does Sage support Grassmannians over finite fields?

[Jackson Walters]

Does Sage support computing Gr_{F_q}(k,r), the space of k-hyperplanes through the origin in GF(q**r)? I looked around for things like Schubert cells and didn't see anything immediately relevant. I've started to write some stuff for k=2. I may do a PR if it's not already available.

Thanks,

Jackson

[Dima Pasechnik]

We have functionality to build Grassmann graphs. See GrassmannGraph in

<https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html#sage.graphs.graph_generators.graphgenerators.paleygraph>

[Jackson Walters]

Thanks, that's helpful.

G = graphs.GrassmannGraph(5, 4, 2); G

gives 806 vertices, as expected, but why do the vertices seem to have six points?

G.vertices()[0]

{(0, 0, 0, 1), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 0, 4), (1, 0, 0, 0), (1, 0, 0, 3)}

I don't see anything about this in the documentation. Shouldn't it be just two basis vectors that span the subspace?

Thanks,

Jackson

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[Jackson Walters]

I guess they are projective lines with q+1 points arising from designs.ProjectiveGeometryDesign. However, the blocks (the 2-subspaces) should be in one-to-one correspondence with the vertices.

[Jackson Walters]

I suppose it comes down to this:

from sage.combinat.designs import design_catalog as designs

PG = designs.ProjectiveGeometryDesign(4 - 1, 2 - 1, 5)

PG.blocks()[0]

[(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 0, 3), (1, 0, 0, 4), (0, 0, 0, 1)]

[dim...@gmail.com]

On Tue, Mar 25, 2025 at 06:04:30PM -0400, Jackson Walters wrote:
> Thanks, that's helpful.
>
> G = graphs.GrassmannGraph(5, 4, 2); G
>
> gives 806 vertices, as expected, but why do the vertices seem to have six
> points?
>
> G.vertices()[0]
> {(0, 0, 0, 1), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 0, 4), (1, 0, 0, 0), (1,
> 0, 0, 3)}
>
> I don't see anything about this in the documentation. Shouldn't it be just
> two basis vectors that span the subspace?

The 3rd parameter, the dimension of the subspaces, is equal to
2. Thus these are all the q+1 (projective) points in the projective line over
GF(q), with q=5. It was a quick and not optimised coding.

If you would like to change this to a more economic representation,
please open a PR.
As the documentation doesn't say anything about a particular way the
subspaces are represented, it can be changed quickly, no deprecation
would be needed.

Dima

>
> Thanks,
> Jackson
>
> On Tue, Mar 25, 2025 at 5:33 PM Dima Pasechnik <dim...@gmail.com> wrote:
>
> > We have functionality to build Grassmann graphs. See GrassmannGraph in
> >
> > <
> > https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html#sage.graphs.graph_generators.graphgenerators.paleygraph
> > >
> >
> >
> > On 25 March 2025 12:46:56 GMT-05:00, Jackson Walters <
> > jackson...@gmail.com> wrote:
> >
> >> Does Sage support computing Gr_{F_q}(k,r), the space of k-hyperplanes
> >> through the origin in GF(q**r)? I looked around for things like Schubert
> >> cells and didn't see anything immediately relevant. I've started to write
> >> some stuff for k=2. I may do a PR if it's not already available.
> >>
> >> Thanks,
> >> Jackson
> >>
> >> --
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> > <https://groups.google.com/d/msgid/sage-devel/68A3261B-8643-43BC-BFBF-AB2DBF9D89FB%40gmail.com?utm_medium=email&utm_source=footer>
> > .

> >
>
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