Editing Invariant theory (section)

== The nineteenth-century origins ==
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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. 
|source={{harvtxt|Weyl|1939b|loc=p.489}}
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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]], "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.)<ref name="Wolfson 2008 pp. 37–46">{{cite journal | last=Wolfson | first=Paul R. | title=George Boole and the origins of invariant theory | journal=Historia Mathematica | publisher=Elsevier BV | volume=35 | issue=1 | year=2008 | issn=0315-0860 | doi=10.1016/j.hm.2007.06.004 | pages=37–46| doi-access= }}</ref>

Classically, the term "invariant theory" refers to the study of invariant [[algebraic form]]s (equivalently, [[symmetric tensor]]s) for the [[Group action (mathematics)|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.

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&nbsp;''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''&nbsp;=&nbsp;2.

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= 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. 
|source={{harvtxt|Kung|Rota|1984|loc=p.27}}
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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 (mathematician)|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).