Orthogonal Grassmannian
1
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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
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
$endgroup$
add a comment |
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 ?
differential-geometry algebraic-geometry representation-theory schubert-calculus
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
$endgroup$
$begingroup$
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
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
Jan 9 '18 at 10:40
add a comment |
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 ?
differential-geometry algebraic-geometry representation-theory schubert-calculus
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
$endgroup$
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
differential-geometry algebraic-geometry representation-theory schubert-calculus
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
edited Jan 5 at 11:17
Matt Samuel
39.2k63770
39.2k63770
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
asked Jan 9 '18 at 10:21
icmes imrficmes imrf
705
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
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
Jan 9 '18 at 10:40
add a comment |
$begingroup$
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
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
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
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
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
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
Jan 9 '18 at 10:40
add a comment |
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$begingroup$
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
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
Jan 9 '18 at 10:40