Grassmannians¶

Index convention: the displayed Ambient Index is the software's ambient Chern gcd. A smooth conic has geometric index 2 even when this field is 1; the zero-dimensional linear section is a point, for which this gcd does not define a geometric Fano index. See the catalog conventions.

This section contains complete intersections inside this geometric family, systematically organized by dimension and ambient index.

For the kept-root beta convention and copyable browser/CLI syntax, see How to Use.

Reproduce an entry in the software¶

For a Grassmannian Gr(k,n), run A(n-1) with Kept simple roots k. Each displayed / O(d) is one row in Bundle K: enter d, or enter d1;d2;... for several summands. For example, Gr(2,6) / O(1)+O(2) is A5, kept roots 2, and Bundle K 1;2 (or [[1],[2]] in Python). The card's dimension is k(n-k) minus the number of rows, and its ambient index is n - sum(d). This convention also covers the dual presentation Gr(n-k,n) shown in the catalog.

8-folds¶

Geometry	Ambient Index	Basis Rank
Gr(2,6)	6	15

Gr(2,6)

Ambient Space: $A5$
Dimension: $8$
Ambient Index: $6$
Basis Rank: $15$
Eigenvalues: $- 6 \sqrt{3}$ (mult: 1), $6 \sqrt{3}$ (mult: 1), $- 6 i$ (mult: 1), $6 i$ (mult: 1), $- 3 \sqrt{3} - 3 i$ (mult: 1), $- 3 \sqrt{3} + 3 i$ (mult: 1), $3 \sqrt{3} - 3 i$ (mult: 1), $3 \sqrt{3} + 3 i$ (mult: 1), $- 6 \sqrt{- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}}$ (mult: 1), $- 6 \sqrt{- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}}$ (mult: 1), $6 \sqrt{- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}}$ (mult: 1), $6 \sqrt{- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}}$ (mult: 1), $0$ (mult: 3)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0\\6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0\\0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0\\0 & 0 & 6 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\0 & 0 & 0 & 0 & 6 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 6 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 6 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0\end{bmatrix} \]

7-folds¶

Geometry	Ambient Index	Basis Rank
Gr(2,6) / O(1)	5	12

Gr(2,6) / O(1)

Ambient Space: $A5$
Dimension: $7$
Ambient Index: $5$
Basis Rank: $12$
Eigenvalues: $-5$ (mult: 1), $- \frac{5 \sqrt{5}}{4} + \frac{5}{4} - 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1), $- \frac{5 \sqrt{5}}{4} + \frac{5}{4} + 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1), $\frac{5}{4} + \frac{5 \sqrt{5}}{4} - 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $\frac{5}{4} + \frac{5 \sqrt{5}}{4} + 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $5 \cdot 3^{\frac{3}{5}}$ (mult: 1), $- \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - \frac{5 \cdot 3^{\frac{3}{5}}}{4} - 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - \frac{5 \cdot 3^{\frac{3}{5}}}{4} + 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{5 \cdot 3^{\frac{3}{5}}}{4} + \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1), $- \frac{5 \cdot 3^{\frac{3}{5}}}{4} + \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} + 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 & 0 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\\0 & 0 & 5 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\0 & 0 & 0 & 0 & 5 & 10 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & -5 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 10 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix} \]

6-folds¶

Geometry	Ambient Index	Basis Rank
Gr(2,5)	5	10
Gr(3,5)	5	10
Gr(2,6) / O(1)+O(1)	4	10
Gr(2,6) / O(1)+O(2)	3	10

Gr(2,5)

Ambient Space: $A4$
Dimension: $6$
Ambient Index: $5$
Basis Rank: $10$
Eigenvalues: $\frac{5}{2} - \frac{5 \sqrt{5}}{2}$ (mult: 1), $\frac{5}{2} + \frac{5 \sqrt{5}}{2}$ (mult: 1), $\frac{5}{2} - 5 \sqrt{- \frac{5}{4} - \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} - 5 \sqrt{- \frac{5}{4} + \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} + 5 \sqrt{- \frac{5}{4} - \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} + 5 \sqrt{- \frac{5}{4} + \frac{\sqrt{5}}{2}}$ (mult: 1), $- \frac{15}{4} + \frac{5 \sqrt{5}}{4} - 5 \sqrt{- \frac{5}{8} + \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} - \frac{5 \sqrt{5}}{4} - 5 \sqrt{- \frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} - \frac{5 \sqrt{5}}{4} + 5 \sqrt{- \frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} + \frac{5 \sqrt{5}}{4} + 5 \sqrt{- \frac{5}{8} + \frac{\sqrt{5}}{8}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\0 & 0 & 5 & 5 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix} \]

