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Yosemite Sam
Yosemite Sam
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Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))?

The first question in the title might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of $GL(n,\mathbb{C})$GL(n,C) (or $SL(n,\mathbb{C})$SL(n,C) if you prefer) by finite subgroups.

Is Is there anything I can say about such athe quotient (set of cosets)?

The simplestbabiest case would be the standard representationrep of the dihedral/symmetric group $S_3$.

What What is $GL(2,\mathbb{C})/S_3$ (or$GL(2,C)/S_3$? Or $SL(2,\mathbb{C})/S_3$$SL(2,C)/S_3$ if it makes things easier)?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what is the quotient is.

Any ideas or references?

Can I recognize the quotient of a group by a closed subgroup (for example the standard representation of S3 in GL(2,C))?

The first question in the title might be too general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of $GL(n,\mathbb{C})$ (or $SL(n,\mathbb{C})$ if you prefer) by finite subgroups.

Is there anything I can say about such a quotient (set of cosets)?

The simplest case would be the standard representation of the dihedral/symmetric group $S_3$.

What is $GL(2,\mathbb{C})/S_3$ (or $SL(2,\mathbb{C})/S_3$ if it makes things easier)?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what is the quotient.

Any ideas or references?

Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))

The first question might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups. Is there anything I can say about the quotient (set of cosets)?

The babiest case would be the standard rep of the dihedral/symmetric group $S_3$. What is $GL(2,C)/S_3$? Or $SL(2,C)/S_3$ if it makes things easier?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what the quotient is.

Any ideas or references?

Minor edits.
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Sean Lawton
Sean Lawton
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Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))?

The first question in the title might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C)$GL(n,\mathbb{C})$ (or SL(n,C)$SL(n,\mathbb{C})$ if you prefer) by finite subgroups. Is

Is there anything I can say about thesuch a quotient (set of cosets)?

The babiestsimplest case would be the standard reprepresentation of the dihedral/symmetric group $S_3$. What

What is $GL(2,C)/S_3$? Or$GL(2,\mathbb{C})/S_3$ $SL(2,C)/S_3$(or $SL(2,\mathbb{C})/S_3$ if it makes things easier)?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what is the quotient is.

Any ideas or references?

Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))

The first question might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups. Is there anything I can say about the quotient (set of cosets)?

The babiest case would be the standard rep of the dihedral/symmetric group $S_3$. What is $GL(2,C)/S_3$? Or $SL(2,C)/S_3$ if it makes things easier?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what the quotient is.

Any ideas or references?

Can I recognize the quotient of a group by a closed subgroup (for example the standard representation of S3 in GL(2,C))?

The first question in the title might be too general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of $GL(n,\mathbb{C})$ (or $SL(n,\mathbb{C})$ if you prefer) by finite subgroups.

Is there anything I can say about such a quotient (set of cosets)?

The simplest case would be the standard representation of the dihedral/symmetric group $S_3$.

What is $GL(2,\mathbb{C})/S_3$ (or $SL(2,\mathbb{C})/S_3$ if it makes things easier)?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what is the quotient.

Any ideas or references?

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Yosemite Sam
Yosemite Sam
  • 1.9k
  • 1
  • 14
  • 27

Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))

The first question might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups. Is there anything I can say about the quotient (set of cosets)?

The babiest case would be the standard rep of the dihedral/symmetric group $S_3$. What is $GL(2,C)/S_3$? Or $SL(2,C)/S_3$ if it makes things easier?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what the quotient is.

Any ideas or references?