Symmetry group of the geometric realization of a simplicial complex
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Question: Given $Delta$ an abstract simplicial complex, can one find a geometric realization of $Delta$ whose symmetry group is isomorphic to $Aut(Delta)$?
Relevant definitions:
Let $Delta$ an abstract simplicial complex on the vertex set $V$.
An automorphism of $Delta$ is a bijection $f:V to V$ whose induced map on the whole complex sends elements of $Delta$ to elements of $Delta$. These automprhisms form a group $Aut(Delta)$.
A geometric realization of $Delta$ is a (geometric) simplicial complex whose underlying set is $Delta$.
For any geometric simplicial complex embedded in $mathbb{R}^n$ one can define its symmetry group as
the set of all isometries of $mathbb{R}^n$ that map the complex to itself.
$ $
Thanks!
combinatorics geometry
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
add a comment |
up vote
2
down vote
favorite
Question: Given $Delta$ an abstract simplicial complex, can one find a geometric realization of $Delta$ whose symmetry group is isomorphic to $Aut(Delta)$?
Relevant definitions:
Let $Delta$ an abstract simplicial complex on the vertex set $V$.
An automorphism of $Delta$ is a bijection $f:V to V$ whose induced map on the whole complex sends elements of $Delta$ to elements of $Delta$. These automprhisms form a group $Aut(Delta)$.
A geometric realization of $Delta$ is a (geometric) simplicial complex whose underlying set is $Delta$.
For any geometric simplicial complex embedded in $mathbb{R}^n$ one can define its symmetry group as
the set of all isometries of $mathbb{R}^n$ that map the complex to itself.
$ $
Thanks!
combinatorics geometry
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
4
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
2
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
1
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Question: Given $Delta$ an abstract simplicial complex, can one find a geometric realization of $Delta$ whose symmetry group is isomorphic to $Aut(Delta)$?
Relevant definitions:
Let $Delta$ an abstract simplicial complex on the vertex set $V$.
An automorphism of $Delta$ is a bijection $f:V to V$ whose induced map on the whole complex sends elements of $Delta$ to elements of $Delta$. These automprhisms form a group $Aut(Delta)$.
A geometric realization of $Delta$ is a (geometric) simplicial complex whose underlying set is $Delta$.
For any geometric simplicial complex embedded in $mathbb{R}^n$ one can define its symmetry group as
the set of all isometries of $mathbb{R}^n$ that map the complex to itself.
$ $
Thanks!
combinatorics geometry
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
Question: Given $Delta$ an abstract simplicial complex, can one find a geometric realization of $Delta$ whose symmetry group is isomorphic to $Aut(Delta)$?
Relevant definitions:
Let $Delta$ an abstract simplicial complex on the vertex set $V$.
An automorphism of $Delta$ is a bijection $f:V to V$ whose induced map on the whole complex sends elements of $Delta$ to elements of $Delta$. These automprhisms form a group $Aut(Delta)$.
A geometric realization of $Delta$ is a (geometric) simplicial complex whose underlying set is $Delta$.
For any geometric simplicial complex embedded in $mathbb{R}^n$ one can define its symmetry group as
the set of all isometries of $mathbb{R}^n$ that map the complex to itself.
$ $
Thanks!
combinatorics geometry
combinatorics geometry
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
asked Dec 9 '13 at 5:41
Alexandru Papiu
736
736
Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
4
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
2
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
1
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18
add a comment |
Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
4
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
2
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
1
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18
Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
4
4
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
2
2
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
1
1
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18
add a comment |
1 Answer
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Put the vertices at the corners of a $(|V|−1)$-simplex.
answered Dec 6 at 1:26
apt1002
1,748615
add a comment |
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1 Answer
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up vote
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Put the vertices at the corners of a $(|V|−1)$-simplex.
answered Dec 6 at 1:26
apt1002
1,748615
add a comment |
up vote
0
down vote
Put the vertices at the corners of a $(|V|−1)$-simplex.
answered Dec 6 at 1:26
apt1002
1,748615
add a comment |
up vote
0
down vote
up vote
0
down vote
Put the vertices at the corners of a $(|V|−1)$-simplex.
answered Dec 6 at 1:26
apt1002
1,748615
Put the vertices at the corners of a $(|V|−1)$-simplex.
answered Dec 6 at 1:26
apt1002
1,748615
answered Dec 6 at 1:26
apt1002
1,748615
answered Dec 6 at 1:26
apt1002
1,748615
answered Dec 6 at 1:26
apt1002
1,748615
1,748615
add a comment |
add a comment |
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Could you define more specific of geometric realization?
– Siqi He
Dec 9 '13 at 5:46
4
I don't really know this subject, but here's a guess. Can you simply put the vertices at the corners of a $(|V|-1)$-simplex?
– apt1002
Dec 9 '13 at 5:59
2
apt1002's suggestion solves the problem.
– Jim Belk
Dec 9 '13 at 6:30
Ah I see, that would indeed work, thanks!
– Alexandru Papiu
Mar 2 '14 at 16:14
1
@apt1002 You should definitely write this up as an answer so that this question can be marked as answered! I would upvote.
– M. Winter
Dec 1 at 11:18