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Does a spherical building embedsembed in a building of type $A_n$?

I'm interested in the question in the title.:

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding workswork? For instance, does an automorphism of $B$ extendsextend to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit2Edit 2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$, which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Henche,Hence the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possiblepossibly infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for anysome field $k$. Hence thisthese buildings are embeddable in $A_{2n}$. Type

Type $E_n$, $n=6,7,8$, should not be a problem either, since they correspond to algebraic groups. Is this also correct?

Does a spherical building embeds in a building of type $A_n$?

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$, which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Henche, the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possible infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for any field $k$. Hence this buildings are embeddable in $A_{2n}$. Type $E_n$, $n=6,7,8$ should not be a problem either, since they correspond to algebraic groups. Is this also correct?

Does a spherical building embed in a building of type $A_n$?

I'm interested in the question in the title:

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding work? For instance, does an automorphism of $B$ extend to an automorphism of $\tilde B$?

Edit: I forgot that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit 2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$ which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Hence the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possibly infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for some field $k$. Hence these buildings are embeddable in $A_{2n}$.

Type $E_n$, $n=6,7,8$, should not be a problem either, since they correspond to algebraic groups. Is this also correct?

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Luc
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I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$, which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Henche, the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possible infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for any field $k$. Hence this buildings are embeddable in $A_{2n}$. Type $E_n$, $n=6,7,8$ should not be a problem either, since they correspond to algebraic groups. Is this also correct?

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Edit2: As Dima Pasechnik pointed out, if the building $B$ has residues of type $A_2$, which are non-Desarguesian projective planes, then we cannot have such an embedding, since projective spaces of dimension $\geq 3$ must be Desarguesian. This is a problem if $B$ is of type $B_3$ or $F_4$. In the former case, it is known that there are non-embeddable buildings (see Tits' "Buildings of spherical type and BN-pairs", chapter 9). Henche, the following question:

What about type $F_4$?

Buildings of type $B_n$ with $n\geq 4$ are embeddable (see Tits' "Buildings of spherical type and BN-pairs", chapter 8) into a projective space of possible infinite rank. So, I assume that the answer to my quesion in this case is "No in general" (because of the infinite rank). Is this correct?

Buildings of type $D_n$ should be embeddable, since they correspond to the group $SO_{2n}(k)$ for any field $k$. Hence this buildings are embeddable in $A_{2n}$. Type $E_n$, $n=6,7,8$ should not be a problem either, since they correspond to algebraic groups. Is this also correct?

added 186 characters in body
Source Link
Luc
  • 265
  • 1
  • 7

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

I'm interested in the question in the title.

Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$?

By embedding I mean an isometric embedding with respect to their $CAT(1)$ metric. If the question has a positive answer, then how does this embedding works? For instance, does an automorphism of $B$ extends to an automorphism of $\tilde B$?

Edit: I forgot, that there are strange polygons. I would like to restrict the question to the case of irreducible thick buildings of rank at least 3. In particular, they are Moufang.

Source Link
Luc
  • 265
  • 1
  • 7
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