Monomial Bases in Lie Theory

Authors: Xin Fang and Ghislain Fourier

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*w_1 + w_2 + 2*w_3 + w_4
  of dimension 8750
  with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6, x7, x8, x9, x10])
over abstract Lie algebra of type A4 over QQ
where the used birational sequence consists of the following roots:
  [a_1, a_2, a_3, a_4, a_1, a_1 + a_2, a_1 + a_2 + a_3, a_2, a_2 + a_3, a_3]
  

julia> basis_lie_highest_weight_ffl(:A, 3, [1,1,1])
Monomial basis of a highest weight module
  of highest weight w_1 + w_2 + w_3
  of dimension 64
  with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6])
over abstract Lie algebra of type A3 over QQ
where the used birational sequence consists of the following roots:
  [a_1 + a_2 + a_3, a_2 + a_3, a_1 + a_2, a_3, a_2, a_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 w_1 + w_2 + w_3
  of dimension 512
  with monomial ordering neglex([x1, x2, x3, x4, x5, x6, x7, x8, x9])
over abstract Lie algebra of type B3 over QQ
where the used birational sequence consists of the following roots:
  [a_3, a_2, a_3, a_2, a_1, a_2, a_3, a_2, a_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 w_1 + w_2 + w_3 + w_4
  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 abstract Lie algebra of type D4 over QQ
where the used birational sequence consists of the following roots:
  [a_4, a_3, a_2 + a_3 + a_4, a_2 + a_3, a_2 + a_4, a_2, a_1 + 2*a_2 + a_3 + a_4, a_1 + a_2 + a_3 + a_4, a_1 + a_2 + a_4, a_1 + a_
  2 + a_3, a_1 + a_2, a_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 w_1 + w_2 + w_3
  of dimension 512
  with monomial ordering degrevlex([x1, x2, x3, x4, x5, x6, x7, x8, x9])
over abstract Lie algebra of type C3 over QQ
where the used birational sequence consists of the following roots:
  [a_3, a_2, a_3, a_2, a_1, a_2, a_3, a_2, a_1]