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Let $\pi$ be a Lakshmibai-Seshadri $(L-S)$ path of weight $\lambda$ .Let $B_{\pi}$ be the set of paths obtained from $\pi$ by applying the root operatos.It is known that the formal character of $B_{\pi}$ is equal to the character of simple module of highest weight $ \lambda $ of a symmetrizable kac moody algebra $L$. $B_{\pi}$ is equal to the set of paths obtained from $\lambda $ by applying lowering operators corresponding to different simple roots of $ L $. let $f^a_{\alpha_1}.f^b_{\alpha_2}...f^t_{\alpha_n}.\pi$ be a path in $B_{\pi}$.under what conditions on $a$,$b$,... $t$ the path $f^a_{\alpha_1}.f^b_{\alpha_2}...f^t_{\alpha_n}.\pi$ will be zero where $a$,$b$,$c$... $t$ are non negative integers?

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up vote 1 down vote accepted

The condition is that in the crystal graph for the higest weight $\lambda$ one can take, starting at the node for the highest weight, $t$ downwards steps labelled $\alpha_n$, followed by..., followed by $b$ downwards steps labelled $\alpha_2$, followed by $a$ downwards steps labelled $\alpha_1$. Since you want to know this for all possible parameters, you are asking for the complete structure of the crystal graph. There are many ways to describe or construct this crystal graph implcitly, but none of them is really easy and explicit (unless you are in a classical type, and even there rather intricate compinatorial objects are involved in the description). In other words, you are just asking too much.

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Is it easier to construct crystal graph at least for classical case?could you please explain how to construct it? – Rekha Biswal Mar 29 '13 at 9:00

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