k=3

The cohomology of $C_2$-equivariant Grassmannians $\text{Gr}_k(\mathbb{R}^{p,q})$ when $k=3$

$k=3$	$q=1$	$q=2$	$q=3$	$q=4$	$q=5$	$q=6$
$p=6$	$\text{Gr}_3(\mathbb{R}^{6,1})$	$\text{Gr}_3(\mathbb{R}^{6,2})$	$\text{Gr}_3(\mathbb{R}^{6,3})$
$p=7$	$\text{Gr}_3(\mathbb{R}^{7,1})$	2 possible	14 possible
$p=8$	$\text{Gr}_3(\mathbb{R}^{8,1})$	4 possible	168 possible	?
$p=9$	2 possible	?	?	?
$p=10$	4 possible	?	?	?	?
$p=11$	8 possible	?	?	?	?
$p=12$	24 possible	?	?	?	?	?

The cohomology of $\text{Gr}_3(\mathbb{R}^{6,1})$

Poincaré polynomial: $$x^{9} y^{3} + x^{8} y^{2} + 2 x^{7} y^{2} + 3 x^{6} y^{2} + 2 x^{5} y^{2} + x^{5} y + x^{4} y^{2} + 2 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{6,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_3(\mathbb{R}^{6,2})$

Poincaré polynomial: $$x^{9} y^{4} + x^{8} y^{4} + x^{7} y^{4} + x^{7} y^{3} + 3 x^{6} y^{3} + 2 x^{5} y^{3} + x^{5} y^{2} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{6,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{2,2}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_3(\mathbb{R}^{6,3})$

For details of this computation, see Section 6 of this paper .

Poincare polynomial: $$x^{9} y^{5} + x^{8} y^{4} + 2 x^{7} y^{4} + x^{6} y^{4} + 2 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{6,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{2,2}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{3,3}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{4,3}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{6,4}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_3(\mathbb{R}^{7,1})$

Poincaré polynomial: $$x^{12} y^{3} + x^{11} y^{3} + x^{10} y^{3} + x^{10} y^{2} + x^{9} y^{3} + 2 x^{9} y^{2} + 4 x^{8} y^{2} + 4 x^{7} y^{2} + 4 x^{6} y^{2} + x^{6} y + 2 x^{5} y^{2} + 2 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{7,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2\oplus\Sigma^{12,3}\mathbb{M}_2.$$

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{7,2})$.

There are 2 possibilities.
Here are their Poincaré polynomials: $$x^{12} y^{5} + x^{11} y^{5} + 2 x^{10} y^{4} + 3 x^{9} y^{4} + 3 x^{8} y^{4} + x^{8} y^{3} + x^{7} y^{4} + 3 x^{7} y^{3} + 4 x^{6} y^{3} + x^{6} y^{2} + 2 x^{5} y^{3} + 2 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{5} + x^{11} y^{5} + 2 x^{10} y^{4} + 3 x^{9} y^{4} + 3 x^{8} y^{4} + x^{8} y^{3} + 4 x^{7} y^{3} + 5 x^{6} y^{3} + 2 x^{5} y^{3} + 2 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 2 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{7,3})$.

There are 14 possibilities.
Here are their Poincaré polynomials: $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + x^{5} y^{5} + 4 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + x^{5} y^{5} + 3 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{4} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{4} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + x^{4} y^{4} + x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + x^{4} y^{4} + x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$ $$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$ Here are the corresponding generator grids to these 14 possibilities:

The cohomology of $\text{Gr}_3(\mathbb{R}^{8,1})$

Poincaré polynomial: $$x^{15} y^{3} + x^{14} y^{3} + 2 x^{13} y^{3} + 2 x^{12} y^{3} + x^{12} y^{2} + 2 x^{11} y^{3} + 2 x^{11} y^{2} + x^{10} y^{3} + 4 x^{10} y^{2} + x^{9} y^{3} + 5 x^{9} y^{2} + 6 x^{8} y^{2} + 5 x^{7} y^{2} + x^{7} y + 4 x^{6} y^{2} + 2 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{8,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,2}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2\oplus\Sigma^{12,2}\mathbb{M}_2\oplus\Sigma^{12,3}\mathbb{M}_2\oplus\Sigma^{13,3}\mathbb{M}_2\oplus\Sigma^{14,3}\mathbb{M}_2\oplus\Sigma^{15,3}\mathbb{M}_2.$$

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{8,2})$.

