Mixing
Is the quotient of standard parabolic subgroups isomorphic to a Schubert variety
$begingroup$
Let $G$ be a reductive algebraic group over an algebraically closed field.
Let $B subseteq P_2 subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 subseteq G/P_2$ is a closed subvariety.
Is $P_1/P_2$ a Schubert variety defined in $G/P_2$? If not, what are the intersection of $P_1/P_2$ with the Schubert cells, also defined in $G/P_2$?
More precisely, let $W_{P_i}$ be the subgroup of the Weyl group $W$ of $G$ defined with a maximal torus $Tsubseteq B$, such that $W_{P_i} simeq N_{P_i}(T)/T$, for $i=1,2$. Let $$W_{P_i}^{text{min}} = {w in W | ell(ww') = ell(w)+ell(w'), text{ for all } w' in W_{P_i}},$$
for $i=1,2$. Let $w_0$ be the longest element in $W$. Then there is a unique element in $W_{P_i}^{text{min}}$ with maximal length, which is denoted as $w_i$, satisfying $w_0 = w_i cdot w_{P_i}$, where $w_{P_i}$ is the longest element in $W_{P_i}$, for $i=1,2$.
Since $W_{P_2}$ is a subgroup of $W_{P_1}$, there exists $w_1' in W_{P_1}$, such that $w_{P_1} = w_1' cdot w_{P_2}$. So $w_2 = w_1 cdot w_1'$.
Let $X_i(cdot)$ denote the Schubert variety defined in $G/P_i$ for $i=1,2$. Then $G/P_1 simeq X_1(w_1)$ and $G/P_2 simeq X_2(w_1 cdot w_1')$. This makes me suspect that $P_1/P_2$ and $X_2(w_1')$ are related in some sense. Is it true?
Thank you very much.
P.S.
- In a Coxeter group, we say $x = y cdot z$ if $x = yz$ and $ell(x)=ell(y)+ell(z)$.
2.For the definition of all the terms, please check: Billey and Lakshmibai's Singular Loci of Schubert Varieties, Chapter 2.
algebraic-geometry algebraic-groups schubert-calculus
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
$endgroup$
add a comment |
$begingroup$
Let $G$ be a reductive algebraic group over an algebraically closed field.
Let $B subseteq P_2 subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 subseteq G/P_2$ is a closed subvariety.
Is $P_1/P_2$ a Schubert variety defined in $G/P_2$? If not, what are the intersection of $P_1/P_2$ with the Schubert cells, also defined in $G/P_2$?
More precisely, let $W_{P_i}$ be the subgroup of the Weyl group $W$ of $G$ defined with a maximal torus $Tsubseteq B$, such that $W_{P_i} simeq N_{P_i}(T)/T$, for $i=1,2$. Let $$W_{P_i}^{text{min}} = {w in W | ell(ww') = ell(w)+ell(w'), text{ for all } w' in W_{P_i}},$$
for $i=1,2$. Let $w_0$ be the longest element in $W$. Then there is a unique element in $W_{P_i}^{text{min}}$ with maximal length, which is denoted as $w_i$, satisfying $w_0 = w_i cdot w_{P_i}$, where $w_{P_i}$ is the longest element in $W_{P_i}$, for $i=1,2$.
Since $W_{P_2}$ is a subgroup of $W_{P_1}$, there exists $w_1' in W_{P_1}$, such that $w_{P_1} = w_1' cdot w_{P_2}$. So $w_2 = w_1 cdot w_1'$.
Let $X_i(cdot)$ denote the Schubert variety defined in $G/P_i$ for $i=1,2$. Then $G/P_1 simeq X_1(w_1)$ and $G/P_2 simeq X_2(w_1 cdot w_1')$. This makes me suspect that $P_1/P_2$ and $X_2(w_1')$ are related in some sense. Is it true?
Thank you very much.
P.S.
- In a Coxeter group, we say $x = y cdot z$ if $x = yz$ and $ell(x)=ell(y)+ell(z)$.
2.For the definition of all the terms, please check: Billey and Lakshmibai's Singular Loci of Schubert Varieties, Chapter 2.
algebraic-geometry algebraic-groups schubert-calculus
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
$endgroup$
$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23
add a comment |
$begingroup$
Let $G$ be a reductive algebraic group over an algebraically closed field.
Let $B subseteq P_2 subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 subseteq G/P_2$ is a closed subvariety.
Is $P_1/P_2$ a Schubert variety defined in $G/P_2$? If not, what are the intersection of $P_1/P_2$ with the Schubert cells, also defined in $G/P_2$?
