```
julia> basis_lie_highest_weight_operators(:A, 4)
10-element Vector{Tuple{Int64, Vector{QQFieldElem}}}:
(1, [1, 0, 0, 0])
(2, [0, 1, 0, 0])
(3, [0, 0, 1, 0])
(4, [0, 0, 0, 1])
(5, [1, 1, 0, 0])
(6, [0, 1, 1, 0])
(7, [0, 0, 1, 1])
(8, [1, 1, 1, 0])
(9, [0, 1, 1, 1])
(10, [1, 1, 1, 1])
```

```
julia> basis_lie_highest_weight(:A, 4, [2,1,2,1], [1,2,3,4,1,5,8,2,6,3]; monomial_ordering=:degrevlex)
Monomial basis of a highest weight module
of highest weight [2, 1, 2, 1]
of dimension 8750
with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6, x7, x8, x9, x10])
over Lie algebra of type A4
where the used birational sequence consists of the following roots (given as coefficients w.r.t. alpha_i):
[1, 0, 0, 0]
[0, 1, 0, 0]
[0, 0, 1, 0]
[0, 0, 0, 1]
[1, 0, 0, 0]
[1, 1, 0, 0]
[1, 1, 1, 0]
[0, 1, 0, 0]
[0, 1, 1, 0]
[0, 0, 1, 0]
and the basis was generated by Minkowski sums of the bases of the following highest weight modules:
[1, 0, 0, 0]
[0, 1, 0, 0]
[0, 0, 1, 0]
[0, 0, 0, 1]
```

```
julia> basis_lie_highest_weight_ffl(:A, 3, [1,1,1])
Monomial basis of a highest weight module
of highest weight [1, 1, 1]
of dimension 64
with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6])
over Lie algebra of type A3
where the used birational sequence consists of the following roots (given as coefficients w.r.t. alpha_i):
[1, 1, 1]
[0, 1, 1]
[1, 1, 0]
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
and the basis was generated by Minkowski sums of the bases of the following highest weight modules:
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
```

```
julia> basis_lie_highest_weight_string(:B, 3, [1,1,1], [3,2,3,2,1,2,3,2,1])
Monomial basis of a highest weight module
of highest weight [1, 1, 1]
of dimension 512
with monomial ordering neglex([x1, x2, x3, x4, x5, x6, x7, x8, x9])
over Lie algebra of type B3
where the used birational sequence consists of the following roots (given as coefficients w.r.t. alpha_i):
[0, 0, 1]
[0, 1, 0]
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
and the basis was generated by Minkowski sums of the bases of the following highest weight modules:
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
```

```
julia> basis_lie_highest_weight_lusztig(:D, 4, [1,1,1,1], [4,3,2,4,3,2,1,2,4,3,2,1])
Monomial basis of a highest weight module
of highest weight [1, 1, 1, 1]
of dimension 4096
with monomial ordering wdegrevlex([x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12], [1, 1, 3, 2, 2, 1, 5, 4, 3, 3, 2, 1])
over Lie algebra of type D4
where the used birational sequence consists of the following roots (given as coefficients w.r.t. alpha_i):
[0, 0, 0, 1]
[0, 0, 1, 0]
[0, 1, 1, 1]
[0, 1, 1, 0]
[0, 1, 0, 1]
[0, 1, 0, 0]
[1, 2, 1, 1]
[1, 1, 1, 1]
[1, 1, 0, 1]
[1, 1, 1, 0]
[1, 1, 0, 0]
[1, 0, 0, 0]
and the basis was generated by Minkowski sums of the bases of the following highest weight modules:
[1, 0, 0, 0]
[0, 1, 0, 0]
[0, 0, 1, 0]
[0, 0, 0, 1]
[0, 0, 1, 1]
```

```
julia> basis_lie_highest_weight_nz(:C, 3, [1,1,1], [3,2,3,2,1,2,3,2,1])
Monomial basis of a highest weight module
of highest weight [1, 1, 1]
of dimension 512
with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6, x7, x8, x9])
over Lie algebra of type C3
where the used birational sequence consists of the following roots (given as coefficients w.r.t. alpha_i):
[0, 0, 1]
[0, 1, 0]
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
and the basis was generated by Minkowski sums of the bases of the following highest weight modules:
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
```