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| '''Invariant theory''' is a branch of [[abstract algebra]] dealing with [[group action|actions]] of [[group (mathematics)|groups]] on [[algebraic variety|algebraic varieties]] from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of [[polynomial function]]s that do not change, or are ''invariant'', under the transformations from a given [[linear group]].
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| Invariant theory of [[finite group]]s has intimate connections with [[Galois theory]]. One of the first major results was the main theorem on the [[symmetric function]]s that described the invariants of the [[symmetric group]] ''S''<sub>''n''</sub> acting on the [[polynomial ring]] '''R'''[''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>] by [[permutation]]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 (algebra)|characteristic]], ideologically close to [[modular representation theory]], is an area of active study, with links to [[algebraic topology]].
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| Invariant theory of [[infinite group]]s is inextricably linked with the development of [[linear algebra]], especially, the theories of [[quadratic form]]s and [[determinant]]s. 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 group]]s has its roots in invariant theory.
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| [[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 group]]s on [[affine variety|affine]] and [[projective variety|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]]s.
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| == The nineteenth-century origins ==
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| {{quote box
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| |align=left
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| |width=33%
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| |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 [[Arthur Cayley|Cayley's]] Jovian head.
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| |source={{harvtxt|Weyl|1939b|loc=p.489}}
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| }}
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| Classically, the term "invariant theory" refers to the study of invariant [[algebraic form]]s (equivalently, [[symmetric tensor]]s) for the [[group action|action]] of [[linear transformation]]s. This was a major field of study in the latter part of the nineteenth century. Current theories relating to the [[symmetric group]] and [[symmetric function]]s, [[commutative algebra]], [[moduli space]]s and the [[representations of Lie groups]] are rooted in this area.
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| In greater detail, given a finite-dimensional [[vector space]] ''V'' of dimension ''n'' we can consider the [[symmetric algebra]] ''S''(''S''<sup>''r''</sup>(''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''<sup>''r''</sup>(''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 form]]s where ''n'' = 2.
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| Other work included that of [[Felix Klein]] in computing the invariant rings of finite group actions on <math>\mathbf{C}^2</math> (the [[binary polyhedral group]]s, classified by the [[ADE classification]]); these are the coordinate rings of [[du Val singularities]].
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| {{quote box
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| |align=right
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| |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.
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| |source={{harvtxt|Kung|Rota|1984|loc=p.27}}
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| }}
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| 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).
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| ==Hilbert's theorems==
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| {{harvtxt|Hilbert|1890}} proved that if ''V'' is a finite dimensional representation of the complex algebraic group ''G'' = SL<sub>''n''</sub>(''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]] ρ from ''R'' to ''R''<sup>''G''</sup> with the properties
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| *''ρ''(1) = 1
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| *''ρ''(''a'' + ''b'') = ''ρ''(''a'') + ''ρ''(''b'')
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| *''ρ''(''ab'') = ''a'' ''ρ''(''b'') whenever ''a'' is an invariant.
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| 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]].
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| 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''<sub>1</sub>,...,''i''<sub>''n''</sub> 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''<sup>''G''</sup> of invariants. Suppose that ''x'' is some homogeneous invariant of degree ''d'' > 0. Then
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| :''x'' = ''a''<sub>1</sub>''i''<sub>1</sub> + ... + ''a''<sub>n</sub>''i''<sub>n</sub>
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| for some ''a''<sub>''j''</sub> in the ring ''R'' because ''x'' is in the ideal ''I''. We can assume that ''a''<sub>''j''</sub> is homogeneous of degree ''d'' − deg ''i''<sub>''j''</sub> for every ''j'' (otherwise, we replace ''a''<sub>''j''</sub> by its homogeneous component of degree ''d'' − deg ''i''<sub>''j''</sub>; if we do this for every ''j'', the equation ''x'' = ''a''<sub>1</sub>''i''<sub>1</sub> + ... + ''a''<sub>''n''</sub>''i''<sub>n</sub> will remain valid). Now, applying the Reynolds operator to ''x'' = ''a''<sub>1</sub>''i''<sub>1</sub> + ... + ''a''<sub>''n''</sub>''i''<sub>n</sub> gives
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| :''x'' = ρ(''a''<sub>1</sub>)''i''<sub>1</sub> + ... + ''ρ''(''a''<sub>''n''</sub>)''i''<sub>''n''</sub> | |
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| We are now going to show that ''x'' lies in the ''R''-algebra generated by ''i''<sub>1</sub>,...,''i''<sub>''n''</sub>.
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| First, let us do this in the case when the elements ρ(''a''<sub>''k''</sub>) all have degree less than ''d''. In this case, they are all in the ''R''-algebra generated by ''i''<sub>1</sub>,...,''i''<sub>''n''</sub> (by our induction assumption). Therefore ''x'' is also in this ''R''-algebra (since ''x'' = ''ρ''(''a''<sub>1</sub>)''i''<sub>1</sub> + ... + ρ(''a''<sub>n</sub>)''i''<sub>n</sub>).