Gr(3,5)

Ambient Space: $A4$
Dimension: $6$
Ambient Index: $5$
Basis Rank: $10$
Eigenvalues: $\frac{5}{2} - \frac{5 \sqrt{5}}{2}$ (mult: 1), $\frac{5}{2} + \frac{5 \sqrt{5}}{2}$ (mult: 1), $\frac{5}{2} - 5 \sqrt{- \frac{5}{4} - \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} - 5 \sqrt{- \frac{5}{4} + \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} + 5 \sqrt{- \frac{5}{4} - \frac{\sqrt{5}}{2}}$ (mult: 1), $\frac{5}{2} + 5 \sqrt{- \frac{5}{4} + \frac{\sqrt{5}}{2}}$ (mult: 1), $- \frac{15}{4} + \frac{5 \sqrt{5}}{4} - 5 \sqrt{- \frac{5}{8} + \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} - \frac{5 \sqrt{5}}{4} - 5 \sqrt{- \frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} - \frac{5 \sqrt{5}}{4} + 5 \sqrt{- \frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{15}{4} + \frac{5 \sqrt{5}}{4} + 5 \sqrt{- \frac{5}{8} + \frac{\sqrt{5}}{8}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 5 & 5 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix} \]

Gr(2,6) / O(1)+O(1)

Ambient Space: $A5$
Dimension: $6$
Ambient Index: $4$
Basis Rank: $10$
Eigenvalues: $- 4 \cdot 3^{\frac{3}{4}}$ (mult: 1), $4 \cdot 3^{\frac{3}{4}}$ (mult: 1), $- 4 \cdot 3^{\frac{3}{4}} i$ (mult: 1), $4 \cdot 3^{\frac{3}{4}} i$ (mult: 1), $- 2 \sqrt{2} - 2 \sqrt{2} i$ (mult: 1), $- 2 \sqrt{2} + 2 \sqrt{2} i$ (mult: 1), $2 \sqrt{2} - 2 \sqrt{2} i$ (mult: 1), $2 \sqrt{2} + 2 \sqrt{2} i$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 4 & 8 & 0 & 0 & 0 & 0\\4 & 0 & 0 & 0 & 0 & 0 & 8 & 4 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 0 & 0 & 0 & 4 & 8 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 4 & -4 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 12 & 8 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix} \]

Gr(2,6) / O(1)+O(2)

Ambient Space: $A5$
Dimension: $6$
Ambient Index: $3$
Basis Rank: $10$
Eigenvalues: $9 \cdot 2^{\frac{2}{3}}$ (mult: 1), $- \frac{9 \cdot 2^{\frac{2}{3}}}{2} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $- \frac{9 \cdot 2^{\frac{2}{3}}}{2} + \frac{9 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $- 3 \cdot 2^{\frac{2}{3}}$ (mult: 1), $\frac{3 \cdot 2^{\frac{2}{3}}}{2} - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $\frac{3 \cdot 2^{\frac{2}{3}}}{2} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $0$ (mult: 4)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 6 & 18 & 0 & 0 & 0 & 0 & 72 & 0\\3 & 0 & 0 & 0 & 24 & 30 & 0 & 0 & 0 & 72\\0 & 3 & 0 & 0 & 0 & 0 & 12 & 6 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 30 & 18 & 0 & 0\\0 & 0 & 3 & 3 & 0 & 0 & 0 & 0 & 6 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 12 & 0\\0 & 0 & 0 & 0 & 3 & 6 & 0 & 0 & 0 & 12\\0 & 0 & 0 & 0 & 3 & -3 & 0 & 0 & 0 & -6\\0 & 0 & 0 & 0 & 0 & 0 & 9 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix} \]

5-folds¶

Geometry	Ambient Index	Basis Rank
Gr(2,5) / O(1)	4	8
Gr(3,5) / O(1)	4	8
Gr(2,5) / O(2)	3	8
Gr(3,5) / O(2)	3	8
Gr(2,6) / O(1)+O(1)+O(1)	3	8
Gr(2,5) / O(3)	2	8
Gr(3,5) / O(3)	2	8
Gr(2,6) / O(1)+O(1)+O(2)	2	8

