theory of invariants
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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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
dealing with
actions Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fi ...
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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, 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 functions that do not change, or are ''invariant'', under the transformations from a given
linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...
. For example, if we consider the action of the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
''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 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 ...
, 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
\pi:G \to GL(V), which induces a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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 In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element of ...
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 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 ...
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 groups has intimate connections with
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
. One of the first major results was the main theorem on the
symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
s that described the invariants of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite 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 permutations of the variables. More generally, the
Chevalley–Shephard–Todd theorem 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 vector space is a polynomial ring if and only if the group is generated by pseudo ...
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 ...
, 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 group In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order. Examples * (Z, +), the group of integers with addition is infi ...
s 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 matrices ...
, especially, the theories of quadratic forms 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 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 ...
of
semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s 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 David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his
geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
. 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_n ...
s on
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
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 Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...
and his school. A prominent example of this circle of ideas is given by the theory of standard monomials.


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 expone ...
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 form In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s (equivalently,
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
s) for the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s. This was a major field of study in the latter part of the nineteenth century. Current theories relating to the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
and
symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
s,
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 spaces and the
representations of Lie groups In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vecto ...
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 In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
''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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
in computing the invariant rings of finite group actions on \mathbf^2 (the
binary polyhedral group In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries ...
s, 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, 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 In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the ...
of ''G'' acting on the ring of polynomials ''R'' = ''S''(''V'') is finitely generated. His proof used the Reynolds operator 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 In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...
. 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 In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
is due to
David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
, 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 spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s and monopoles.


See also

* Gram's theorem *
Representation theory of finite groups The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
* Molien series *
Invariant (mathematics) In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects ...
*
Invariant of a binary form In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables ''x'' and ''y'' that remains invariant under the special linear group acting on the variables ''x'' and ''y''. T ...
*
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