Flag varieties and sections¶
These entries include orthogonal, symplectic, exceptional and partial flag varieties, and complete intersections in them. Each displayed matrix comes from exact gwflags virtual localization using the full automatically supported curve list.
For a worked full-flag example, see Fl(1,2,3).
Matrices represent multiplication by \(c_1(TY)\) on the returned ambient sector, with every Novikov variable set to \(1\). Columns are inputs and rows are outputs. The ordered basis below uses reduced Schubert words; () is the unit. Basis indices in the download are zero-based.
Download all symbolic matrices, bases, gradings and curve classes or browse Download matrices and inputs. See How to Use for input conventions.
| Geometry | Dimension | Ambient index | Basis rank |
|---|---|---|---|
| Fl(1,4,5) | 7 | 4 | 20 |
| Fl(1,4,5) / O(1,1)+O(1,1) | 5 | 2 | 12 |
| G2/P1 | 5 | 5 | 6 |
| G2/P2 | 5 | 3 | 6 |
| LG(3,6) / O(1) | 5 | 3 | 6 |
| OG(3,7) / O(2) | 5 | 4 | 6 |
| OG(3,7) / O(3) | 5 | 3 | 6 |
| OG(3,7) / O(4) | 5 | 2 | 6 |
| Fl(1,2,5) / O(0,1)+O(0,2)+O(1,0) (GM-20) | 4 | 1 | 9 |
| Fl(1,3,4) / O(1,1) | 4 | 2 | 9 |
| G2/P2 / O(1) | 4 | 2 | 5 |
| LG(3,6) / O(1)+O(1) | 4 | 2 | 5 |
| OG(3,7) / O(2)+O(2) | 4 | 2 | 5 |
| B2/B / O(1,1) | 3 | 1 | 6 |
| Fl(1,3,4) / O(1,1)+O(1,1) | 3 | 1 | 6 |
Fl(1,4,5)
- Ambient Space:
A4 - Keep Nodes:
1,4 - Bundle K: empty (the flag variety itself)
- Dimension: \(7\)
- Ambient Index: \(4\)
- Basis Rank: \(20\)
- Exact data identifier:
6d625eb08919
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8\\4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 0 & 0 & 0 & 0 & 0 & 8 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 4 & 8 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (4) |
1 |
| 4 | (1 4) |
2 |
| 5 | (2 1) |
2 |
| 6 | (3 4) |
2 |
| 7 | (1 3 4) |
3 |
| 8 | (2 1 4) |
3 |
| 9 | (2 3 4) |
3 |
| 10 | (3 2 1) |
3 |
| 11 | (1 2 3 4) |
4 |
| 12 | (2 1 3 4) |
4 |
| 13 | (3 2 1 4) |
4 |
| 14 | (4 3 2 1) |
4 |
| 15 | (1 2 1 3 4) |
5 |
| 16 | (2 3 2 1 4) |
5 |
| 17 | (3 4 3 2 1) |
5 |
| 18 | (1 2 3 2 1 4) |
6 |
| 19 | (2 3 4 3 2 1) |
6 |
| 20 | (1 2 3 4 3 2 1) |
7 |
Fl(1,4,5) / O(1,1)+O(1,1)
- Ambient Space:
A4 - Keep Nodes:
1,4 - Bundle K:
1,1;1,1 - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(12\)
- Exact data identifier:
9cb29c69a3f5
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 2 & 2 & 0 & 0 & 0 & 28 & 28 & 16 & 0 & 0 & 12\\2 & 0 & 0 & 2 & 0 & 6 & 0 & 0 & 0 & 4 & 16 & 0\\2 & 0 & 0 & 2 & 6 & 0 & 0 & 0 & 0 & 8 & 20 & 0\\0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 2 & 0 & 0 & 0 & 0 & 6 & 2 & 6 & 0 & 0 & 4\\0 & 0 & 2 & 0 & 0 & 0 & 2 & 6 & 0 & 0 & 0 & 4\\0 & 0 & 0 & 2 & -3 & 2 & 0 & 0 & 0 & -3 & -6 & 0\\0 & 0 & 0 & 2 & 5 & 0 & 0 & 0 & 0 & 3 & 6 & 0\\0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 2 & 6 & 0\\0 & 0 & 0 & 0 & 0 & 0 & -4 & -16 & 4 & 0 & 0 & -2\\0 & 0 & 0 & 0 & 0 & 0 & 8 & 12 & 2 & 0 & 0 & 2\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 6 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (4) |
1 |
| 4 | (1 4) |
2 |
| 5 | (2 1) |
2 |
| 6 | (3 4) |
2 |
| 7 | (1 3 4) |
3 |
| 8 | (2 1 4) |
3 |
| 9 | (2 3 4) |
3 |
| 10 | (1 2 3 4) |
4 |
| 11 | (2 1 3 4) |
4 |
| 12 | (1 2 1 3 4) |
5 |
G2/P1
- Ambient Space:
G2 - Keep Nodes:
