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LieTypes :: weylAlcove

weylAlcove -- the dominant integral weights of level less than or equal to l

Synopsis

Description

Let g be a Lie algebra, and let l be a nonnegative integer. Choose a Cartan subalgebra h and a base Δ= {α1,...,αn} of simple roots of g. These choices determine a highest root θ. (See highestRoot). Let hR* be the real span of Δ, and let (,) denote the Killing form, normalized so that (θ,θ)=2. The fundamental Weyl chamber is C+ = {λ∈hR* : (λ,αi) >= 0, i=1,...,n }. The fundamental Weyl alcove is the subset of the fundamental Weyl chamber such that (λ,θ) ≤l. This function computes the set of integral weights in the fundamental Weyl alcove.

In the example below, we see that the Weyl alcove of sl3 at level 3 contains 10 integral weights.

i1 : g=simpleLieAlgebra("A",2)

o1 = g

o1 : LieAlgebra
i2 : weylAlcove(3,g)

o2 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1},
     ------------------------------------------------------------------------
     {3, 0}}

o2 : List

Ways to use weylAlcove :