Gr(2,5) / O(1)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $4$
Basis Rank: $8$
Eigenvalues: $- 2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} + 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} - 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} - 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} + 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \cdot 2^{\frac{3}{4}} i \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $2 \cdot 2^{\frac{3}{4}} i \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \cdot 2^{\frac{3}{4}} \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $2 \cdot 2^{\frac{3}{4}} \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 4 & 4 & 0 & 0\\4 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 0 & 4 & 4 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 8 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix} \]

Gr(3,5) / O(1)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $4$
Basis Rank: $8$
Eigenvalues: $- 2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} + 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} - 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} - 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $2 \sqrt[4]{2} \sqrt[4]{-11 + 5 \sqrt{5}} + 2 \sqrt[4]{2} i \sqrt[4]{-11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \cdot 2^{\frac{3}{4}} i \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $2 \cdot 2^{\frac{3}{4}} i \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $- 2 \cdot 2^{\frac{3}{4}} \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1), $2 \cdot 2^{\frac{3}{4}} \sqrt[4]{11 + 5 \sqrt{5}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 4 & 4 & 0 & 0\\4 & 0 & 0 & 0 & 0 & 0 & 4 & 0\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 4\\0 & 4 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 4 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 4 & 4 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 8 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix} \]

Gr(2,5) / O(2)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $3$
Basis Rank: $8$
Eigenvalues: $- 3 \sqrt[3]{-22 + 10 \sqrt{5}}$ (mult: 1), $\frac{3 \sqrt[3]{-22 + 10 \sqrt{5}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{-22 + 10 \sqrt{5}}}{2}$ (mult: 1), $\frac{3 \sqrt[3]{-22 + 10 \sqrt{5}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{-22 + 10 \sqrt{5}}}{2}$ (mult: 1), $3 \sqrt[3]{22 + 10 \sqrt{5}}$ (mult: 1), $- \frac{3 \sqrt[3]{22 + 10 \sqrt{5}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{22 + 10 \sqrt{5}}}{2}$ (mult: 1), $- \frac{3 \sqrt[3]{22 + 10 \sqrt{5}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{22 + 10 \sqrt{5}}}{2}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 6 & 12 & 0 & 0 & 0 & 24\\3 & 0 & 0 & 0 & 18 & 12 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 6 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 12 & 0\\0 & 0 & 3 & 3 & 0 & 0 & 0 & 6\\0 & 0 & 0 & 3 & 0 & 0 & 0 & 6\\0 & 0 & 0 & 0 & 6 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix} \]

Gr(3,5) / O(2)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $3$
Basis Rank: $8$
Eigenvalues: $- 3 \sqrt[3]{-22 + 10 \sqrt{5}}$ (mult: 1), $\frac{3 \sqrt[3]{-22 + 10 \sqrt{5}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{-22 + 10 \sqrt{5}}}{2}$ (mult: 1), $\frac{3 \sqrt[3]{-22 + 10 \sqrt{5}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{-22 + 10 \sqrt{5}}}{2}$ (mult: 1), $3 \sqrt[3]{22 + 10 \sqrt{5}}$ (mult: 1), $- \frac{3 \sqrt[3]{22 + 10 \sqrt{5}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{22 + 10 \sqrt{5}}}{2}$ (mult: 1), $- \frac{3 \sqrt[3]{22 + 10 \sqrt{5}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{22 + 10 \sqrt{5}}}{2}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 12 & 6 & 0 & 0 & 0 & 24\\3 & 0 & 0 & 0 & 12 & 18 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 12 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 6 & 0\\0 & 0 & 3 & 0 & 0 & 0 & 0 & 6\\0 & 0 & 3 & 3 & 0 & 0 & 0 & 6\\0 & 0 & 0 & 0 & 3 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix} \]

Gr(2,6) / O(1)+O(1)+O(1)

