Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$).

Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote the highest-weight vector.

For a word $w=i_1...i_l$ let $e_w=e_{i_1}...e_{i_l}$ and $f_w=f_{i_1}...f_{i_l}$. Let $m(w)_i$ denote the number of times $i$ occurs in $w$, and let $m(w)=(m(w)_1,...,m(w)_n)$.

Let $w_1,w_2$ be two words with $m(w_1)=m(w_2)$. Then $e_{w_1}f_{w_2}v\in L(\Lambda)_\Lambda$, hence there exists a scalar $c_{w_1,w_2}$ such that $e_{w_1}f_{w_2}v=c_{w_1,w_2}v$.

Is there a formula for computing $c_{w_1,w_2}$?


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