Timeline for Does it make sense that "Representations of groups over finite ring" ?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2011 at 4:38 | vote | accept | Lee | ||
| Jun 29, 2011 at 4:38 | vote | accept | Lee | ||
| Jun 29, 2011 at 4:38 | |||||
| Jun 29, 2011 at 3:17 | comment | added | S. Carnahan♦ | Your revised question amounts to studying maps of the form $\mathbb{F}_q[H] \to M_n(R)$, where $H$ is the Heisenberg group, and $R$ is your finite ring. This certainly makes sense, and Curtis and Reiner's book might be a decent reference. | |
| Jun 29, 2011 at 1:08 | history | edited | Lee |
edited tags
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| Jun 29, 2011 at 1:01 | history | edited | Lee | CC BY-SA 3.0 |
made it more concretely
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| Jun 29, 2011 at 0:32 | comment | added | Lee | @Richard, David and Dan: I was meant to say maps from groups to matrices over a finite ring. I'll edit my question. | |
| Jun 28, 2011 at 22:51 | comment | added | Dan Ramras | Representations of groups over finite fields (i.e. homomorphisms from a group to GL_n (F), where F is a finite field) is a big subject. David Benson has good books about it. But I'm still unclear on whether the OP was talking about maps from groups to matrices over a finite ring, or whether something else was intended. | |
| Jun 28, 2011 at 20:59 | comment | added | Johannes Hahn | @David: That's not necessarily true. In modular representation theory $\mathcal{O}[G]$ (where $\mathcal{O}$ is a discrete valuation) and its modules are useful and important objects linking the representation theory over the quotient field (which is usually chosen to have characteristic 0) and the residue field (which is usually chosen to have positive characteristic) of $\mathcal{O}$. | |
| Jun 28, 2011 at 19:33 | answer | added | David White | timeline score: 2 | |
| Jun 28, 2011 at 18:55 | comment | added | David White | I'm a bit confused by the pronouns. I think of representation theory of a group $G$ over $\mathbb{C}$ as the usually studied representation theory. Is the question whether we can replace $\mathbb{C}$ by a finite ring or whether we can replace $G$ by a finite ring? In the former case I think you gain nothing since your ring would need inverses to have a good theory and a finite division ring is a field. I'm searching now for a reference for the latter interpretation | |
| Jun 28, 2011 at 16:54 | comment | added | Richard Rast | I believe Lee was wondering whether there was some good reason we shouldn't do representations over finite rings (there isn't), and if not, he would like a good reference to go read about them. I don't have one, but someone should post one. | |
| Jun 28, 2011 at 14:07 | comment | added | Theo Johnson-Freyd | Dear Lee, There is, of course, much to be said about the representation theory of groups over finite rings. Indeed, there's much more than can be said in a MathOverflow post, and as such, your question is a bit more open-ended than would be best for this site. Please read mathoverflow.net/howtoask and consider revising this question to be more specific: what do you already know, and what in particular would you like to know? | |
| Jun 28, 2011 at 13:42 | history | edited | Lee | CC BY-SA 3.0 |
added 28 characters in body
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| Jun 28, 2011 at 13:36 | history | asked | Lee | CC BY-SA 3.0 |