Ambient Space: $A5$
Dimension: $5$
Ambient Index: $3$
Basis Rank: $8$
Eigenvalues: $9$ (mult: 1), $-3$ (mult: 1), $\frac{3}{2} - \frac{3 \sqrt{3} i}{2}$ (mult: 1), $\frac{3}{2} + \frac{3 \sqrt{3} i}{2}$ (mult: 1), $- \frac{9}{2} - \frac{9 \sqrt{3} i}{2}$ (mult: 1), $- \frac{9}{2} + \frac{9 \sqrt{3} i}{2}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 3 & 9 & 0 & 0 & 0 & 6\\3 & 0 & 0 & 0 & 9 & 9 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 0 & 9 & 0\\0 & 0 & 3 & 3 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 0 & 3\\0 & 0 & 0 & 0 & 5 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 9 & 0\end{bmatrix} \]

Gr(2,5) / O(3)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $2$
Basis Rank: $8$
Eigenvalues: $- 6 i \sqrt{- \frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $6 i \sqrt{- \frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $- 6 \sqrt{\frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $6 \sqrt{\frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $0$ (mult: 4)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 36 & 0 & 0 & 2232 & 1224 & 0 & 26568\\2 & 0 & 54 & 96 & 0 & 0 & 2988 & 0\\0 & 2 & 0 & 0 & 72 & 42 & 0 & 1008\\0 & 2 & 0 & 0 & 96 & 54 & 0 & 1224\\0 & 0 & 2 & 2 & 0 & 0 & 54 & 0\\0 & 0 & 0 & 2 & 0 & 0 & 42 & 0\\0 & 0 & 0 & 0 & 4 & 2 & 0 & 36\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix} \]

Gr(3,5) / O(3)

Ambient Space: $A4$
Dimension: $5$
Ambient Index: $2$
Basis Rank: $8$
Eigenvalues: $- 6 i \sqrt{- \frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $6 i \sqrt{- \frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $- 6 \sqrt{\frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $6 \sqrt{\frac{33}{2} + \frac{15 \sqrt{5}}{2}}$ (mult: 1), $0$ (mult: 4)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 36 & 0 & 0 & 1224 & 2232 & 0 & 26568\\2 & 0 & 96 & 54 & 0 & 0 & 2988 & 0\\0 & 2 & 0 & 0 & 54 & 96 & 0 & 1224\\0 & 2 & 0 & 0 & 42 & 72 & 0 & 1008\\0 & 0 & 2 & 0 & 0 & 0 & 42 & 0\\0 & 0 & 2 & 2 & 0 & 0 & 54 & 0\\0 & 0 & 0 & 0 & 2 & 4 & 0 & 36\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix} \]

Note: This catalog is automatically generated.

Gr(2,6) / O(1)+O(1)+O(2)

Ambient Space: $A5$
Dimension: $5$
Ambient Index: $2$
Basis Rank: $8$
Eigenvalues: $- 12 \sqrt{3}$ (mult: 1), $12 \sqrt{3}$ (mult: 1), $- 4 i$ (mult: 1), $4 i$ (mult: 1), $0$ (mult: 4)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 16 & 0 & 0 & 336 & 288 & 0 & 768\\2 & 0 & 16 & 36 & 0 & 0 & 480 & 0\\0 & 2 & 0 & 0 & 12 & 12 & 0 & 48\\0 & 2 & 0 & 0 & 32 & 28 & 0 & 80\\0 & 0 & 2 & 2 & 0 & 0 & 12 & 0\\0 & 0 & 0 & 2 & 0 & 0 & 24 & 0\\0 & 0 & 0 & 0 & \frac{10}{3} & \frac{8}{3} & 0 & \frac{16}{3}\\0 & 0 & 0 & 0 & 0 & 0 & 6 & 0\end{bmatrix} \]

4-folds¶

Geometry	Ambient Index	Basis Rank
Gr(4,5)	5	5
Gr(2,4)	4	6
Gr(2,5) / O(1)+O(1)	3	6
Gr(3,5) / O(1)+O(1)	3	6
Gr(2,5) / O(1)+O(2)	2	6
Gr(3,5) / O(1)+O(2)	2	6
Gr(2,5) / O(1)+O(3)	1	6
Gr(2,5) / O(2)+O(2)	1	6
Gr(3,5) / O(1)+O(3)	1	6
Gr(3,5) / O(2)+O(2)	1	6

