Defining Flag Varieties \(G/P\) in Our Software¶
This guide explains how to construct complex flag varieties \(G/P\) using our software by specifying a Cartan Type (to define the complex semisimple Lie group \(G\)) and Keep Nodes (to define the parabolic subgroup \(P\)).
1. The Lie-Theoretic Setup¶
Let \(G\) be a complex semisimple Lie group. We use its simply connected form to label line bundles by fundamental weights: \(SL(n)\), \(Spin(n)\), or \(Sp(2n)\) in the classical cases. The orthogonal descriptions below can also be written with \(SO(n)\), which has the same flag varieties. Its Lie algebra \(\mathfrak{g}\) is classified by a Cartan Type and a Dynkin diagram; simple roots are indexed by Bourbaki node numbers.
A standard parabolic subgroup \(P \subset G\) is determined by a subset of the simple roots. In our software, you specify \(P\) by providing the Keep Nodes (using the --keep argument in the CLI or "Keep Nodes" in the GUI). Geometrically, specifying a node \(k\) in "Keep Nodes" corresponds to crossing out that node \(k\) from the Dynkin diagram. This omits the \(k\)-th simple root from the Levi factor of \(P\) (which is equivalent to keeping the \(k\)-th fundamental weight). Thus, if you specify a single node \(k\) in Keep Nodes, node \(k\) is crossed out, making \(P\) a maximal parabolic subgroup, and \(G/P\) is a generalized Grassmannian.
2. Dictionary of Standard Geometric Spaces¶
Below is the explicit mapping from classical geometric spaces to their corresponding (Cartan Type, Keep Nodes) inputs.
Projective Spaces \(\mathbb{P}^n\)¶
The complex projective space \(\mathbb{P}^n\) is the space of 1-dimensional subspaces in \(\mathbb{C}^{n+1}\).
- Group: \(G = SL(n+1)\)
- Cartan Type: $A_n$
- Keep Nodes: 1 (or n for the dual projective space \(\mathbb{P}^{n\ast}\))
- Example: To construct \(\mathbb{P}^4\), use Algebra = A4, Keep Nodes = 1.
Grassmannians \(Gr(k, n)\)¶
The Grassmannian \(Gr(k, n)\) is the space of \(k\)-dimensional subspaces in \(\mathbb{C}^n\).
- Group: \(G = SL(n)\)
- Cartan Type: $A_{n-1}$
- Keep Nodes: k
- Example: To construct \(Gr(2, 5)\), use Algebra = A4, Keep Nodes = 2.
Lagrangian Grassmannians \(LG(n, 2n)\)¶
The Lagrangian Grassmannian \(LG(n, 2n)\) parameterizes \(n\)-dimensional isotropic subspaces of a \(2n\)-dimensional symplectic vector space.
- Group: \(G = Sp(2n)\)
- Cartan Type: $C_n$
- Keep Nodes: n
- Example: To construct \(LG(3, 6)\), use Algebra = C3, Keep Nodes = 3.
Odd Orthogonal Grassmannians \(OG(n, 2n+1)\)¶
This space parameterizes \(n\)-dimensional isotropic subspaces of a \((2n+1)\)-dimensional orthogonal vector space.
- Group: \(G = SO(2n+1)\)
- Cartan Type: $B_n$
- Keep Nodes: n
- Example: To construct \(OG(3, 7)\), use Algebra = B3, Keep Nodes = 3.
Even Orthogonal Grassmannians \(OG(n, 2n)\) (Spinor Varieties)¶
This space has two connected components, each parameterizing one family of \(n\)-dimensional isotropic subspaces in a \(2n\)-dimensional orthogonal vector space.
- Group: \(G = SO(2n)\)
- Cartan Type: D_n
- Keep Nodes: n (or n-1 for the other connected component)
- Example: To construct \(OG(4, 8)\), use Algebra = D4, Keep Nodes = 4.
3. References¶
On the maximal orthogonal Grassmannians above, O(1) denotes the primitive
spinor generator. The Plücker determinant line bundle is O(2). This matters
when translating section degrees into Bundle K; the two conventions must not
be interchanged.
For rigorous foundations of Lie algebras, root systems, Dynkin diagrams, and the construction of homogeneous spaces, please refer to the following standard texts:
- Fulton, W., & Harris, J. (1991). Representation Theory: A First Course. Springer.
- Humphreys, J. E. (1972). Introduction to Lie Algebras and Representation Theory. Springer.