Timeline for When is there a $g$-module isomorphism between a semi-simple Lie algebra $g$ and an exterior power of its standard representation?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 27, 2010 at 0:12 | comment | added | J Lodder | Above, "$\binom{n}{k}$ for $n \geq 2$" should be "$\binom{n}{k}$ for $2 \leq k \leq n-2$." | |
| Sep 27, 2010 at 0:02 | comment | added | J Lodder | Among the exceptional Lie algebras, the only one with dimension being a binomial coefficient $\binom {n}{k}$ for $n \geq 2$ is $E_6$ with $78 = \binom{13}{2}$. However, the smallest non-trivial rep. of $E_6$ has dimension 27, which rules out 13. Among the classical Lie algebras, as $sl(n)$-modules, $sl(n)$ is essentially iso. to $I^{\otimes 2}$ (with the trivial rep. deleted), and as $sp(n)$-modules, $sp(n)$ is iso. to the second symmetric power of $I$. For $k \geq 2$ and $g$ complex, simple, the only solutions to the question are from the $so(n)$ family. | |
| Sep 20, 2010 at 10:47 | comment | added | Jim Humphreys | Yes, I shouldn't have focused on type A alone, so I edited that. The point is just that one can't take for granted anything about irreducibility of higher exterior or symmetric powers. | |
| Sep 20, 2010 at 10:35 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
deleted 19 characters in body
|
| Sep 19, 2010 at 23:00 | comment | added | Victor Protsak | Higher exterior powers of the "natural representation" are also irreducible for the orthogonal Lie algebra, i.e. for B and D types. | |
| Sep 19, 2010 at 21:48 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
added 696 characters in body
|
| Sep 19, 2010 at 19:51 | history | answered | Jim Humphreys | CC BY-SA 2.5 |