1 - Bundle K: empty (the flag variety itself)
- Dimension: \(5\)
- Ambient Index: \(5\)
- Basis Rank: \(6\)
- Exact data identifier:
fd2bd6eae26b
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 5 & 0\\5 & 0 & 0 & 0 & 0 & 5\\0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 10 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (2 1) |
2 |
| 4 | (1 2 1) |
3 |
| 5 | (2 1 2 1) |
4 |
| 6 | (1 2 1 2 1) |
5 |
G2/P2
- Ambient Space:
G2 - Keep Nodes:
2 - Bundle K: empty (the flag variety itself)
- Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Exact data identifier:
834882311808
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 3 & 0 & 0 & 6\\3 & 0 & 0 & 3 & 0 & 0\\0 & 9 & 0 & 0 & 3 & 0\\0 & 0 & 6 & 0 & 0 & 3\\0 & 0 & 0 & 9 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (2) |
1 |
| 3 | (1 2) |
2 |
| 4 | (2 1 2) |
3 |
| 5 | (1 2 1 2) |
4 |
| 6 | (2 1 2 1 2) |
5 |
LG(3,6) / O(1)
- Ambient Space:
C3 - Keep Nodes:
3 - Bundle K:
1 - Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Exact data identifier:
141c91366f21
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 6 & 0 & 0 & 0\\3 & 0 & 0 & 3 & 0 & 0\\0 & 6 & 0 & 0 & 6 & 0\\0 & 0 & 12 & 0 & 0 & 6\\0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
| 6 | (2 1 3 2 3) |
5 |
OG(3,7) / O(2)
- Ambient Space:
B3 - Keep Nodes:
3 - Bundle K:
2 - Dimension: \(5\)
- Ambient Index: \(4\)
- Basis Rank: \(6\)
- Exact data identifier:
76928fabba80
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 8 & 0 & 0\\4 & 0 & 0 & 0 & 16 & 0\\0 & 4 & 0 & 0 & 0 & 8\\0 & 0 & 8 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
| 6 | (2 1 3 2 3) |
5 |
OG(3,7) / O(3)
- Ambient Space:
B3 - Keep Nodes:
3 - Bundle K:
3 - Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Exact data identifier:
29e94da02c49
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 36 & 0 & 0 & 216\\3 & 0 & 0 & 63 & 0 & 0\\0 & 3 & 0 & 0 & 63 & 0\\0 & 0 & 6 & 0 & 0 & 36\\0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
| 6 | (2 1 3 2 3) |
5 |
OG(3,7) / O(4)
- Ambient Space:
B3 - Keep Nodes:
3 - Bundle K:
4 - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(6\)
- Exact data identifier:
10e047830d24
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 96 & 0 & 26688 & 0 & 1603584\\2 & 0 & 512 & 0 & 78976 & 0\\0 & 2 & 0 & 416 & 0 & 26688\\0 & 0 & 4 & 0 & 512 & 0\\0 & 0 & 0 & 2 & 0 & 96\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
| 6 | (2 1 3 2 3) |
5 |
Fl(1,2,5) / O(0,1)+O(0,2)+O(1,0) (GM-20)
- Ambient Space:
A4 - Keep Nodes:
1,2 - Bundle K:
0,1;0,2;1,0 - Dimension: \(4\)
- Ambient Index: \(1\)
- Basis Rank: \(9\)
- Exact data identifier:
e1322438fd01
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 12 & 16 & 96 & 84 & 156 & 512 & 672 & 880\\1 & -1 & 6 & 4 & 0 & 8 & 72 & 72 & 160\\1 & 1 & 0 & 12 & 20 & 20 & 132 & 168 & 176\\0 & 1 & 2 & 0 & 0 & 4 & 8 & 8 & 36\\0 & 2 & 1 & 6 & -1 & 10 & 4 & 8 & 36\\0 & 0 & 1 & 0 & 1 & -2 & 16 & 16 & 24\\0 & 0 & 0 & 1 & 3 & 1 & 5 & 12 & 4\\0 & 0 & 0 & 1 & -1 & 2 & -2 & -6 & 4\\0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (2) |
1 |
| 4 | (1 2) |
2 |
| 5 | (2 1) |
2 |
| 6 | (3 2) |
2 |
| 7 | (1 2 1) |
3 |
| 8 | (1 3 2) |
3 |
| 9 | (1 3 2 1) |
4 |
The full matrix includes nonzero contributions from curve classes of anticanonical degree 5.
