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

## 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 :

• weylAlcove(String,ZZ,ZZ)
• weylAlcove(ZZ,LieAlgebra)