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Consider the action of $GL(n,k)$ on the set $M\times M$ where $M$ is the set of all $n$-by-$n$ matrices over $k$ given by $g\cdot(h,l) \mapsto (ghg^{-1}, glg^t)$.

Individually these actions are well-studied and good descriptions are known.

I would like to know if a decent description of this simultaneous action is known.

Here I elaborate what I mean: if we look at only conjugacy action, i.e., the first component only, one can describe representatives of each orbit using rational canonical form theory. There are similar results for the second component action, e.g., for a symmetric matrix the orbit representative could be taken as a diagonal (Gram-Schmidt like results) matrix.

I need "good" orbit representatives for the simultaneous action. For me a matrix is good if it is sparse; having a lot of zeros.

Thanks a lot!

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    $\begingroup$ What do you mean by a decent description? $\endgroup$
    – abx
    Commented Jun 16, 2015 at 10:07
  • $\begingroup$ @abx I edited the question to make it a bit more precise. Thanks! $\endgroup$
    – Anupam Singh
    Commented Jun 16, 2015 at 10:31
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    $\begingroup$ Should $ghg^t$ be $glg^t$? With this change, I think it's a reasonable question, but since it's so close to the wild problem of classifying pairs of matrices up to simultaneous conjugacy, I doubt if there is a very satisfactory answer. $\endgroup$
    – Mark Wildon
    Commented Jun 16, 2015 at 10:37
  • $\begingroup$ As with Mark's comment, I assume that you mean $glg^t$ for the second component, but are you sure that you don't also want to assume that $l$ is symmetric to start out with? (Otherwise, you can split $l$ into its symmetric and skew-symmetric parts, and they will transform separately.) You do seem to be assuming that $l$ is symmetric further down in the question. $\endgroup$
    – Robert Bryant
    Commented Jun 16, 2015 at 11:10
  • $\begingroup$ Yes, the second component should be $glg^t$. And no I am not assuming $l$ to be symmetric in general. Thank you for your attention! $\endgroup$
    – Anupam Singh
    Commented Jun 17, 2015 at 5:14

1 Answer 1

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The following paper describes all invariants for the diagonal conjugation action on $m$-tuples of matrices. This is not exactly what you want (because you second action is not conjugation), but you can try to adapt the methods to describe a full set of invariants.

  • MR0419491 (54 #7512)
    Procesi, C. The invariant theory of n×n matrices. Advances in Math. 19 (1976), no. 3, 306–381.

  • MR0419490 (54 #7511) Reviewed Procesi, Claudio The invariants of n×n matrices. Bull. Amer. Math. Soc. 82 (1976), no. 6, 891–892.

From the review: The classical groups $GL(n,\mathbb C)$, and their maximal compact subgroups act by conjugation on m-tuples of $n\times n$ complex matrices $(X_1,\dots,X_m)$. The author announces without proof results about the corresponding invariants. $\dots$

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    $\begingroup$ It would be interesting to try to work out the polynomial invariants for the OP's problem along these lines. This would still not answer his question, because he is looking for canonical forms for pairs under this action, which, I fear, will end up being a wild problem. $\endgroup$
    – Amritanshu Prasad
    Commented Jun 16, 2015 at 15:53
  • $\begingroup$ I am also aware about classification under second action only. Here is a reference: ams.org/mathscinet/search/… $\endgroup$
    – Anupam Singh
    Commented Jun 17, 2015 at 5:20
  • $\begingroup$ another reference: ams.org/mathscinet/search/… $\endgroup$
    – Anupam Singh
    Commented Jun 17, 2015 at 5:25

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