Timeline for Simple modules for $U_q(\mathfrak{sl}_n)$ at roots of unity
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Mar 3, 2011 at 13:57 | vote | accept | M T | ||
| Mar 2, 2011 at 13:34 | comment | added | Jim Humphreys |
The $\mathfrak{sl}_3$ example has been worked out in these settings, say for primes $p \geq 3$ or your parameter $N \geq 3$ in the quantum case, but without explicit constructions of simple modules. The key highest weights lie in two alcoves for the affine Weyl group relative to $N$. Weights in the closure of the lower alcove give characters and dimensions as in Cartan-Weyl theory, but a weight inside the top alcove gives a non-simple "Weyl module". Then subtract the simple module with reflected highest weight in lower alcove as in Lusztig's Conjecture (A-J-S, Asterisque 220).
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| Mar 2, 2011 at 0:11 | comment | added | M T | Many thanks, I will chase those references. I wonder if it is possible to explicitly write down the irreps on which the $E^N$ and $F^N$ act as zero and the $K^N$ as 1, where $N$ is the order of $q$ (i.e. irreps of the `small quantum group'). Surely for $N=3$ at least... Hopefully some of the references will make this clearer to me. | |
| Mar 1, 2011 at 23:52 | history | answered | Jim Humphreys | CC BY-SA 2.5 |