Gr(4,5)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $5$
Basis Rank: $5$
Eigenvalues: $5$ (mult: 1), $- \frac{5 \sqrt{5}}{4} - \frac{5}{4} - 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{5 \sqrt{5}}{4} - \frac{5}{4} + 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$ (mult: 1), $- \frac{5}{4} + \frac{5 \sqrt{5}}{4} - 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1), $- \frac{5}{4} + \frac{5 \sqrt{5}}{4} + 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 5\\5 & 0 & 0 & 0 & 0\\0 & 5 & 0 & 0 & 0\\0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 5 & 0\end{bmatrix} \]

Gr(2,4)

Ambient Space: $A3$
Dimension: $4$
Ambient Index: $4$
Basis Rank: $6$
Eigenvalues: $- 4 \sqrt{2}$ (mult: 1), $4 \sqrt{2}$ (mult: 1), $- 4 \sqrt{2} i$ (mult: 1), $4 \sqrt{2} i$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 0 & 4 & 0\\4 & 0 & 0 & 0 & 0 & 4\\0 & 4 & 0 & 0 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 0\\0 & 0 & 4 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0\end{bmatrix} \]

Gr(2,5) / O(1)+O(1)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $3$
Basis Rank: $6$
Eigenvalues: $- 3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $\frac{3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $\frac{3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- \frac{3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $- \frac{3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 3 & 6 & 0 & 0\\3 & 0 & 0 & 0 & 6 & 0\\0 & 3 & 0 & 0 & 0 & 0\\0 & 3 & 0 & 0 & 0 & 3\\0 & 0 & 3 & \frac{9}{2} & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0\end{bmatrix} \]

Gr(3,5) / O(1)+O(1)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $3$
Basis Rank: $6$
Eigenvalues: $- 3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $\frac{3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $\frac{3 \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- \frac{3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} - \frac{3 \sqrt{3} i \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1), $- \frac{3 \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2} + \frac{3 \sqrt{3} i \sqrt[3]{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 6 & 3 & 0 & 0\\3 & 0 & 0 & 0 & 3 & 0\\0 & 3 & 0 & 0 & 0 & 3\\0 & 3 & 0 & 0 & 0 & 0\\0 & 0 & 9 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix} \]

Gr(2,5) / O(1)+O(2)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $2$
Basis Rank: $6$
Eigenvalues: $- 4 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $4 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- 4 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $4 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 12 & 0 & 0 & 96 & 0\\2 & 0 & 12 & 20 & 0 & 48\\0 & 2 & 0 & 0 & 8 & 0\\0 & 2 & 0 & 0 & 16 & 0\\0 & 0 & 2 & 3 & 0 & 6\\0 & 0 & 0 & 0 & 4 & 0\end{bmatrix} \]

Gr(3,5) / O(1)+O(2)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $2$
Basis Rank: $6$
Eigenvalues: $- 4 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $4 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- 4 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $4 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 12 & 0 & 0 & 48 & 0\\2 & 0 & 20 & 12 & 0 & 48\\0 & 2 & 0 & 0 & 8 & 0\\0 & 2 & 0 & 0 & 4 & 0\\0 & 0 & 6 & 4 & 0 & 12\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix} \]

Gr(2,5) / O(1)+O(3)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $1$
Basis Rank: $6$
Eigenvalues: $-18$ (mult: 4), $\frac{261}{2} - \frac{135 \sqrt{5}}{2}$ (mult: 1), $\frac{261}{2} + \frac{135 \sqrt{5}}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 2772 & 200988 & 310284 & 23784192 & 430673760\\1 & 57 & 5328 & 8244 & 644490 & 11892096\\0 & 1 & 18 & 57 & 4824 & 91692\\0 & 1 & 48 & 57 & 5832 & 109296\\0 & 0 & 1 & \frac{3}{2} & 57 & 1386\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix} \]

Gr(2,5) / O(2)+O(2)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $1$
Basis Rank: $6$
Eigenvalues: $-12$ (mult: 4), $76 - 40 \sqrt{5}$ (mult: 1), $76 + 40 \sqrt{5}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 1080 & 46080 & 71424 & 3301632 & 39398400\\1 & 32 & 1848 & 2872 & 135552 & 1650816\\0 & 1 & 8 & 32 & 1648 & 20736\\0 & 1 & 28 & 32 & 2048 & 25344\\0 & 0 & 1 & \frac{3}{2} & 32 & 540\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix} \]

