# Word/Loop in $L(\Lambda)$

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 $$c_{w_1,w_2}$$?