Orthogonal Grassmannian












1












$begingroup$


The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to a $SO(n)/P_k$ where $P_k$ is the maximal parabolic subgroup with respect to a simple root ?










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$endgroup$












  • $begingroup$
    I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:27










  • $begingroup$
    The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
    $endgroup$
    – icmes imrf
    Jan 9 '18 at 10:33












  • $begingroup$
    Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:40
















1












$begingroup$


The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to a $SO(n)/P_k$ where $P_k$ is the maximal parabolic subgroup with respect to a simple root ?










share|cite|improve this question











$endgroup$












  • $begingroup$
    I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:27










  • $begingroup$
    The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
    $endgroup$
    – icmes imrf
    Jan 9 '18 at 10:33












  • $begingroup$
    Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:40














1












1








1





$begingroup$


The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to a $SO(n)/P_k$ where $P_k$ is the maximal parabolic subgroup with respect to a simple root ?










share|cite|improve this question











$endgroup$




The orthogonal Grassmannian $OG(k,n)$ is the set of all isotropic $k$ dimensional subspaces of a $n$ dimensional subspaces with respect to a non-degenerate symmetric bilinear form. Is it isomorphic to a $SO(n)/P_k$ where $P_k$ is the maximal parabolic subgroup with respect to a simple root ?







differential-geometry algebraic-geometry representation-theory schubert-calculus






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 11:17









Matt Samuel

38.7k63769




38.7k63769










asked Jan 9 '18 at 10:21









icmes imrficmes imrf

705




705












  • $begingroup$
    I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:27










  • $begingroup$
    The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
    $endgroup$
    – icmes imrf
    Jan 9 '18 at 10:33












  • $begingroup$
    Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:40


















  • $begingroup$
    I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:27










  • $begingroup$
    The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
    $endgroup$
    – icmes imrf
    Jan 9 '18 at 10:33












  • $begingroup$
    Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
    $endgroup$
    – Dirk Liebhold
    Jan 9 '18 at 10:40
















$begingroup$
I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
$endgroup$
– Dirk Liebhold
Jan 9 '18 at 10:27




$begingroup$
I would almost say yes, up to the point that it is $SO(n)/P_k$ where $P_k$ is parabolic (that is a very standard one-line argument using Witt's theorem). However, $P_k$ is not maximal in general, so could you clarify what "maximal with respect to a simple root" means?
$endgroup$
– Dirk Liebhold
Jan 9 '18 at 10:27












$begingroup$
The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
$endgroup$
– icmes imrf
Jan 9 '18 at 10:33






$begingroup$
The maximal parabolic associated to the simple root $alpha_k$ is the subgroup generated by the Borel subgroup $B$ and ${n_{alpha}: alpha in S setminus {alpha_k}}$ where $S$ is the set of simple roots and $n_{alpha}$ is a representative of $s_{alpha}$ in $N_G(T)$.
$endgroup$
– icmes imrf
Jan 9 '18 at 10:33














$begingroup$
Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
$endgroup$
– Dirk Liebhold
Jan 9 '18 at 10:40




$begingroup$
Did you write down the stabilizer of a space in $OG(k,n)$ in matrix notation? It is a block triangular matrix with two blocks, so it should contain every Coxeter generator but one. Sorry, my knowledge of the general terminology, without writing it down with matrices, is not good enough to answer in general.
$endgroup$
– Dirk Liebhold
Jan 9 '18 at 10:40










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