There are 4 possibilities.
Here are their Poincaré polynomials: $$x^{15} y^{6} + x^{14} y^{5} + 2 x^{13} y^{5} + 2 x^{12} y^{5} + x^{12} y^{4} + x^{11} y^{5} + 3 x^{11} y^{4} + 5 x^{10} y^{4} + 5 x^{9} y^{4} + x^{9} y^{3} + 3 x^{8} y^{4} + 3 x^{8} y^{3} + x^{7} y^{4} + 4 x^{7} y^{3} + x^{7} y^{2} + 4 x^{6} y^{3} + 2 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{15} y^{6} + x^{14} y^{5} + 2 x^{13} y^{5} + 2 x^{12} y^{5} + x^{12} y^{4} + x^{11} y^{5} + 3 x^{11} y^{4} + 5 x^{10} y^{4} + 5 x^{9} y^{4} + x^{9} y^{3} + 2 x^{8} y^{4} + 4 x^{8} y^{3} + x^{7} y^{4} + 5 x^{7} y^{3} + 4 x^{6} y^{3} + 2 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{15} y^{6} + x^{14} y^{5} + 2 x^{13} y^{5} + 2 x^{12} y^{5} + x^{12} y^{4} + x^{11} y^{5} + 3 x^{11} y^{4} + 5 x^{10} y^{4} + 5 x^{9} y^{4} + x^{9} y^{3} + 3 x^{8} y^{4} + 3 x^{8} y^{3} + 5 x^{7} y^{3} + x^{7} y^{2} + 5 x^{6} y^{3} + x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{15} y^{6} + x^{14} y^{5} + 2 x^{13} y^{5} + 2 x^{12} y^{5} + x^{12} y^{4} + x^{11} y^{5} + 3 x^{11} y^{4} + 5 x^{10} y^{4} + 5 x^{9} y^{4} + x^{9} y^{3} + 2 x^{8} y^{4} + 4 x^{8} y^{3} + 6 x^{7} y^{3} + 5 x^{6} y^{3} + x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 4 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 4 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{9,1})$.

There are 2 possibilities.
Here are their Poincaré polynomials: $$x^{18} y^{3} + x^{17} y^{3} + 2 x^{16} y^{3} + 3 x^{15} y^{3} + 3 x^{14} y^{3} + x^{14} y^{2} + 3 x^{13} y^{3} + 2 x^{13} y^{2} + 3 x^{12} y^{3} + 4 x^{12} y^{2} + 2 x^{11} y^{3} + 5 x^{11} y^{2} + x^{10} y^{3} + 7 x^{10} y^{2} + x^{9} y^{3} + 7 x^{9} y^{2} + 7 x^{8} y^{2} + x^{8} y + 5 x^{7} y^{2} + 2 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{18} y^{3} + x^{17} y^{3} + 2 x^{16} y^{3} + 3 x^{15} y^{3} + 3 x^{14} y^{3} + x^{14} y^{2} + 3 x^{13} y^{3} + 2 x^{13} y^{2} + 3 x^{12} y^{3} + 4 x^{12} y^{2} + 2 x^{11} y^{3} + 5 x^{11} y^{2} + x^{10} y^{3} + 7 x^{10} y^{2} + 8 x^{9} y^{2} + 8 x^{8} y^{2} + 5 x^{7} y^{2} + 2 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 2 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{10,1})$.

There are 4 possibilities.
Here are their Poincaré polynomials: $$x^{21} y^{3} + x^{20} y^{3} + 2 x^{19} y^{3} + 3 x^{18} y^{3} + 4 x^{17} y^{3} + 4 x^{16} y^{3} + x^{16} y^{2} + 5 x^{15} y^{3} + 2 x^{15} y^{2} + 4 x^{14} y^{3} + 4 x^{14} y^{2} + 4 x^{13} y^{3} + 5 x^{13} y^{2} + 3 x^{12} y^{3} + 7 x^{12} y^{2} + 2 x^{11} y^{3} + 8 x^{11} y^{2} + x^{10} y^{3} + 9 x^{10} y^{2} + x^{9} y^{3} + 8 x^{9} y^{2} + x^{9} y + 7 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{21} y^{3} + x^{20} y^{3} + 2 x^{19} y^{3} + 3 x^{18} y^{3} + 4 x^{17} y^{3} + 4 x^{16} y^{3} + x^{16} y^{2} + 5 x^{15} y^{3} + 2 x^{15} y^{2} + 4 x^{14} y^{3} + 4 x^{14} y^{2} + 4 x^{13} y^{3} + 5 x^{13} y^{2} + 3 x^{12} y^{3} + 7 x^{12} y^{2} + 2 x^{11} y^{3} + 8 x^{11} y^{2} + 10 x^{10} y^{2} + x^{9} y^{3} + 9 x^{9} y^{2} + 7 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{21} y^{3} + x^{20} y^{3} + 2 x^{19} y^{3} + 3 x^{18} y^{3} + 4 x^{17} y^{3} + 4 x^{16} y^{3} + x^{16} y^{2} + 5 x^{15} y^{3} + 2 x^{15} y^{2} + 4 x^{14} y^{3} + 4 x^{14} y^{2} + 4 x^{13} y^{3} + 5 x^{13} y^{2} + 3 x^{12} y^{3} + 7 x^{12} y^{2} + 2 x^{11} y^{3} + 8 x^{11} y^{2} + x^{10} y^{3} + 9 x^{10} y^{2} + 9 x^{9} y^{2} + x^{9} y + 8 x^{8} y^{2} + x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{21} y^{3} + x^{20} y^{3} + 2 x^{19} y^{3} + 3 x^{18} y^{3} + 4 x^{17} y^{3} + 4 x^{16} y^{3} + x^{16} y^{2} + 5 x^{15} y^{3} + 2 x^{15} y^{2} + 4 x^{14} y^{3} + 4 x^{14} y^{2} + 4 x^{13} y^{3} + 5 x^{13} y^{2} + 3 x^{12} y^{3} + 7 x^{12} y^{2} + 2 x^{11} y^{3} + 8 x^{11} y^{2} + 10 x^{10} y^{2} + 10 x^{9} y^{2} + 8 x^{8} y^{2} + x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 4 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{11,1})$.

There are 8 possibilities.
Here are their Poincaré polynomials: $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + 2 x^{11} y^{3} + 10 x^{11} y^{2} + x^{10} y^{3} + 10 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 8 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + x^{11} y^{3} + 11 x^{11} y^{2} + x^{10} y^{3} + 11 x^{10} y^{2} + x^{9} y^{3} + 8 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + 2 x^{11} y^{3} + 10 x^{11} y^{2} + 11 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 9 x^{9} y^{2} + x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + x^{11} y^{3} + 11 x^{11} y^{2} + 12 x^{10} y^{2} + x^{9} y^{3} + 9 x^{9} y^{2} + x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + 2 x^{11} y^{3} + 10 x^{11} y^{2} + x^{10} y^{3} + 10 x^{10} y^{2} + x^{10} y + 9 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + x^{11} y^{3} + 11 x^{11} y^{2} + x^{10} y^{3} + 11 x^{10} y^{2} + 9 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + 2 x^{11} y^{3} + 10 x^{11} y^{2} + 11 x^{10} y^{2} + x^{10} y + 10 x^{9} y^{2} + x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{3} + x^{23} y^{3} + 2 x^{22} y^{3} + 3 x^{21} y^{3} + 4 x^{20} y^{3} + 5 x^{19} y^{3} + 6 x^{18} y^{3} + x^{18} y^{2} + 6 x^{17} y^{3} + 2 x^{17} y^{2} + 6 x^{16} y^{3} + 4 x^{16} y^{2} + 6 x^{15} y^{3} + 5 x^{15} y^{2} + 5 x^{14} y^{3} + 7 x^{14} y^{2} + 4 x^{13} y^{3} + 8 x^{13} y^{2} + 3 x^{12} y^{3} + 10 x^{12} y^{2} + x^{11} y^{3} + 11 x^{11} y^{2} + 12 x^{10} y^{2} + 10 x^{9} y^{2} + x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 8 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{12,1})$.

There are 24 possibilities.
Here are their Poincaré polynomials: $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 2 x^{11} y^{3} + 11 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 10 x^{10} y^{2} + x^{9} y^{3} + 2 x^{10} y + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 2 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 10 x^{10} y^{2} + x^{9} y^{3} + 2 x^{10} y + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 11 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + x^{11} y^{3} + 13 x^{11} y^{2} + x^{10} y^{3} + 11 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 13 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 12 x^{10} y^{2} + x^{9} y^{3} + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 14 x^{11} y^{2} + x^{10} y^{3} + 12 x^{10} y^{2} + x^{9} y^{3} + 8 x^{9} y^{2} + 3 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 2 x^{11} y^{3} + 11 x^{11} y^{2} + x^{11} y + 11 x^{10} y^{2} + x^{9} y^{3} + 2 x^{10} y + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 2 x^{11} y^{3} + 12 x^{11} y^{2} + 11 x^{10} y^{2} + x^{9} y^{3} + 2 x^{10} y + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + x^{11} y^{3} + 12 x^{11} y^{2} + x^{11} y + 12 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + x^{11} y^{3} + 13 x^{11} y^{2} + 12 x^{10} y^{2} + x^{9} y^{3} + x^{10} y + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 13 x^{11} y^{2} + x^{11} y + 13 x^{10} y^{2} + x^{9} y^{3} + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 14 x^{11} y^{2} + 13 x^{10} y^{2} + x^{9} y^{3} + 9 x^{9} y^{2} + 2 x^{9} y + 7 x^{8} y^{2} + 3 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 2 x^{11} y^{3} + 11 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 10 x^{10} y^{2} + 2 x^{10} y + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 2 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 10 x^{10} y^{2} + 2 x^{10} y + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 11 x^{10} y^{2} + x^{10} y + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + x^{11} y^{3} + 13 x^{11} y^{2} + x^{10} y^{3} + 11 x^{10} y^{2} + x^{10} y + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 13 x^{11} y^{2} + x^{10} y^{3} + x^{11} y + 12 x^{10} y^{2} + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 14 x^{11} y^{2} + x^{10} y^{3} + 12 x^{10} y^{2} + 9 x^{9} y^{2} + 3 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 2 x^{11} y^{3} + 11 x^{11} y^{2} + x^{11} y + 11 x^{10} y^{2} + 2 x^{10} y + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 2 x^{11} y^{3} + 12 x^{11} y^{2} + 11 x^{10} y^{2} + 2 x^{10} y + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + x^{11} y^{3} + 12 x^{11} y^{2} + x^{11} y + 12 x^{10} y^{2} + x^{10} y + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + x^{11} y^{3} + 13 x^{11} y^{2} + 12 x^{10} y^{2} + x^{10} y + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 3 x^{12} y^{3} + 12 x^{12} y^{2} + 13 x^{11} y^{2} + x^{11} y + 13 x^{10} y^{2} + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{27} y^{3} + x^{26} y^{3} + 2 x^{25} y^{3} + 3 x^{24} y^{3} + 4 x^{23} y^{3} + 5 x^{22} y^{3} + 7 x^{21} y^{3} + 7 x^{20} y^{3} + x^{20} y^{2} + 8 x^{19} y^{3} + 2 x^{19} y^{2} + 8 x^{18} y^{3} + 4 x^{18} y^{2} + 8 x^{17} y^{3} + 5 x^{17} y^{2} + 7 x^{16} y^{3} + 7 x^{16} y^{2} + 7 x^{15} y^{3} + 8 x^{15} y^{2} + 5 x^{14} y^{3} + 10 x^{14} y^{2} + 4 x^{13} y^{3} + 11 x^{13} y^{2} + 2 x^{12} y^{3} + 13 x^{12} y^{2} + 14 x^{11} y^{2} + 13 x^{10} y^{2} + 10 x^{9} y^{2} + 2 x^{9} y + 8 x^{8} y^{2} + 2 x^{8} y + 5 x^{7} y^{2} + 3 x^{7} y + 4 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 24 possibilities: