deGraafRepresentation(lambda,g)Let $V$ be an irreducible $\mathfrak{g}$-module with highest weight $\lamdba$. Then $V$ may be constructed as follows. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$ (see universalEnvelopingAlgebra. Let $A(\lambda)$ be the (infinite-dimensional) Verma module $U(\mathfrak{g})/J$, where $J$ is the left ideal $\{ x_1,\ldots,x_l,h_1-\lambda(H_1),\ldots,h_n-\lambda(H_n)\rangle$. Then $V \cong A(\lambda)/I$, and the action of $X \in \mathfrak{g}$ on elements of the basis of $U(N^{-})/I$ is left multiplication.
We cannot implement the algorithm outlined above in a naive way because the AssociativeAlgebras package does not currently support quotients of quotients of free algebras. Instead, following de Graaf, we exploit the isomorphism $A(\lambda) \cong U(N^{-})$, and proceed as follows.
$\quad$ 1. We create $U(\mathfrak{g})$ and $U(N^{-})$ (see universalEnvelopingAlgebra and uNminus)
$\quad$ 2. We compute the Gröbner basis $G$ of $I$, and a basis of the quotient $U(N^{-})/I$ (see deGraafBases)
$\quad$ 3. For each $X$ in a basis of $\mathfrak{g}$, and each $B_i$ in de Graaf's basis of $V$:
$\quad$$\quad$ a. Multiply $X.B_i$ in $U(\mathfrak{g})$
$\quad$$\quad$ b. Map this to $U(N^{-})$ under the map sending $x_i \mapsto 0$ and $h_i \mapsto h_i-\lambda(h_i)$
$\quad$$\quad$ c. At top level, we reduce by $I$ in $U(N^{-})$.
$\quad$ This gives the action of $X$ on de Graaf's basis of $V$. We extract the coefficients to build the matrix.
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The object deGraafRepresentation is a method function.
The source of this document is in LieAlgebraRepresentations/documentation.m2:2124:0.