We will see how to compute polytopes whose lattice points parametrize a basis of highest weight modules for Lie algebras of type $A$. The inequalities for these polytopes can be derived from Dyck paths on the set of roots of the Lie algebra. The polytopes are handled with Polymake
.
using Pkg
Pkg.add(url="https://gitlab.com/johannesflake/polybases.jl.git")
using Oscar, PolyBases
Updating git-repo `https://gitlab.com/johannesflake/polybases.jl.git` Resolving package versions... No Changes to `~/.julia/environments/v1.6/Project.toml` No Changes to `~/.julia/environments/v1.6/Manifest.toml`
Let's start with the $\mathfrak{sl}_3$-module for the highest weight whose coordinates in terms of the fundamental weights are $(1,2)$.
polytope, pts = PolyBases.DyckPathA.dyck_poly( (1,2,0) );
display(pts)
display(size(pts))
pm::Matrix<long> 0 0 0 0 0 1 0 0 2 0 1 0 0 1 1 0 1 2 0 2 0 0 2 1 0 3 0 1 0 0 1 0 1 1 0 2 1 1 0 1 1 1 1 2 0
(15, 3)
We get $15$ lattice points and the following polytope:
PM = OSCAR.Polymake
PM.visual(polytope; FacetTransparency=0.5, ViewPoint=PM.Vector([3,3,3]) )
... will show an interactive visualization of the polytope similar to the following picture:
We can compare this with the results we obtain from GAP
:
G = Oscar.GAP; GG = Oscar.GAP.Globals;
L = GG.SimpleLieAlgebra(G.g"A", 2, GG.Rationals)
GAP: <Lie algebra of dimension 8 over Rationals>
dim = GG.DimensionOfHighestWeightModule(L, G.@gap [1,2])
display(dim)
dim == size(pts, 1)
15
true
Of course, this was only a toy example. More adventures await you.
polytope2, pts2 = PolyBases.DyckPathA.dyck_poly( (1,2,1,0) );
display(size(pts2, 1))
display(polytope2)
175
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