nLab
invariant theory

Idea

Invariant theory studies invariants: algebraic entities – for instance elements in a ring – invariant under some group action.

In geometric invariant theory one regards the algebraic objects as formally dual to a geometric space and interprets the invariants as functions on a quotient space.

Examples

Let $V$ be a (graded) vector space equipped with the action $\rho$ of a group $G$. This induces an action on the symmetric tensor powers ${\mathrm{Sym}}^{n}V$. A linear map out of sums of such symmetric powers is called a polynomial on $V$. It is an invariant polynomial if it is invariant under the group action, hence if for every $g\in G$ we have (writing it for a homogeneous polynomial for convenience)

$$f({\rho }_g(x_1),\cdots ,{\rho }_g(x))=f(x_1,\cdots ,x_n).$$f(\rho_g(x_1), \cdots, \rho_g(x)) = f(x_1, \cdots, x_n) \,.

For instance if $G$ is a Lie group and $V=𝔤$ is its Lie algebra, there is a canonical adjoint action $\rho =\mathrm{Ad}$ of $G$ on ${\mathrm{Sym}}^n𝔤$. The corresponding invariant polynomials play a central role in Lie theory, notably via Chern-Weil theory. In this case the $\mathrm{Ad}$-invariance is often expressed in its differential form (obtained by differentiating the above equation at the neutral element), where it says that for all $y\in 𝔤$ we have

$$f([y,x_1],\cdots ,x_n)+\cdots +f(x_1,\cdots ,[y,x_n])=0.$$f([y,x_1], \cdots, x_n) + \cdots + f(x_1, \cdots, [y,x_n]) = 0 \,.

References

• Jean Dieudonné, James B. Carrell, Invariant theory, old and new, Advances in Mathematics 4 (1970) 1-80. Also published as a book (1971).

• Hanspeter Kraft, Claudio Procesi, Classical invariant theory – A primer (pdf)

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

• Igor Dolgachev, Lectures on invariant theory, ps

• William Crawley-Boevey, Lectures on representation theory and invariant theory (pdf)

• David Mumford, John Fogarty, Frances Clare Kirwan, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer-Verlag

• Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp.

• Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 1989, том 55, с. 137–309 pdf

• B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809 MR0311837 doi