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| In the general case, we cannot be sure that the elements ρ(''a''<sub>''k''</sub>) all have degree less than ''d''. But we can replace each ρ(''a''<sub>''k''</sub>) by its homogeneous component of degree ''d'' − deg ''i''<sub>''j''</sub>. As a result, these modified ρ(''a''<sub>''k''</sub>) 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''<sub>''k''</sub> > 0). The equation ''x'' = ρ(''a''<sub>1</sub>)''i''<sub>1</sub> + ... + ρ(''a''<sub>n</sub>)''i''<sub>n</sub> still holds for our modified ρ(''a''<sub>''k''</sub>), so we can again conclude that ''x'' lies in the ''R''-algebra generated by ''i''<sub>1</sub>,...,''i''<sub>''n''</sub>.
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| Hence, by induction on the degree, all elements of ''R''<sup>''G''</sup> are in the ''R''-algebra generated by ''i''<sub>1</sub>,...,''i''<sub>''n''</sub>.
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| == Geometric invariant theory ==
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| 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.
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| ==See also==
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| * [[Gram's theorem]]
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| * [[invariant theory of finite groups]]
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| * [[representation theory of finite groups]]
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| * [[Molien series]]
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| * [[invariant (mathematics)]]
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| ==References==
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| *{{Citation | last1=Dieudonné | first1=Jean A. | last2=Carrell | first2=James B. | title=Invariant theory, old and new | doi=10.1016/0001-8708(70)90015-0 | year=1970 | journal=Advances in Mathematics | issn=0001-8708 | volume=4 | pages=1–80 | mr=0255525}} Reprinted as {{Citation | last1=Dieudonné | first1=Jean A. | last2=Carrell | first2=James B. | title=Invariant theory, old and new | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-215540-6 | doi=10.1016/0001-8708(70)90015-0 | year=1971 | mr=0279102 | journal=Advances in Mathematics | volume=4 | pages=1–80}}
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| *{{Citation | last1=Dolgachev | first1=Igor | title=Lectures on invariant theory | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-52548-0 | doi=10.1017/CBO9780511615436 | year=2003 | volume=296 | mr=2004511}}
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| *{{Citation | last=Grace|first= J. H.|last2= Young|first2= Alfred | title=The algebra of invariants | location=Cambridge | publisher=Cambridge University Press | year=1903 |url=http://archive.org/details/algebraofinvaria00graciala}}
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| *{{Citation | author=Grosshans, Frank D. | authorlink = Frank Grosshans | title=Algebraic homogeneous spaces and invariant theory | location=New York | publisher=Springer | year=1997 | isbn=3-540-63628-5}}
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| *{{Citation | last1=Kung | first1=Joseph P. S. | last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | title=The invariant theory of binary forms | url=http://www.ams.org/journals/bull/1984-10-01/S0273-0979-1984-15188-7 | doi=10.1090/S0273-0979-1984-15188-7 | year=1984 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=10 | issue=1 | pages=27–85 | mr=722856}}
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| *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | doi=10.1007/BF01208503 | year=1890 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=36 | issue=4 | pages=473–534 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0036&DMDID=DMDLOG_0045}}
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| *{{citation|last=Hilbert|first=D.|title=Über die vollen Invariantensysteme (On Full Invariant Systems)|journal=Math. Annalen|volume=42|issue=3|page=313|year=1893|doi=10.1007/BF01444162 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0042&DMDID=DMDLOG_0034}}
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| *{{Citation | author=Neusel, Mara D.; and Smith, Larry | title=Invariant Theory of Finite Groups | location=Providence, RI | publisher=American Mathematical Society | year=2002 | isbn=0-8218-2916-5}} A recent resource for learning about modular invariants of finite groups.
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| *{{Citation | author=Olver, Peter J. | title=Classical invariant theory | location=Cambridge | publisher=Cambridge University Press | year=1999 | 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.
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| *{{springer|id=i/i052350|title=Invariants, theory of|first=V.L. |last=Popov}}
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| *{{Citation | author=Springer, T. A. | title=Invariant Theory | location=New York | publisher=Springer | year=1977 | isbn=0-387-08242-5}} An older but still useful survey.
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| *{{Citation | author=[[Bernd Sturmfels|Sturmfels, Bernd]] | title=Algorithms in Invariant Theory | location=New York | publisher=Springer | year=1993 | isbn=0-387-82445-6}} A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups. Their Invariants and Representations | url=http://books.google.com/books?isbn=0691057567 | publisher=[[Princeton University Press]] | isbn=978-0-691-05756-9 | year=1939 | mr=0000255}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Invariants | url=http://projecteuclid.org/euclid.dmj/1077491405 | year=1939b | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=5 | pages=489–502 | mr=0000030 | doi=10.1215/S0012-7094-39-00540-5 | issue=3}}
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| == External links ==
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| *H. Kraft, C. Procesi, [http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf Classical Invariant Theory, a Primer]
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| [[Category:Invariant theory|*]]
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