For the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism $$ \text{Gr}(N,k) = O(N)/(O(1O(N)/(O(k) \times O(N-k)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(1U(N)/(U(k) \times U(N-k)) $$ 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?
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For the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism $$ \text{Gr}(N,k) = O(N)/(O(1) \times O(N-1)O(N-k)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(1) \times U(N-1)U(N-k)) $$ 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? |
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Finite Field Grassmannians as Homogeneous SpacesFor the real Grassmannian Gr$(N,k)$ we have the well-known isomorphism $$ \text{Gr}(N,k) = O(N)/(O(1) \times O(N-1)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(1) \times U(N-1)) $$ 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?
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