nLab
invariant theory

Contents

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

f(ρ g(x 1),,ρ g(x))=f(x 1,,x n). f(\rho_g(x_1), \cdots, \rho_g(x)) = f(x_1, \cdots, x_n) \,.

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

f([y,x 1],,x n)++f(x 1,,[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

Revised on March 1, 2013 20:46:22 by Zoran Škoda (161.53.130.104)