Timeline for Non-trivial representation of second-smallest dimension
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2013 at 8:53 | vote | accept | CommunityBot | ||
| Jan 11, 2013 at 12:41 | answer | added | Jim Humphreys | timeline score: 2 | |
| Jan 11, 2013 at 12:41 | answer | added | Mikhail Borovoi | timeline score: 2 | |
| Jan 11, 2013 at 10:46 | history | edited | user30435 | CC BY-SA 3.0 |
edited title; edited title
|
| Jan 11, 2013 at 10:43 | comment | added | user30435 | You are right, it is better the title to be changed. | |
| Jan 11, 2013 at 7:43 | comment | added | user30035 | There would be less confusion if the title of this question was "Representation of third-smallest dimension" ;-) [if we're counting the trivial 1-d rep -- and why not?] | |
| Jan 10, 2013 at 22:27 | comment | added | Mikhail Borovoi | The answer is yes, see my answer to mathoverflow.net/questions/118472/…. | |
| Jan 10, 2013 at 21:08 | comment | added | Jim Humphreys |
The question itself just involves classical ideas, so it might be answered in the literature(?); anyway it's really about the Lie algebra, which may be a tag to add. As robot observes, Weyl's dimension polynomial gives bigger values for non-fundamental weights. The fundamental irreducibles are close to the exterior powers, with dimensions given as a difference of two binomial coefficients. A quick calculation for $m=3$ gives (I hope)` dimensions 6, 14, 14, but after that the later ones grow faster. Presumably the answer to your question is yes, but it needs an argument.
|
|
| Jan 10, 2013 at 18:59 | comment | added | Mikhail Borovoi | @robot: OP wants to know the second smallest dimension of a nontrivial irreducible representation of a group of type $C_m$. | |
| Jan 10, 2013 at 17:55 | comment | added | Vít Tuček | Well, minimal was not the right word. What I meant to say that the representation with minimal dimension among all representations is the same as the representation which has the smallest dimension amongst fundamental representations. | |
| Jan 10, 2013 at 17:52 | comment | added | Vít Tuček | It follows from the Weyl dimension formula that the fundamental representations have minimal dimensions. So you only have to check the dimensions of these. | |
| Jan 10, 2013 at 16:50 | history | asked | user30435 | CC BY-SA 3.0 |