Timeline for Subalgebras of matrices
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2022 at 7:37 | comment | added | David Roberts♦ | The link in David Speyer's comment is broken, here's a replacement: arxiv.org/abs/math/0608491 | |
| Jun 11, 2010 at 3:12 | comment | added | Victor Protsak | Mariano, I think that we (and MO in general) are quite a bit above OP's sophistication level. | |
| Jun 10, 2010 at 6:32 | comment | added | Mariano Suárez-Álvarez | Hmm. Again, I was not trying to say anything new about anything nor do I suspect any form of non-triviality coming out of the construction. I was just pointing one usual approach to construct a paramtrization of objects, that does, in all likelyhood in this instance, not work. You, as we can see, are well aware of the construction and its limitations---the OP, on the other hand, might well not be: s/he was my audience :) | |
| Jun 10, 2010 at 6:32 | comment | added | Bugs Bunny | I have just asked it as a question... | |
| Jun 10, 2010 at 6:12 | comment | added | Bugs Bunny | The group is quite large. Besides I cannot think of any continuous invariant of $d$-dimensional algebras, can you? This is why I suspect trivality... You may do much better by taking a quotient of a single component but describing all the components is an interesting task in its own. | |
| Jun 10, 2010 at 6:07 | comment | added | Bugs Bunny | Sure! d-dimensional subalgebras form a closed $GL$-equivariant subset of the grassmanian whose equations and tangent spaces, you can easily describe. But all you say there is "subalgebra" but in a funny geometric language, which some people like nevertheless. But can you say anything new? For instance, can you show that your quotient is not just a bunch of reduced points:-)? | |
| Jun 9, 2010 at 20:38 | comment | added | Mariano Suárez-Álvarez | @BB: I did not expect sunshine, but was only describing the obvious approach to obtaining a set of moduli :) | |
| Jun 9, 2010 at 18:04 | comment | added | Bugs Bunny | Mariano, this quotient space will bring you no sunshine. Think, for example, of n-tuples of matrices: we all know and love the basic invariants (traces of products) but the problem of classifying them is as wild as a hare. | |
| Jun 8, 2010 at 20:32 | comment | added | Mariano Suárez-Álvarez | Ah! Thanks for the reference, David. | |
| Jun 8, 2010 at 15:44 | comment | added | Victor Protsak | I think that at least part of the problem is that the geometric invariant theory quotient only captures generic orbits, whereas for the true classification, you need to know $\textit{all}$ orbits. So the quotient in the sense of classifying all isomorphism types will be a horrible non-separated mess with pieces of different dimensions. This may conceivably happen even if the generic orbits have a manageable description (e.g. think of $AB=BA$ where $A$ and $B$ are semisimple). | |
| Jun 8, 2010 at 14:09 | comment | added | David E Speyer | This variety has been studied; see Remark 1.2 in front.math.ucdavis.edu/0608.5491 for a survey of the literature. | |
| Jun 8, 2010 at 12:40 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
added 451 characters in body
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| Jun 8, 2010 at 12:28 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 2.5 |