Computing monomial bases 1¶
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.
In [1]:
using Oscar, PolyBases
Let's start with the $\mathfrak{sl}_3$-module for the highest weight whose coordinates in terms of the fundamental weights are $(1,2)$.
In [2]:
polytope, pts = PolyBases.dyck_poly( (1,2,0) );
display(pts)
display(size(pts))
We get $15$ lattice points and the following polytope:
In [ ]:
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:
In [4]:
G = Oscar.GAP; GG = Oscar.GAP.Globals;
In [5]:
L = GG.SimpleLieAlgebra(G.g"A", 2, GG.Rationals)
Out[5]:
In [6]:
dim = GG.DimensionOfHighestWeightModule(L, G.@gap [1,2])
display(dim)
dim == size(pts, 1)
Out[6]:
Of course, this was only a toy example. More adventures await you.
In [7]:
polytope, pts = PolyBases.dyck_poly( (1,2,1,0) );
display(size(pts, 1))
display(polytope)