Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $G$ be a semisimple complex Lie group (or perhaps a reductive algebraic group over $\mathbb{C}$) and $V$ an irreducible finite-dimensional representation of $G$, determined by its highest weight. Is there a complete description of the orbits of $V$, or $\mathbb{P}(V)$, under $G$?

In the case of the adjoint representation of $SL_n$ or $GL_n$, I expect to recover the Jordan canonical form of a matrix, or something of the sort.

share|improve this question
    
In the case of the dual of the adjoint representation, this is intensively studied under the keyword "coadjoint orbit." –  Qiaochu Yuan Jun 19 '13 at 0:28
3  
If you want an idea of how hopeless this question is, look at $GL_n$ acting on pairs of matrices, by conjugation. I admit that that's a reducible representation but irreducibility is not going to help much. –  Allen Knutson Jun 19 '13 at 1:00
    
Allen's cautionary example is a good one. Another example, one that illustrates the complexity by relating it to something geometric, is to look at the (irreducible) representation of $\mathrm{GL}(3,\mathbb{C})$ on the homogeneous polynomials of degree $d$ in $3$ variables. The orbits in this case are essentially the projective equivalence classes of plane curves of degree $d$, which is known to be an enormously complex zoo, with no hope of a normal form. Even the very classical case $d=3$ is nontrivial. –  Robert Bryant Jun 20 '13 at 12:06
add comment

1 Answer

As Allen has indicated, there is no hope to solve this question even for $G=GL_n.$ However, there are some representations that are mild enough to be analogous to the vector (or standard) representation of $GL_n$ on $\mathbb{C}^n$ (all non-zero vectors belong to the same orbit) or to the adjoint representation of $GL_n$, where the classification problem is equivalent to determining the conjugacy classes of matrices. The following classes of "well-behaved" representations, listed in the order of increasing complexity, have been extensively studied: multiplicity-free actions, prehomogeneous vector spaces, Kostant--Rallis actions. In particular, $\mathbb{C}^n, S^{2}\mathbb{C}^n$ and $\Lambda{^2}\mathbb{C}^n$ are multiplicity-free and $M_{n,n}$ is a Kostant--Rallis action. For all these classes, the invariant algebra is either trivial (scalars) or a free commutative algebra. Unfortunately, most representations fall outside of these classes, even if you restrict attention to the irreducible polynomial representations of $GL_n.$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.