Gr(3,5) / O(1)+O(3)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $1$
Basis Rank: $6$
Eigenvalues: $-18$ (mult: 4), $\frac{261}{2} - \frac{135 \sqrt{5}}{2}$ (mult: 1), $\frac{261}{2} + \frac{135 \sqrt{5}}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 2772 & 310284 & 200988 & 11892096 & 430673760\\1 & 57 & 8244 & 5328 & 322245 & 11892096\\0 & 1 & 57 & 48 & 2916 & 109296\\0 & 1 & 57 & 18 & 2412 & 91692\\0 & 0 & 3 & 2 & 57 & 2772\\0 & 0 & 0 & 0 & 1 & 0\end{bmatrix} \]

Gr(3,5) / O(2)+O(2)

Ambient Space: $A4$
Dimension: $4$
Ambient Index: $1$
Basis Rank: $6$
Eigenvalues: $-12$ (mult: 4), $76 - 40 \sqrt{5}$ (mult: 1), $76 + 40 \sqrt{5}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 1080 & 71424 & 46080 & 1650816 & 39398400\\1 & 32 & 2872 & 1848 & 67776 & 1650816\\0 & 1 & 32 & 28 & 1024 & 25344\\0 & 1 & 32 & 8 & 824 & 20736\\0 & 0 & 3 & 2 & 32 & 1080\\0 & 0 & 0 & 0 & 1 & 0\end{bmatrix} \]

3-folds¶

Geometry	Ambient Index	Basis Rank
Gr(3,4)	4	4
Gr(4,5) / O(1)	4	4
Gr(2,4) / O(1)	3	4
Gr(4,5) / O(2)	3	4
Gr(2,4) / O(2)	2	4
Gr(2,5) / O(1)+O(1)+O(1)	2	4
Gr(3,5) / O(1)+O(1)+O(1)	2	4
Gr(4,5) / O(3)	2	4
Gr(2,4) / O(3)	1	4
Gr(2,5) / O(1)+O(1)+O(2)	1	4
Gr(3,5) / O(1)+O(1)+O(2)	1	4
Gr(4,5) / O(4)	1	4

Gr(3,4)

Ambient Space: $A3$
Dimension: $3$
Ambient Index: $4$
Basis Rank: $4$
Eigenvalues: $-4$ (mult: 1), $4$ (mult: 1), $- 4 i$ (mult: 1), $4 i$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 4\\4 & 0 & 0 & 0\\0 & 4 & 0 & 0\\0 & 0 & 4 & 0\end{bmatrix} \]

Gr(4,5) / O(1)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $4$
Basis Rank: $4$
Eigenvalues: $-4$ (mult: 1), $4$ (mult: 1), $- 4 i$ (mult: 1), $4 i$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 0 & 4\\4 & 0 & 0 & 0\\0 & 4 & 0 & 0\\0 & 0 & 4 & 0\end{bmatrix} \]

Gr(2,4) / O(1)

Ambient Space: $A3$
Dimension: $3$
Ambient Index: $3$
Basis Rank: $4$
Eigenvalues: $3 \cdot 2^{\frac{2}{3}}$ (mult: 1), $- \frac{3 \cdot 2^{\frac{2}{3}}}{2} - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $- \frac{3 \cdot 2^{\frac{2}{3}}}{2} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 3 & 0\\3 & 0 & 0 & 3\\0 & 6 & 0 & 0\\0 & 0 & 3 & 0\end{bmatrix} \]

Gr(4,5) / O(2)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $3$
Basis Rank: $4$
Eigenvalues: $3 \cdot 2^{\frac{2}{3}}$ (mult: 1), $- \frac{3 \cdot 2^{\frac{2}{3}}}{2} - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $- \frac{3 \cdot 2^{\frac{2}{3}}}{2} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}$ (mult: 1), $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 6 & 0\\3 & 0 & 0 & 6\\0 & 3 & 0 & 0\\0 & 0 & 3 & 0\end{bmatrix} \]

Gr(2,4) / O(2)

Ambient Space: $A3$
Dimension: $3$
Ambient Index: $2$
Basis Rank: $4$
Eigenvalues: $-8$ (mult: 1), $8$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 8 & 0 & 16\\2 & 0 & 8 & 0\\0 & 4 & 0 & 8\\0 & 0 & 2 & 0\end{bmatrix} \]