More precisely, let $W_{P_i}$ be the subgroup of the Weyl group $W$ of $G$ defined with a maximal torus $Tsubseteq B$, such that $W_{P_i} simeq N_{P_i}(T)/T$, for $i=1,2$. Let $$W_{P_i}^{text{min}} = {w in W | ell(ww') = ell(w)+ell(w'), text{ for all } w' in W_{P_i}},$$
for $i=1,2$. Let $w_0$ be the longest element in $W$. Then there is a unique element in $W_{P_i}^{text{min}}$ with maximal length, which is denoted as $w_i$, satisfying $w_0 = w_i cdot w_{P_i}$, where $w_{P_i}$ is the longest element in $W_{P_i}$, for $i=1,2$.
Since $W_{P_2}$ is a subgroup of $W_{P_1}$, there exists $w_1' in W_{P_1}$, such that $w_{P_1} = w_1' cdot w_{P_2}$. So $w_2 = w_1 cdot w_1'$.
Let $X_i(cdot)$ denote the Schubert variety defined in $G/P_i$ for $i=1,2$. Then $G/P_1 simeq X_1(w_1)$ and $G/P_2 simeq X_2(w_1 cdot w_1')$. This makes me suspect that $P_1/P_2$ and $X_2(w_1')$ are related in some sense. Is it true?
Thank you very much.
P.S.
- In a Coxeter group, we say $x = y cdot z$ if $x = yz$ and $ell(x)=ell(y)+ell(z)$.
2.For the definition of all the terms, please check: Billey and Lakshmibai's Singular Loci of Schubert Varieties, Chapter 2.
algebraic-geometry algebraic-groups schubert-calculus
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
$endgroup$
Let $G$ be a reductive algebraic group over an algebraically closed field.
Let $B subseteq P_2 subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 subseteq G/P_2$ is a closed subvariety.
Is $P_1/P_2$ a Schubert variety defined in $G/P_2$? If not, what are the intersection of $P_1/P_2$ with the Schubert cells, also defined in $G/P_2$?
More precisely, let $W_{P_i}$ be the subgroup of the Weyl group $W$ of $G$ defined with a maximal torus $Tsubseteq B$, such that $W_{P_i} simeq N_{P_i}(T)/T$, for $i=1,2$. Let $$W_{P_i}^{text{min}} = {w in W | ell(ww') = ell(w)+ell(w'), text{ for all } w' in W_{P_i}},$$
for $i=1,2$. Let $w_0$ be the longest element in $W$. Then there is a unique element in $W_{P_i}^{text{min}}$ with maximal length, which is denoted as $w_i$, satisfying $w_0 = w_i cdot w_{P_i}$, where $w_{P_i}$ is the longest element in $W_{P_i}$, for $i=1,2$.
Since $W_{P_2}$ is a subgroup of $W_{P_1}$, there exists $w_1' in W_{P_1}$, such that $w_{P_1} = w_1' cdot w_{P_2}$. So $w_2 = w_1 cdot w_1'$.
Let $X_i(cdot)$ denote the Schubert variety defined in $G/P_i$ for $i=1,2$. Then $G/P_1 simeq X_1(w_1)$ and $G/P_2 simeq X_2(w_1 cdot w_1')$. This makes me suspect that $P_1/P_2$ and $X_2(w_1')$ are related in some sense. Is it true?
Thank you very much.
P.S.
- In a Coxeter group, we say $x = y cdot z$ if $x = yz$ and $ell(x)=ell(y)+ell(z)$.
2.For the definition of all the terms, please check: Billey and Lakshmibai's Singular Loci of Schubert Varieties, Chapter 2.
algebraic-geometry algebraic-groups schubert-calculus
algebraic-geometry algebraic-groups schubert-calculus
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
edited Jan 2 at 0:48
Matt Samuel
37.5k63665
37.5k63665
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
asked Jul 22 '17 at 3:48
IzumiEternalIzumiEternal
329
329
$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23
add a comment |
$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23
$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23
add a comment |
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$begingroup$
Have you tried looking at examples? $SL_3$ with $P_2=B$ and $P_1=G_W$ where $W$ is the point of the cooridinate plane $(e_1,e_2)$ in $Gr(2,3)$ might be reasonable. Then $P_1/P_2cong mathbf{P}^1$ and $G/P_2=G/B$ is the variety of complete flags in a 3-dim vec space (this has 6 cells, and only 2 cells of dimension 1).
$endgroup$
– Eoin
Sep 4 '17 at 6:03
$begingroup$
@Eoin: Thank you very much for the comment. But I am sorry I don't quite get it. In this example, is $P_1/P_2$ a Schubert variety ( a union of one cell of dimension $1$ and one cell of dimension $0$) or not? Thanks again.
$endgroup$
– IzumiEternal
Sep 20 '17 at 17:23