Timeline for Can the image of a Schur functor always be made an irreducible representation?
Current License: CC BY-SA 2.5
13 events
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| Jul 20, 2010 at 0:44 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 11 characters in body
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| Jul 19, 2010 at 18:45 | vote | accept | Qiaochu Yuan | ||
| Jul 19, 2010 at 17:17 | comment | added | Jim Humphreys | The restated question still isn't quite clear to me, being worded in terms of an irreducible representation V of a finite (rather than general linear) group. Anyway, the arXiv preprint by Guralnick and Tiep is located at math.GR/0502080, to appear in the electronic journal Representation Theory. | |
| Jul 19, 2010 at 14:32 | answer | added | David E Speyer | timeline score: 3 | |
| Jul 19, 2010 at 4:30 | answer | added | Theo Johnson-Freyd | timeline score: 1 | |
| Jul 18, 2010 at 22:57 | comment | added | moonface | The answer is no, for the sixth symmetric power in characteristic zero. But I don't know if there is an easy proof. See "Symmetric powers and a problem of Kollar and Larsen," by Guralnick and Thiep. | |
| Jul 18, 2010 at 22:40 | comment | added | Qiaochu Yuan | I have edited the statement of the question to reflect my motivation more accurately. | |
| Jul 18, 2010 at 22:40 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
Improved the statement of the question.
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| Jul 18, 2010 at 22:31 | comment | added | Qiaochu Yuan | @Ben: Thanks. @Jim: The motivation is vaguer than the question, which is why I wasn't sure whether to put it in. For any representation V of a finite group G we know that V^{\otimes k} decomposes into parts corresponding to the irreps of GL(V), which by Schur-Weyl duality can be labeled by some irreps of S_k. I want to know if this is the best one can do for a "generic" group G, e.g. whether each of these representations is actually irreducible for some finite group G and some representation V. If I've got the statement of the problem right, this is equivalent to f being unbounded. | |
| Jul 18, 2010 at 22:08 | comment | added | Jim Humphreys |
Ben's edit makes the question more precise, but I'm unclear about the motivation for it. Would there be interesting consequences (for general linear groups or for finite groups) if $f$ can be computed explicitly or if the boundedness is somehow shown to be true or false? The question is intriguing, but also puzzling. Classical Schur-Weyl theory gives good algorithmic knowledge of the constituents of tensor powers of $V$` for general linear groups over $\mathbb{C}$; but neither this nor the finite subgroups are known in closed form as $n$ grows (eventually all finite groups appear).
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| Jul 18, 2010 at 22:03 | comment | added | Ben Webster♦ | I hope you don't mind too much that I made a little edit for clarity. I found the original statement very confusing. | |
| Jul 18, 2010 at 21:26 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
added 43 characters in body
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| Jul 18, 2010 at 20:49 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |