Subtori of groups of type E6
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Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_bark$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.
Does there exist a smaller semisimple subgroup $H$ of $G$ such that $T$ is a subtorus of $H$? What is the type of such $H$?
For example, can you find $H$ of type $A_2times A_2times A_2$ (probably too optimistic), or maybe $D_6$, or $F_4$?
ag.algebraic-geometry algebraic-groups root-systems
asked 4 hours ago
Cehiju
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Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_bark$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.
Does there exist a smaller semisimple subgroup $H$ of $G$ such that $T$ is a subtorus of $H$? What is the type of such $H$?
For example, can you find $H$ of type $A_2times A_2times A_2$ (probably too optimistic), or maybe $D_6$, or $F_4$?
ag.algebraic-geometry algebraic-groups root-systems
asked 4 hours ago
Cehiju
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_bark$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.
Does there exist a smaller semisimple subgroup $H$ of $G$ such that $T$ is a subtorus of $H$? What is the type of such $H$?
For example, can you find $H$ of type $A_2times A_2times A_2$ (probably too optimistic), or maybe $D_6$, or $F_4$?
ag.algebraic-geometry algebraic-groups root-systems
asked 4 hours ago
Cehiju
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_bark$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.
Does there exist a smaller semisimple subgroup $H$ of $G$ such that $T$ is a subtorus of $H$? What is the type of such $H$?
For example, can you find $H$ of type $A_2times A_2times A_2$ (probably too optimistic), or maybe $D_6$, or $F_4$?
ag.algebraic-geometry algebraic-groups root-systems
ag.algebraic-geometry algebraic-groups root-systems
asked 4 hours ago
Cehiju
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Cehiju
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Cehiju
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Cehiju
63
asked 4 hours ago
Cehiju
63
63
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Cehiju is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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1 Answer
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This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E_6$, $A_1 + A_5$, and $A_2 + A_2 + A_2$ (as you hoped, corresponding to removing the root $alpha_4$ in Bourbaki's notation).
answered 4 hours ago
LSpice
2,64322126
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E_6$, $A_1 + A_5$, and $A_2 + A_2 + A_2$ (as you hoped, corresponding to removing the root $alpha_4$ in Bourbaki's notation).
answered 4 hours ago
LSpice
2,64322126
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
add a comment |Â
up vote
2
down vote
This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E_6$, $A_1 + A_5$, and $A_2 + A_2 + A_2$ (as you hoped, corresponding to removing the root $alpha_4$ in Bourbaki's notation).
answered 4 hours ago
LSpice
2,64322126
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E_6$, $A_1 + A_5$, and $A_2 + A_2 + A_2$ (as you hoped, corresponding to removing the root $alpha_4$ in Bourbaki's notation).
answered 4 hours ago
LSpice
2,64322126
This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E_6$, $A_1 + A_5$, and $A_2 + A_2 + A_2$ (as you hoped, corresponding to removing the root $alpha_4$ in Bourbaki's notation).
answered 4 hours ago
LSpice
2,64322126
answered 4 hours ago
LSpice
2,64322126
answered 4 hours ago
LSpice
2,64322126
answered 4 hours ago
LSpice
2,64322126
2,64322126
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
add a comment |Â
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
Thank you. Do you know in which form these results apply to non-closed $k$? This is my main interest relative to this situation.
– Cehiju
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
For split $E_6$, all the split forms can be realised. Since Borel–de Siebenthal theory is constructive, it shouldn't be too hard to describe the other forms, but probably you'd want at least to specify some particular ground field to have any hope of, e.g., describing all possible tori in the first place.
– LSpice
3 hours ago
add a comment |Â
Cehiju is a new contributor. Be nice, and check out our Code of Conduct.
Cehiju is a new contributor. Be nice, and check out our Code of Conduct.
Cehiju is a new contributor. Be nice, and check out our Code of Conduct.
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