Invariant theory

Invariant theory is a branch of

is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SL_n.

Introduction

Let ${\displaystyle G}$ be a group, and ${\displaystyle V}$ a finite-dimensional vector space over a field ${\displaystyle k}$ (which in classical invariant theory was usually assumed to be the complex numbers). A representation of ${\displaystyle G}$ in ${\displaystyle V}$ is a group homomorphism ${\displaystyle \pi :G\to GL(V)}$, which induces a

of ${\displaystyle G}$ on ${\displaystyle V}$. If ${\displaystyle k[V]}$ is the space of polynomial functions on ${\displaystyle V}$, then the group action of ${\displaystyle G}$ on ${\displaystyle V}$ produces an action on ${\displaystyle k[V]}$ by the following formula:

${\displaystyle (g\cdot f)(x):=f(g^{-1}(x))\qquad \forall x\in V,g\in G,f\in k[V].}$

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 ${\displaystyle g\cdot f=f}$ for all ${\displaystyle g\in G}$. This space of invariant polynomials is denoted ${\displaystyle k[V]^{G}}$.

First problem of invariant theory:^[1] Is ${\displaystyle k[V]^{G}}$ a finitely generated algebra over ${\displaystyle k}$?

For example, if ${\displaystyle G=SL_{n}}$ and ${\displaystyle V=M_{n}}$ the space of square matrices, and the action of ${\displaystyle G}$ on ${\displaystyle V}$ is given by left multiplication, then ${\displaystyle k[V]^{G}}$ is isomorphic to a polynomial algebra 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, ${\displaystyle k[V]^{G}}$ is finitely generated over ${\displaystyle k}$.

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

) is finitely generated over ${\displaystyle k[V]}$.

Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group ${\displaystyle S_{n}}$ acting on the polynomial ring ${\displaystyle R[x_{1},\ldots ,x_{n}}$] by permutations 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, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology.

Invariant theory of

has its roots in invariant theory.

.

Examples

Simple examples of invariant theory come from computing the invariant monomials from a group action. For example, consider the ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$-action on ${\displaystyle \mathbb {C} [x,y]}$ sending

${\displaystyle {\begin{aligned}x\mapsto -x&&y\mapsto -y\end{aligned}}}$

Then, since ${\displaystyle x^{2},xy,y^{2}}$ are the lowest degree monomials which are invariant, we have that

${\displaystyle \mathbb {C} [x,y]^{\mathbb {Z} /2\mathbb {Z} }\cong \mathbb {C} [x^{2},xy,y^{2}]\cong {\frac {\mathbb {C} [a,b,c]}{(ac-b^{2})}}}$

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, "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.)^[2]

Classically, the term "invariant theory" refers to the study of invariant

are rooted in this area.

In greater detail, given a finite-dimensional

where n = 2.

Other work included that of Felix Klein in computing the invariant rings of finite group actions on ${\displaystyle \mathbf {C} ^{2}}$ (the

.

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

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

.

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 i₁,...,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 = a₁i₁ + ... + a_ni_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 = a₁i₁ + ... + a_ni_n will remain valid). Now, applying the Reynolds operator to x = a₁i₁ + ... + a_ni_n gives

x = ρ(a₁)i₁ + ... + ρ(a_n)i_n

We are now going to show that x lies in the R-algebra generated by i₁,...,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 i₁,...,i_n (by our induction assumption). Therefore, x is also in this R-algebra (since x = ρ(a₁)i₁ + ... + 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(a₁)i₁ + ... + 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 i₁,...,i_n.

Hence, by induction on the degree, all elements of R^G are in the R-algebra generated by i₁,...,i_n.

Geometric invariant theory

The modern formulation of

, 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 instantons and monopoles.

See also

References

• MR 0279102
• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296,
  MR 2004511
• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press
• ISBN 3-540-63628-5
• ISSN 0025-5831
• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen, 42 (3): 313,
  doi:10.1007/BF01444162
• Kung, Joseph P. S.;
  MR 0722856
• ISBN 0-8218-2916-5
  A recent resource for learning about modular invariants of finite groups.
• Omega process
  starting at page 87.
• Popov, V.L. (2001) [1994], "Invariants, theory of", Encyclopedia of Mathematics, EMS Press
• Springer, T. A. (1977), Invariant Theory, New York: Springer,
  ISBN 0-387-08242-5
  An older but still useful survey.
• ISBN 0-387-82445-6
  A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.
• MR 0000255
• MR 0000030

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