynolds operator In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountere ...
ρ from ''R'' to ''R''''G'' with the properties *''ρ''(1) = 1 *''ρ''(''a'' + ''b'') = ''ρ''(''a'') + ''ρ''(''b'') *''ρ''(''ab'') = ''a'' ''ρ''(''b'') whenever ''a'' is an invariant. Hilbert constructed the Reynolds operator explicitly using Cayley's omega process Ω, though now it is more common to construct ρ indirectly as follows: for compact groups ''G'', the Reynolds operator is given by taking the average over ''G'', and non-compact reductive groups can be reduced to the case of compact groups using Weyl's unitarian trick. Given the Reynolds operator, Hilbert's theorem is proved as follows. The ring ''R'' is a polynomial ring so is graded by degrees, and the ideal ''I'' is defined to be the ideal generated by the homogeneous invariants of positive degrees. By Hilbert's basis theorem the ideal ''I'' is finitely generated (as an ideal). Hence, ''I'' is finitely generated ''by finitely many invariants of G'' (because if we are given any – possibly infinite – subset ''S'' that generates a finitely generated ideal ''I'', then ''I'' is already generated by some finite subset of ''S''). Let ''i''1,...,''i''''n'' be a finite set of invariants of ''G'' generating ''I'' (as an ideal). The key idea is to show that these generate the ring ''R''''G'' of invariants. Suppose that ''x'' is some homogeneous invariant of degree ''d'' > 0. Then :''x'' = ''a''1''i''1 + ... + ''a''n''i''n for some ''a''''j'' in the ring ''R'' because ''x'' is in the ideal ''I''. We can assume that ''a''''j'' is homogeneous of degree ''d'' − deg ''i''''j'' for every ''j'' (otherwise, we replace ''a''''j'' by its homogeneous component of degree ''d'' − deg ''i''''j''; if we do this for every ''j'', the equation ''x'' = ''a''1''i''1 + ... + ''a''''n''''i''n will remain valid). Now, applying the Reynolds operator to ''x'' = ''a''1''i''1 + ... + ''a''''n''''i''n gives :''x'' = ρ(''a''1)''i''1 + ... + ''ρ''(''a''''n'')''i''''n'' We are now going to show that ''x'' lies in the ''R''-algebra generated by ''i''1,...,''i''''n''. First, let us do this in the case when the elements ρ(''a''''k'') all have degree less than ''d''. In this case, they are all in the ''R''-algebra generated by ''i''1,...,''i''''n'' (by our induction assumption). Therefore, ''x'' is also in this ''R''-algebra (since ''x'' = ''ρ''(''a''1)''i''1 + ... + ρ(''a''n)''i''n). In the general case, we cannot be sure that the elements ρ(''a''''k'') all have degree less than ''d''. But we can replace each ρ(''a''''k'') by its homogeneous component of degree ''d'' − deg ''i''''j''. As a result, these modified ρ(''a''''k'') are still ''G''-invariants (because every homogeneous component of a ''G''-invariant is a ''G''-invariant) and have degree less than ''d'' (since deg ''i''''k'' > 0). The equation ''x'' = ρ(''a''1)''i''1 + ... + ρ(''a''n)''i''n still holds for our modified ρ(''a''''k''), so we can again conclude that ''x'' lies in the ''R''-algebra generated by ''i''1,...,''i''''n''. Hence, by induction on the degree, all elements of ''R''''G'' are in the ''R''-algebra generated by ''i''1,...,''i''''n''.


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. 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. One motivation was to construct
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
and equivariant topology, and was used to construct moduli spaces of objects in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, such as instantons and
monopoles Monopole may refer to: * Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole * Monopole (mathematics), a connection over a principal bundle G with a section (the Higgs field) ...
.


See also

*
Gram's theorem In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', wi ...
* Representation theory of finite groups * Molien series * Invariant (mathematics) * Invariant of a binary form *
Invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping ...
* First and second fundamental theorems of invariant theory


References

* Reprinted as * * * * * * * A recent resource for learning about modular invariants of finite groups. * An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omega process starting at page 87. * * An older but still useful survey. * 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
* V. L. Popov, E. B. Vinberg, ``Invariant Theory", in ''Algebraic geometry''. IV. Encyclopaedia of Mathematical Sciences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994; vi+284 pp.; {{ISBN, 3-540-54682-0