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

$\require{action}$
 -----    -----    -----      -      -----
|     |  |     |  |     |    | |    |     |
|     |  |        |         |   |   |     |
|     |   -----   |        |     |  |-----
|     |        |  |        |-----|  |   |
|     |  |     |  |     |  |     |  |    |
-----    -----    -----   -     -  -     -

...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.4.0 ...
... which comes with absolutely no warranty whatsoever
(c) 2019-2020 by The Oscar Development Team


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))

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:

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]:
GAP: <Lie algebra of dimension 8 over Rationals>
In [6]:
dim = GG.DimensionOfHighestWeightModule(L, G.@gap [1,2])
display(dim)
dim == size(pts, 1)

$15$
Out[6]:
true

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)

$175$
type
Polytope
AFFINE_HULL
BOUNDARY_LATTICE_POINTS
1 0 0 0 0 0 0
1 0 0 0 0 0 1
1 0 0 0 0 1 0
1 0 0 0 0 1 1
1 0 0 0 0 2 0
1 0 0 0 0 2 1
1 0 0 0 0 3 0
1 0 0 0 1 0 0
1 0 0 0 1 0 1
1 0 0 0 1 1 0
1 0 0 0 1 1 1
1 0 0 0 1 2 0
1 0 0 0 2 0 0
1 0 0 0 2 0 1
1 0 0 0 2 1 0
1 0 0 1 0 0 0
1 0 0 1 0 0 1
1 0 0 1 0 1 0
1 0 0 1 0 1 1
1 0 0 1 0 2 0
1 0 0 1 0 2 1
1 0 0 1 0 3 0
1 0 0 1 1 0 0
1 0 0 1 1 0 1
1 0 0 1 1 1 0
1 0 0 1 1 1 1
1 0 0 1 1 2 0
1 0 0 1 2 0 0
1 0 0 1 2 0 1
1 0 0 1 2 1 0
1 0 0 2 0 0 0
1 0 0 2 0 0 1
1 0 0 2 0 1 0
1 0 0 2 0 1 1
1 0 0 2 0 2 0
1 0 0 2 1 0 0
1 0 0 2 1 0 1
1 0 0 2 1 1 0
1 0 0 2 1 1 1
1 0 0 2 1 2 0
1 0 0 2 2 0 0
1 0 0 2 2 0 1
1 0 0 2 2 1 0
1 0 0 3 0 0 0
1 0 0 3 0 0 1
1 0 0 3 0 1 0
1 0 0 3 1 0 0
1 0 0 3 1 0 1
1 0 0 3 1 1 0
1 0 0 3 2 0 0
1 0 0 3 2 0 1
1 0 0 3 2 1 0
1 0 0 4 0 0 0
1 0 0 4 1 0 0
1 0 0 4 2 0 0
1 0 1 0 0 0 0
1 0 1 0 0 0 1
1 0 1 0 0 1 0
1 0 1 0 0 1 1
1 0 1 0 0 2 0
1 0 1 0 0 2 1
1 0 1 0 0 3 0
1 0 1 0 1 0 0
1 0 1 0 1 0 1
1 0 1 0 1 1 0
1 0 1 0 1 1 1
1 0 1 0 1 2 0
1 0 1 0 2 0 0
1 0 1 0 2 0 1
1 0 1 0 2 1 0
1 0 1 1 0 0 0
1 0 1 1 0 0 1
1 0 1 1 0 1 0
1 0 1 1 0 1 1
1 0 1 1 0 2 0
1 0 1 1 1 0 0
1 0 1 1 1 0 1
1 0 1 1 1 1 0
1 0 1 1 1 1 1
1 0 1 1 1 2 0
1 0 1 1 2 0 0
1 0 1 1 2 0 1
1 0 1 1 2 1 0
1 0 1 2 0 0 0
1 0 1 2 0 0 1
1 0 1 2 0 1 0
1 0 1 2 1 0 0
1 0 1 2 1 0 1
1 0 1 2 1 1 0
1 0 1 2 2 0 0
1 0 1 2 2 0 1
1 0 1 2 2 1 0
1 0 1 3 0 0 0
1 0 1 3 1 0 0
1 0 1 3 2 0 0
1 0 2 0 0 0 0
