Welcome to MathOverflow

Question

3

1

For the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism $$ \text{Gr}(N,k) = O(N)/(O(k) \times O(N-k)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(k) \times U(N-k)) $$ I would like to know if anything like this holds in the finite field setting, ie can the finite field Grassmannians be described as a homogeneous space of an algebraic group over a finite field, or something like this?

edited Apr 2 2012 at 15:16

asked Mar 31 2012 at 13:29

Ago Szekeres
207●6

4

Yes. It's a homogeneous space over GL of the base field, for all fields. – Ben Webster♦ Mar 31 2012 at 14:55

Do you have a reference for this? What's the stabilizer subgroup, and how does it act on GL? – Ago Szekeres Mar 31 2012 at 15:41

4

Wikipedia: en.wikipedia.org/wiki/… Which it's considered good form to check before asking MO questions. – Ben Webster♦ Mar 31 2012 at 16:38

Duly noted, thanks – Ago Szekeres Mar 31 2012 at 16:45

1

Perhaps it should be noted that over finite fields there are "outer forms" of some Grassmannians, e.g., the Grassmannian of $k$-dimensional subspaces of a vector space of dimension $2k$. These are not homogeneous spaces for the "split form" of $\textbf{GL}_{2k}$. Also, I believe the OP mistakenly wrote $O(1) \times O(N-1)$ instead of $O(k) times O(N-k)$. – Jason Starr Apr 2 2012 at 11:31

show 3 more comments

Dima Pasechnik · Answer 1 · 2012-04-02 16:23:53Z UTC

There is a description of these homogeneous spaces over finite fields in terms of graphs in "Distance-Regular Graphs" by Brouwer, Cohen and Neumaier (Springer, 1989) and in "Algebraic combinatorics I: Association schemes" by E.Bannai and T.Ito (Benjamin/Cummings, 1984).

Finite Field Grassmannians as Homogeneous Spaces

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.