Gr(2,5) / O(1)+O(1)+O(1)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $2$
Basis Rank: $4$
Eigenvalues: $- 2 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $2 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- 2 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $2 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 6 & 0 & 8\\2 & 0 & 4 & 0\\0 & 5 & 0 & 6\\0 & 0 & 2 & 0\end{bmatrix} \]

Gr(3,5) / O(1)+O(1)+O(1)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $2$
Basis Rank: $4$
Eigenvalues: $- 2 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $2 i \sqrt{- \frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $- 2 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1), $2 \sqrt{\frac{11}{2} + \frac{5 \sqrt{5}}{2}}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 6 & 0 & 4\\2 & 0 & 6 & 0\\0 & \frac{10}{3} & 0 & 2\\0 & 0 & 6 & 0\end{bmatrix} \]

Gr(4,5) / O(3)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $2$
Basis Rank: $4$
Eigenvalues: $- 6 \sqrt{3}$ (mult: 1), $6 \sqrt{3}$ (mult: 1), $0$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 12 & 0 & 72\\2 & 0 & 30 & 0\\0 & 2 & 0 & 12\\0 & 0 & 2 & 0\end{bmatrix} \]

Gr(2,4) / O(3)

Ambient Space: $A3$
Dimension: $3$
Ambient Index: $1$
Basis Rank: $4$
Eigenvalues: $96$ (mult: 1), $-12$ (mult: 3)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 792 & 21816 & 396576\\1 & 30 & 1170 & 21816\\0 & 2 & 30 & 792\\0 & 0 & 1 & 0\end{bmatrix} \]

Gr(2,5) / O(1)+O(1)+O(2)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $1$
Basis Rank: $4$
Eigenvalues: $-6$ (mult: 2), $16 - 10 \sqrt{5}$ (mult: 1), $16 + 10 \sqrt{5}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 156 & 1440 & 13248\\1 & 10 & 152 & 1440\\0 & \frac{5}{2} & 10 & 156\\0 & 0 & 1 & 0\end{bmatrix} \]

Gr(3,5) / O(1)+O(1)+O(2)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $1$
Basis Rank: $4$
Eigenvalues: $-6$ (mult: 2), $16 - 10 \sqrt{5}$ (mult: 1), $16 + 10 \sqrt{5}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 156 & 2160 & 6624\\1 & 10 & 228 & 720\\0 & \frac{5}{3} & 10 & 52\\0 & 0 & 3 & 0\end{bmatrix} \]

Gr(4,5) / O(4)

Ambient Space: $A4$
Dimension: $3$
Ambient Index: $1$
Basis Rank: $4$
Eigenvalues: $232$ (mult: 1), $-24$ (mult: 3)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 3888 & 504576 & 18323712\\1 & 80 & 13600 & 504576\\0 & 1 & 80 & 3888\\0 & 0 & 1 & 0\end{bmatrix} \]

Surfaces (2-folds)¶

Geometry	Ambient Index	Basis Rank
Gr(3,4) / O(1)	3	3
Gr(4,5) / O(1)+O(1)	3	3
Gr(2,4) / O(1)+O(1)	2	3
Gr(3,4) / O(2)	2	3
Gr(4,5) / O(1)+O(2)	2	3
Gr(2,4) / O(1)+O(2)	1	3
Gr(3,4) / O(3)	1	3
Gr(4,5) / O(1)+O(3)	1	3
Gr(4,5) / O(2)+O(2)	1	3

Gr(3,4) / O(1)

Ambient Space: $A3$
Dimension: $2$
Ambient Index: $3$
Basis Rank: $3$
Eigenvalues: $3$ (mult: 1), $- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}$ (mult: 1), $- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 3\\3 & 0 & 0\\0 & 3 & 0\end{bmatrix} \]

Gr(4,5) / O(1)+O(1)

Ambient Space: $A4$
Dimension: $2$
Ambient Index: $3$
Basis Rank: $3$
Eigenvalues: $3$ (mult: 1), $- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}$ (mult: 1), $- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 0 & 3\\3 & 0 & 0\\0 & 3 & 0\end{bmatrix} \]

Gr(2,4) / O(1)+O(1)

Ambient Space: $A3$
Dimension: $2$
Ambient Index: $2$
Basis Rank: $3$
Eigenvalues: $-4$ (mult: 1), $4$ (mult: 1), $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4 & 0\\2 & 0 & 2\\0 & 4 & 0\end{bmatrix} \]

