On Fri, Mar 23, 2012 at 08:24:06AM -0400, msh...@math.vt.edu wrote:
> It is easy to get the inversion set directly from any reduced word of the
> Weyl group element.
>
> If w = s_{i_l} ... s_{i_2} s_{i_1}
>
> then the set of positive roots alpha such that w alpha is negative, is
>
> \alpha_{i_1}, s_{i_1} \alpha_{i_2}, s_{i_1} s_{i_2} \alpha_{i_3},...
>
> This produces "right inversions". If you want left inversions then
> use w^{-1} (or use the reverse of the reduced word for w).
Good point. And if this is done right (by keeping track of the partial
product s_{i_1}...s_{i_k}) this is roughly O(dr) where d is the rank
of the root system and r is the length of the element. That is linear
in the size of the output, which is as good as it can get.
> If you are going to do this a lot then it can of course be done
> recursively with remembering.
Yup.
Volunteer anyone?
By the way: the reverse bijection could be useful too.
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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