I am learning the Borel–Weil–Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:

  1. How to judge if a weight is singular?
  2. 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:

  1. 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?

  2. 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 $\lambda$?

  • $\begingroup$ Please explain your notation! Also, $w$ most often denotes an element of the Weyl group, so you should use a symbol like $\lambda$ or $\varpi$ to denote a weight. (It's also useful to highlight your question by preceding it with > and a space.) $\endgroup$ Jan 2, 2018 at 20:25
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    $\begingroup$ What is $B_n$? (Are you naming the group by the type of its Dynkin diagram? Fortunately, here, fundamental-group issues don't matter, but in general this doesn't specify the group uniquely.) What are the various $\lambda_i$'s—I mean not just that they are fundamental weights, but specifically which ones in which order? (For example, are you using Bourbaki's or some other numbering?) $\endgroup$
    – LSpice
    Jan 3, 2018 at 14:52
  • $\begingroup$ Note that the question doesn't actually involve Lie groups (or algebraic groups), just the roots and weights. Though the finite dimensional representation theory here is well understood in principle, it can be tricky to work out stabilizers of weights, etc. But as LSpice points out, it's essential first to specify which numbering system for simple roots or associated fundamental weights is being used. In your example for type $B_n$, which is the short simple root? $\endgroup$ Jan 5, 2018 at 1:15

1 Answer 1

  1. 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.

  2. 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 $\lambda$.

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    $\begingroup$ What is "Bourbaki's $\epsilon$-basis"? If the one in the Lie-groups book, then I think your answer to (1) is not true. For example, it seems to me that, in $\mathsf B_2$, in Bourbaki's numbering, we have that $\lambda = \epsilon_1 = \alpha_1 + \alpha_2$ satisfies your condition, but $\langle\lambda, \alpha_2\rangle = 0$. $\endgroup$
    – LSpice
    Jan 4, 2018 at 11:43
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    $\begingroup$ Also, you say that the index (of a weight) equals the length (I guess also of a weight), but what is the definition of the length of a weight? The post defines the index of a weight to be the length of a Weyl-group element bringing it into the fundamental chamber, but it seems that the poster doesn't want to compute it that way. (Are you saying that simply checking the definition, perhaps by counting walls, is the best way to compute it?) $\endgroup$
    – LSpice
    Jan 4, 2018 at 11:46
  • $\begingroup$ @LSpice: I agree with you that the wording of Vit's answer is sometimes imprecise; but in the first question his notion of "coefficient" refers to writing $\lambda$ as a linear combination of the $\epsilon_i$. In either question, I still wonder if there is a more efficient way to do such computations in general. $\endgroup$ Jan 4, 2018 at 14:40
  • $\begingroup$ @Vit: In the second question, note that the weight is assumed to be nonsingular, though "index" makes sense in all cases. $\endgroup$ Jan 4, 2018 at 14:42
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    $\begingroup$ I've edited my answer to address your concerns. Sorry for negligence. $\endgroup$ Jan 5, 2018 at 18:02

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