Gr(3,4) / O(2)

Ambient Space: $A3$
Dimension: $2$
Ambient Index: $2$
Basis Rank: $3$
Eigenvalues: $-4$ (mult: 1), $4$ (mult: 1), $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4 & 0\\2 & 0 & 4\\0 & 2 & 0\end{bmatrix} \]

Gr(4,5) / O(1)+O(2)

Ambient Space: $A4$
Dimension: $2$
Ambient Index: $2$
Basis Rank: $3$
Eigenvalues: $-4$ (mult: 1), $4$ (mult: 1), $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4 & 0\\2 & 0 & 4\\0 & 2 & 0\end{bmatrix} \]

Gr(2,4) / O(1)+O(2)

Ambient Space: $A3$
Dimension: $2$
Ambient Index: $1$
Basis Rank: $3$
Eigenvalues: $12$ (mult: 1), $-4$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 40 & 96\\1 & 4 & 20\\0 & 2 & 0\end{bmatrix} \]

Gr(3,4) / O(3)

Ambient Space: $A3$
Dimension: $2$
Ambient Index: $1$
Basis Rank: $3$
Eigenvalues: $21$ (mult: 1), $-6$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 108 & 756\\1 & 9 & 108\\0 & 1 & 0\end{bmatrix} \]

Gr(4,5) / O(1)+O(3)

Ambient Space: $A4$
Dimension: $2$
Ambient Index: $1$
Basis Rank: $3$
Eigenvalues: $21$ (mult: 1), $-6$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 108 & 756\\1 & 9 & 108\\0 & 1 & 0\end{bmatrix} \]

Gr(4,5) / O(2)+O(2)

Ambient Space: $A4$
Dimension: $2$
Ambient Index: $1$
Basis Rank: $3$
Eigenvalues: $12$ (mult: 1), $-4$ (mult: 2)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 40 & 192\\1 & 4 & 40\\0 & 1 & 0\end{bmatrix} \]

Curves (1-folds)¶

Geometry	Ambient Index	Basis Rank
Gr(3,4) / O(1)+O(1)	2	2
Gr(4,5) / O(1)+O(1)+O(1)	2	2
Gr(2,4) / O(1)+O(1)+O(1)	1	2
Gr(3,4) / O(1)+O(2)	1	2
Gr(4,5) / O(1)+O(1)+O(2)	1	2

Gr(3,4) / O(1)+O(1)

Ambient Space: $A3$
Dimension: $1$
Ambient Index: $2$
Basis Rank: $2$
Eigenvalues: $-2$ (mult: 1), $2$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 2\\2 & 0\end{bmatrix} \]

Gr(4,5) / O(1)+O(1)+O(1)

Ambient Space: $A4$
Dimension: $1$
Ambient Index: $2$
Basis Rank: $2$
Eigenvalues: $-2$ (mult: 1), $2$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 2\\2 & 0\end{bmatrix} \]

Gr(2,4) / O(1)+O(1)+O(1)

Ambient Space: $A3$
Dimension: $1$
Ambient Index: $1$
Basis Rank: $2$
Eigenvalues: $-2$ (mult: 1), $2$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4\\1 & 0\end{bmatrix} \]

Gr(3,4) / O(1)+O(2)

Ambient Space: $A3$
Dimension: $1$
Ambient Index: $1$
Basis Rank: $2$
Eigenvalues: $-2$ (mult: 1), $2$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4\\1 & 0\end{bmatrix} \]

Gr(4,5) / O(1)+O(1)+O(2)

Ambient Space: $A4$
Dimension: $1$
Ambient Index: $1$
Basis Rank: $2$
Eigenvalues: $-2$ (mult: 1), $2$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0 & 4\\1 & 0\end{bmatrix} \]

0-folds¶

Geometry	Ambient Index	Basis Rank
Gr(3,4) / O(1)+O(1)+O(1)	1	1

Gr(3,4) / O(1)+O(1)+O(1)

Ambient Space: $A3$
Dimension: $0$
Ambient Index: $1$
Basis Rank: $1$
Eigenvalues: $0$ (mult: 1)

Quantum Matrix ($y=1$)

\[ \begin{bmatrix}0\end{bmatrix} \]