1 0 2 0 0 0 1
1 0 2 0 0 1 0
1 0 2 0 0 1 1
1 0 2 0 0 2 0
1 0 2 0 1 0 0
1 0 2 0 1 0 1
1 0 2 0 1 1 0
1 0 2 1 0 0 0
1 0 2 1 0 0 1
1 0 2 1 0 1 0
1 0 2 1 1 0 0
1 0 2 1 1 0 1
1 0 2 1 1 1 0
1 0 2 2 0 0 0
1 0 2 2 1 0 0
1 0 3 0 0 0 0
1 0 3 0 0 0 1
1 0 3 0 0 1 0
1 0 3 1 0 0 0
1 1 0 0 0 0 0
1 1 0 0 0 0 1
1 1 0 0 0 1 0
1 1 0 0 0 1 1
1 1 0 0 0 2 0
1 1 0 0 0 2 1
1 1 0 0 0 3 0
1 1 0 0 1 0 0
1 1 0 0 1 0 1
1 1 0 0 1 1 0
1 1 0 0 1 1 1
1 1 0 0 1 2 0
1 1 0 0 2 0 0
1 1 0 0 2 0 1
1 1 0 0 2 1 0
1 1 0 1 0 0 0
1 1 0 1 0 0 1
1 1 0 1 0 1 0
1 1 0 1 0 1 1
1 1 0 1 0 2 0
1 1 0 1 1 0 0
1 1 0 1 1 0 1
1 1 0 1 1 1 0
1 1 0 1 1 1 1
1 1 0 1 1 2 0
1 1 0 1 2 0 0
1 1 0 1 2 0 1
1 1 0 1 2 1 0
1 1 0 2 0 0 0
1 1 0 2 0 0 1
1 1 0 2 0 1 0
1 1 0 2 1 0 0
1 1 0 2 1 0 1
1 1 0 2 1 1 0
1 1 0 2 2 0 0
1 1 0 2 2 0 1
1 1 0 2 2 1 0
1 1 0 3 0 0 0
1 1 0 3 1 0 0
1 1 0 3 2 0 0
1 1 1 0 0 0 0
1 1 1 0 0 0 1
1 1 1 0 0 1 0
1 1 1 0 0 1 1
1 1 1 0 0 2 0
1 1 1 0 1 0 0
1 1 1 0 1 0 1
1 1 1 0 1 1 0
1 1 1 1 0 0 0
1 1 1 1 0 0 1
1 1 1 1 0 1 0
1 1 1 1 1 0 0
1 1 1 1 1 0 1
1 1 1 1 1 1 0
1 1 1 2 0 0 0
1 1 1 2 1 0 0
1 1 2 0 0 0 0
1 1 2 0 0 0 1
1 1 2 0 0 1 0
1 1 2 1 0 0 0
BOUNDED
true
COMBINATORIAL_DIM
6
CONE_AMBIENT_DIM
7
CONE_DIM
7
FACETS
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
1 -1 0 0 0 0 0
2 0 0 0 -1 0 0
1 0 0 0 0 0 -1
3 -1 -1 0 -1 0 0
3 0 0 0 -1 -1 -1
4 -1 -1 -1 0 -1 -1
4 -1 -1 0 -1 -1 -1
FEASIBLE
true
FULL_DIM
true
INEQUALITIES
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
1 -1 0 0 0 0 0
2 0 0 0 -1 0 0
1 0 0 0 0 0 -1
3 -1 -1 0 -1 0 0
3 0 0 0 -1 -1 -1
4 -1 -1 -1 0 -1 -1
4 -1 -1 0 -1 -1 -1
1 0 0 0 0 0 0
INTERIOR_LATTICE_POINTS
LATTICE_POINTS_GENERATORS
<1 0 0 0 0 0 0
1 0 0 0 0 0 1
1 0 0 0 0 1 0
1 0 0 0 0 1 1
1 0 0 0 0 2 0
1 0 0 0 0 2 1
1 0 0 0 0 3 0
1 0 0 0 1 0 0
1 0 0 0 1 0 1
1 0 0 0 1 1 0
1 0 0 0 1 1 1
1 0 0 0 1 2 0
1 0 0 0 2 0 0
1 0 0 0 2 0 1
1 0 0 0 2 1 0
1 0 0 1 0 0 0
1 0 0 1 0 0 1
1 0 0 1 0 1 0
1 0 0 1 0 1 1
1 0 0 1 0 2 0
1 0 0 1 0 2 1
1 0 0 1 0 3 0
1 0 0 1 1 0 0
1 0 0 1 1 0 1
1 0 0 1 1 1 0
1 0 0 1 1 1 1
1 0 0 1 1 2 0
1 0 0 1 2 0 0
1 0 0 1 2 0 1
1 0 0 1 2 1 0
1 0 0 2 0 0 0
1 0 0 2 0 0 1
1 0 0 2 0 1 0
1 0 0 2 0 1 1
1 0 0 2 0 2 0
1 0 0 2 1 0 0
1 0 0 2 1 0 1
1 0 0 2 1 1 0
1 0 0 2 1 1 1
1 0 0 2 1 2 0
1 0 0 2 2 0 0
1 0 0 2 2 0 1
1 0 0 2 2 1 0
1 0 0 3 0 0 0
1 0 0 3 0 0 1
1 0 0 3 0 1 0
1 0 0 3 1 0 0
1 0 0 3 1 0 1
1 0 0 3 1 1 0
1 0 0 3 2 0 0
1 0 0 3 2 0 1