Fl(1,3,4) / O(1,1)
- Ambient Space:
A3 - Keep Nodes:
1,3 - Bundle K:
1,1 - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(9\)
- Exact data identifier:
3ed191207b19
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 2 & 2 & 0 & 0 & 0 & 4 & 8 & 0\\2 & 0 & 0 & 2 & 0 & 4 & 0 & 0 & 4\\2 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 4\\0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 2 & 0 & 0 & 0 & 0 & 2 & 2 & 0\\0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0\\0 & 0 & 0 & 0 & -4 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 6 & 6 & 2 & 0 & 0 & 2\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (3) |
1 |
| 4 | (1 3) |
2 |
| 5 | (2 1) |
2 |
| 6 | (2 3) |
2 |
| 7 | (1 2 3) |
3 |
| 8 | (2 1 3) |
3 |
| 9 | (1 2 1 3) |
4 |
G2/P2 / O(1)
- Ambient Space:
G2 - Keep Nodes:
2 - Bundle K:
1 - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Exact data identifier:
b9a91f8b2a8f
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 6 & 0 & 12 & 0\\2 & 0 & 4 & 0 & 4\\0 & 6 & 0 & 6 & 0\\0 & 0 & 4 & 0 & 2\\0 & 0 & 0 & 6 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (2) |
1 |
| 3 | (1 2) |
2 |
| 4 | (2 1 2) |
3 |
| 5 | (1 2 1 2) |
4 |
LG(3,6) / O(1)+O(1)
- Ambient Space:
C3 - Keep Nodes:
3 - Bundle K:
1;1 - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Exact data identifier:
5a3d521e7212
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 8 & 0 & 8 & 0\\2 & 0 & 8 & 0 & 8\\0 & 4 & 0 & 4 & 0\\0 & 0 & 8 & 0 & 8\\0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
OG(3,7) / O(2)+O(2)
- Ambient Space:
B3 - Keep Nodes:
3 - Bundle K:
2;2 - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Exact data identifier:
176397975a00
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 16 & 0 & 224 & 0\\2 & 0 & 48 & 0 & 224\\0 & 2 & 0 & 24 & 0\\0 & 0 & 4 & 0 & 16\\0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (3) |
1 |
| 3 | (2 3) |
2 |
| 4 | (1 2 3) |
3 |
| 5 | (1 3 2 3) |
4 |
B2/B / O(1,1)
- Ambient Space:
B2 - Keep Nodes:
1,2 - Bundle K:
1,1 - Dimension: \(3\)
- Ambient Index: \(1\)
- Basis Rank: \(6\)
- Exact data identifier:
34fec398819e
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 10 & 10 & 24 & 18 & 24\\1 & -1 & 2 & 8 & 6 & 6\\1 & 3 & -1 & 6 & 2 & 6\\0 & 1 & 2 & -1 & 2 & 2\\0 & 3 & 1 & 3 & -1 & 4\\0 & 0 & 0 & 4 & 3 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (2) |
1 |
| 4 | (1 2) |
2 |
| 5 | (2 1) |
2 |
| 6 | (1 2 1) |
3 |
Fl(1,3,4) / O(1,1)+O(1,1)
- Ambient Space:
A3 - Keep Nodes:
1,3 - Bundle K:
1,1;1,1 - Dimension: \(3\)
- Ambient Index: \(1\)
- Basis Rank: \(6\)
- Exact data identifier:
d3154e614ad6
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 14 & 14 & 72 & 48 & 56\\1 & -1 & 3 & 18 & 14 & 12\\1 & 3 & -1 & 18 & 10 & 12\\0 & 1 & \frac{7}{3} & 2 & 4 & \frac{14}{3}\\0 & 1 & -1 & 0 & -4 & 0\\0 & 0 & 0 & 6 & 4 & 0\end{bmatrix}
\]
Ordered basis and degrees
| Position | Schubert word | Complex codimension |
|---|---|---|
| 1 | () |
0 |
| 2 | (1) |
1 |
| 3 | (3) |
1 |
| 4 | (1 3) |
2 |
| 5 | (2 1) |
2 |
| 6 | (1 2 3) |
3 |