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dual of weyl group

asked 2011-07-09 11:29:38 -0500

updated 2015-01-14 04:36:51 -0500

FrédéricC gravatar image

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.


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answered 2011-07-10 07:29:01 -0500

benjaminfjones gravatar image

updated 2011-07-10 09:05:40 -0500

Yes. In Sage, Weyl groups are constructed as matrix groups acting on the weight (or coweight) lattices. The Weyl groups in question are canonically isomorphic and what you want is an explicit isomorphism between them.

It's not terribly well documented, but there is a hom method for matrix groups that allows you to define a homomorphism by specifying the images of the generators of the domain.

The following code creates the canonical isomorphism phi between W1 the Weyl group acting on weights and W2 the Weyl group acting on coweights:

sage: W1 = RootSystem(['A', Integer(3)]).weight_lattice().weyl_group()
sage: W2 = RootSystem(['A', Integer(3)]).coweight_lattice().weyl_group()
sage: phi = W1.hom(W2.gens())
sage: phi
Homomorphism : Weyl Group of type ['A', 3] (as a matrix group acting on the weight 
lattice) --> Weyl Group of type ['A', 3] (as a matrix group acting on the weight 

sage: (s1, s2, s3) = W1.gens()
sage: phi(s1*s2*s1)
[ 0 -1  0]
[-1  0  0]
[ 1  1  1]

The element phi(s1*s2*s1) will act on the coweight lattice the way you want. The line

phi = W1.hom(W2.gens())

defines a homomorphism (actually an isomorphism) by sending W1.gen(i) to W2.gen(i) for i = 0 ... 2.

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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.

markblunk gravatar imagemarkblunk ( 2011-07-19 10:47:29 -0500 )edit

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Asked: 2011-07-09 11:29:38 -0500

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Last updated: Jul 10 '11