The OSCAR Book Companion
The OSCAR Book Companion
This page contains the code samples from the corresponding chapter in the OSCAR book. You can access the full chapter here.
The interested reader can find more information in this article.
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]