Timeline for Are there positive formulae for the inner product between elements of a Lie algebra representation in the Shapovalov form?
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| Mar 4, 2010 at 1:06 | comment | added | Ben | Uh, sorry, replace $m$ in that formula with the quantum-analog, which keeps track of the lengths of the permutations involved. Similarly, replace $k!$ with $[k]!$. | |
| Mar 3, 2010 at 17:58 | comment | added | Ben | Unfortunately, Ben wants an answer which depends very much on $\lambda$. The formula for the inner product on $U_q^-$ is actually quite easy: $(F_{i_1}\ldots F_{i_k},F_{j_1}\ldots F_{j_l})$ will be zero unless the $i$ sequence is a permutation of the $j$ sequence, and in that case, the answer will be $\frac{m}{(1-t^2)^k}$ where $m$ is the number of permutations sending the $j$ sequence to the $i$ sequence. For example, $(F_i^k,F_i^k)=\frac{k!}{(1-t^2)^k}$. This works because the Shapovalov form is a decategorification of the Hom bifunctor on Khovanov and Lauda's categorification of $U_q^-$. | |
| Mar 3, 2010 at 17:00 | history | answered | Peter Tingley | CC BY-SA 2.5 |