How to compute the index of a given weight? I am learning the Borel–Weil–Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:


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*How to judge if a weight is singular? 

*How to compute the index of a given weight?


To make my questions precise, let $G$ be a semisimple Lie group over $\mathbb C$. Choose a base of simple roots $\alpha_i$, $i=1,\dotsc, n$, and let $\lambda_i$, $i=1,\dotsc, n$ be the corresponding fundamental weights. Now I have a weight $\lambda=\sum_{i=1}^n m_i\lambda_i$. My questions are:


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*Definition: a weight $\lambda$ is called singular, if there is a root $\alpha$ such that $\langle\lambda, \alpha\rangle=0$. How to determine if $\lambda$ is singular?

*Definition: let $w$ be a Weyl group element, the length of $w$ is the smallest number of reflections $s_{\alpha_i}$ occurring in a word for $w$. The index of $\lambda$ is the shortest length of such $w$ that takes $\lambda$ to the fundamental chamber. If $\lambda$ is nonsingular, how to calculate the index of $\lambda$?
For example, let $G=B_n$ and $\lambda=\lambda_{n-2}-(1+n)\lambda_{n-1}+2\lambda_n$. How can I determine whether $\lambda$ is singular? If $\lambda$ is not singular, how should I compute the index of $\lambda$? 
 A: *

*Just evaluate $\langle \lambda, \alpha \rangle$ for all roots $\alpha$. It is sufficient to test the equality for positive roots only. Also, the weight $\lambda$ is regular if and only if it has trivial stabilizer in the Weyl group. For example, if you write $\lambda$ in Bourbaki's "$\epsilon$-basis" then regularity in $A_n$-case means that no two coefficients are the same. For other classical types you need to consider sign changes as well. For exceptionals... well, probably best to use a computer.


*If the weight is regular, then its index equals the length of the Weyl group element that brings it to the fundamental chamber. So you need to calculate the length of a Weyl group element. There are several ways to do that, one of them is actually finding the so called reduced expression for your Weyl group element. The algorithm for that should be in several books. I recommended looking in Combinatorics of Coxeter groups by Anders Björner and Francesco Brenti. GAP implementation of one such algorithm is here.
For singular weights I think you need minimal length representative of cosets $W/W_S$ where $W_S$ is the subgroup corresponding to the Levi part of parabolic subalgebra that stabilizes $\lambda$.
