Monomial Bases in Lie Theory

Authors: Xin Fang and Ghislain Fourier

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]