Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight irreducible representation with a crystal basis $B$. On $B$ I have the crystal operators $\tilde e_1, \dotsc, \tilde e_n, \tilde f_1, \dotsc, \tilde f_n$. I know that the crystal root operators have the same grades as their corresponding Chevalley generators. i.e, $e_i,\tilde e_i:(V_\mu) \to V_{\mu+\alpha_i}$, and $f_i,\tilde f_i:(V_\mu) \to V_{\mu-\alpha_i}$ for $i=1,2,\dotsc,n$.

But beyond this, what relation do the crystal operators have with the corresponding Chevalley generator?

In particular, I am interested to know: if a relation of the form $\textbf{word in crystal operators}\cdot v= 0$ holds in $B$, where $v$ denotes the highest weight vector, then what does it say at the level of representations and the corresponding word in the Chevalley generators?



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