Timeline for Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?
Current License: CC BY-SA 3.0
16 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 20, 2013 at 12:02 | answer | added | Bruce Bartlett | timeline score: 3 | |
| Nov 10, 2012 at 16:51 | vote | accept | Alexander Chervov | ||
| Nov 4, 2012 at 14:53 | comment | added | Allen Knutson | This seems to be two questions. (1) What information can we get from the ring-with-basis of conjugacy classes? Answered several times below: the representation ring, which is not enough to determine the group. (2) How can we categorify the ring-with-basis of conjugacy classes to a category that does determine the group? I don't know, but I like the question! | |
| Oct 28, 2012 at 11:17 | answer | added | Geoff Robinson | timeline score: 4 | |
| Oct 27, 2012 at 22:07 | answer | added | Faisal | timeline score: 6 | |
| Oct 27, 2012 at 21:49 | answer | added | Will Sawin | timeline score: 5 | |
| Oct 27, 2012 at 21:12 | comment | added | Alexander Chervov | @Will I know this, but I cannot relate this with conjunction class product... | |
| Oct 27, 2012 at 21:06 | comment | added | Will Sawin | Because you add representations by adding their characters, you can multiply representations by multiplying their characters, and you can check isomorphism of representations by checking equality of their characters. So the representation ring is the ring generated by the rows of the character table under entrywise addition and multiplication. | |
| Oct 27, 2012 at 21:00 | comment | added | Alexander Chervov | May be I misunderstanding TanakaKrein.... Still why you both so quickly say that "the same info as char table " is it obvious? | |
| Oct 27, 2012 at 20:56 | comment | added | Victor Ostrik | It is my understanding that the ring of conjugacy classes (with basis) carries exactly the same information as character table which in turn is the same information as character ring (with basis). On the other hand the tensor category of representations carries strictly more information: for example nonabelian groups of order 8 have the same character tables (and so the same character rings and conjugacy classes rings) but inequivalent tensor categories of representations (even if one ignores the commutativity constraint). | |
| Oct 27, 2012 at 20:55 | comment | added | Alexander Chervov | Why only on char. table ? | |
| Oct 27, 2012 at 20:52 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
added 4 characters in body
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| Oct 27, 2012 at 20:49 | comment | added | Will Sawin | Both rings are lattices in the center of the group algebra $\mathbb C[G]$. So that's a pretty strong relation between them. There's also a duality between the two lattices, given by the character table / multiplication and the counit. I don't think the ring tells you the category structure, because it doesn't tell you the $\operatorname{Sym}$s and $\wedge$s. The ring depends only on the character table, which is not a complete group invariant. | |
| Oct 27, 2012 at 20:36 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
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| Oct 27, 2012 at 20:31 | history | asked | Alexander Chervov | CC BY-SA 3.0 |