1 0 0 3 2 1 0
1 0 0 4 0 0 0
1 0 0 4 1 0 0
1 0 0 4 2 0 0
1 0 1 0 0 0 0
1 0 1 0 0 0 1
1 0 1 0 0 1 0
1 0 1 0 0 1 1
1 0 1 0 0 2 0
1 0 1 0 0 2 1
1 0 1 0 0 3 0
1 0 1 0 1 0 0
1 0 1 0 1 0 1
1 0 1 0 1 1 0
1 0 1 0 1 1 1
1 0 1 0 1 2 0
1 0 1 0 2 0 0
1 0 1 0 2 0 1
1 0 1 0 2 1 0
1 0 1 1 0 0 0
1 0 1 1 0 0 1
1 0 1 1 0 1 0
1 0 1 1 0 1 1
1 0 1 1 0 2 0
1 0 1 1 1 0 0
1 0 1 1 1 0 1
1 0 1 1 1 1 0
1 0 1 1 1 1 1
1 0 1 1 1 2 0
1 0 1 1 2 0 0
1 0 1 1 2 0 1
1 0 1 1 2 1 0
1 0 1 2 0 0 0
1 0 1 2 0 0 1
1 0 1 2 0 1 0
1 0 1 2 1 0 0
1 0 1 2 1 0 1
1 0 1 2 1 1 0
1 0 1 2 2 0 0
1 0 1 2 2 0 1
1 0 1 2 2 1 0
1 0 1 3 0 0 0
1 0 1 3 1 0 0
1 0 1 3 2 0 0
1 0 2 0 0 0 0
1 0 2 0 0 0 1
1 0 2 0 0 1 0
1 0 2 0 0 1 1
1 0 2 0 0 2 0
1 0 2 0 1 0 0
1 0 2 0 1 0 1
1 0 2 0 1 1 0
1 0 2 1 0 0 0
1 0 2 1 0 0 1
1 0 2 1 0 1 0
1 0 2 1 1 0 0
1 0 2 1 1 0 1
1 0 2 1 1 1 0
1 0 2 2 0 0 0
1 0 2 2 1 0 0
1 0 3 0 0 0 0
1 0 3 0 0 0 1
1 0 3 0 0 1 0
1 0 3 1 0 0 0
1 1 0 0 0 0 0
1 1 0 0 0 0 1
1 1 0 0 0 1 0
1 1 0 0 0 1 1
1 1 0 0 0 2 0
1 1 0 0 0 2 1
1 1 0 0 0 3 0
1 1 0 0 1 0 0
1 1 0 0 1 0 1
1 1 0 0 1 1 0
1 1 0 0 1 1 1
1 1 0 0 1 2 0
1 1 0 0 2 0 0
1 1 0 0 2 0 1
1 1 0 0 2 1 0
1 1 0 1 0 0 0
1 1 0 1 0 0 1
1 1 0 1 0 1 0
1 1 0 1 0 1 1
1 1 0 1 0 2 0
1 1 0 1 1 0 0
1 1 0 1 1 0 1
1 1 0 1 1 1 0
1 1 0 1 1 1 1
1 1 0 1 1 2 0
1 1 0 1 2 0 0
1 1 0 1 2 0 1
1 1 0 1 2 1 0
1 1 0 2 0 0 0
1 1 0 2 0 0 1
1 1 0 2 0 1 0
1 1 0 2 1 0 0
1 1 0 2 1 0 1
1 1 0 2 1 1 0
1 1 0 2 2 0 0
1 1 0 2 2 0 1
1 1 0 2 2 1 0
1 1 0 3 0 0 0
1 1 0 3 1 0 0
1 1 0 3 2 0 0
1 1 1 0 0 0 0
1 1 1 0 0 0 1
1 1 1 0 0 1 0
1 1 1 0 0 1 1
1 1 1 0 0 2 0
1 1 1 0 1 0 0
1 1 1 0 1 0 1
1 1 1 0 1 1 0
1 1 1 1 0 0 0
1 1 1 1 0 0 1
1 1 1 1 0 1 0
1 1 1 1 1 0 0
1 1 1 1 1 0 1
1 1 1 1 1 1 0
1 1 1 2 0 0 0
1 1 1 2 1 0 0
1 1 2 0 0 0 0
1 1 2 0 0 0 1
1 1 2 0 0 1 0
1 1 2 1 0 0 0
>
<>
<>
LINEALITY_DIM
0
LINEALITY_SPACE
(7) (0 1)
N_VERTICES
42
POINTED
true
VERTICES
1 1 0 0 0 3 0
1 1 0 0 0 2 1
1 1 0 2 2 1 0
1 1 0 3 2 0 0
1 1 0 2 2 0 1
1 1 0 0 2 0 1
1 1 0 0 2 1 0
1 1 0 0 2 0 0
1 1 2 1 0 0 0
1 1 2 0 0 0 1
1 1 2 0 0 1 0
1 1 2 0 0 0 0
1 1 0 0 0 0 0
1 1 0 2 0 0 1
1 1 0 0 0 0 1
1 1 0 3 0 0 0
1 0 0 4 0 0 0
1 0 0 0 0 0 1
1 0 0 3 0 0 1
1 0 0 0 0 3 0
1 0 0 1 0 3 0
1 0 1 0 0 3 0
1 0 0 0 0 2 1
1 0 0 1 0 2 1
1 0 1 0 0 2 1
1 0 0 3 2 1 0
1 0 1 2 2 1 0
1 0 0 4 2 0 0
1 0 1 3 2 0 0
1 0 0 3 2 0 1
1 0 1 2 2 0 1
1 0 0 0 2 0 1
1 0 1 0 2 0 1
1 0 0 0 2 1 0
1 0 1 0 2 1 0
1 0 0 0 2 0 0
1 0 1 0 2 0 0
1 0 3 1 0 0 0
1 0 3 0 0 0 1
1 0 3 0 0 1 0
1 0 3 0 0 0 0
1 0 0 0 0 0 0
VERTICES_IN_FACETS
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}
{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}
{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}
{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}
{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}
{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}
{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}
{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}
{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}
{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}
{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}
{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}
{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}
VERTICES_IN_INEQUALITIES
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}
{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}
{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}
{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}
{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}
{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}
{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}
{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}
{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}
{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}
{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}
{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}
{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}
{}
VERTICES_IN_INEQUALITIES
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}
{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}
{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}
{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}
{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}
{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}
{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}
{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}
{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}
{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}
{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}
{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}
{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}
{}