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2012-01-04 08:42:04 -0600 asked a question Producing subgroups of Weyl groups

Let W be a Weyl group, e.g. W = RootSystem('[A, 4]').weight_lattice().weyl_group

Given some elements $S \subset W$, I would like to produce the subgroup generated by $S$. It seems like there are methods in SAGE to do this when W is an abstract group, but I can't see how to do it when $W$ is a Weyl group. Any suggestions?

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2011-07-19 10:47:29 -0600 commented answer dual of weyl group

Thanks for your response. Unfortunately the function I described, while it does map generators (i.e. simple reflections) to generators, is not a homomorphism! It acts as an adjoint operator.

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2011-07-09 11:29:38 -0600 asked a question dual of weyl group

I was wondering if it is possible to, given an element in a Weyl group, produce the corresponding element in the dual Weyl group. As an example, if w in W = RootSystem(['A', 3]).weight_lattice().weyl_group()

Then I would like a function f such that f(w) in RootSystem(['A', 3]).coroot_lattice().weyl_group(), with the obvious duality <w*x,y> = <x,f(w)*(y)>, where x in the weight lattice and y is in the coroot lattice.