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# deGraafBases -- compute the bases produced by de Graaf's algorithm

- Usage:
    `deGraafBasis(lambda,g)`
- Inputs:
  - lambda, a list,
  - g, an instance of the type LieAlgebra,
- Outputs:
  - S, a sequence,

## Description

This function implements the main algorithm in de Graaf, "Constructing representations of split semisimple Lie Algebras", J. Pure Appl. Algebra 164 (2001), no. 1-2, 87-107.

Let $V$ be an irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Then $V$ may be constructed as the quotient of the algebra $U(N^{-})$ (see uNminus) by a left ideal $I$. de Graaf's algorithm produces a Gröbner basis of the ideal $I$, and a basis of the quotient $U(N^{-})/I$.

Note that de Graaf scales his basis monomials. We skip this.

    i1 : g = simpleLieAlgebra("A",2);

    i2 : lambda = {1,1}

    o2 = {1, 1}

    o2 : List

    i3 : deGraafBases(lambda,g)
    Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
    Converting to Naive algorithm.
    max-lev=6
    Finished level 1. {#G,#B}={0, 3}
    Finished level 2. {#G,#B}={2, 5}
    Finished level 3. {#G,#B}={2, 7}
    Finished level 4. {#G,#B}={3, 8}
    Finished level 5. {#G,#B}={5, 8}
    Finished level 6. {#G,#B}={6, 8}

             2            2     2            2
    o3 = ({{Y , {-3, 3}, Y }, {Y , {3, -3}, Y }, {Y Y Y , {-1, -1}, Y Y Y  +
             1            1     2            2     1 2 3             1 2 3
         ------------------------------------------------------------------------
         1 2       2              2       2              2     3             3
         -Y }, {Y Y , {-3, 0}, Y Y }, {Y Y , {0, -3}, Y Y }, {Y , {-2, -2}, Y }},
         2 3     1 3            1 3     2 3            2 3     3             3
         ------------------------------------------------------------------------
                                            2
         {1, Y , Y , Y , Y Y , Y Y , Y Y , Y })
              1   2   3   1 2   1 3   2 3   3

    o3 : Sequence

## Ways to use deGraafBases:

- `deGraafBases(List,LieAlgebra)`

## For the programmer

The object deGraafBases is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:2081:0.