About parabolic subgroup of a Weyl group
1
$begingroup$
Let $W$ be a Weyl group/Coxeter group. Let $Phi$ be the associated root system, fix a positive root system $Phi^+$ and let
$Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup of $W$ generated by $Isubseteq Delta$.
Does $W_I=W_Jimplies I=J$?
Does $sin W$ with $sneq 1$, $s^2=1implies s=s_alpha$ for some root $alpha$?
abstract-algebra group-theory root-systems coxeter-groups weyl-group
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
$endgroup$
add a comment |
1
$begingroup$
Let $W$ be a Weyl group/Coxeter group. Let $Phi$ be the associated root system, fix a positive root system $Phi^+$ and let
$Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup of $W$ generated by $Isubseteq Delta$.
Does $W_I=W_Jimplies I=J$?
Does $sin W$ with $sneq 1$, $s^2=1implies s=s_alpha$ for some root $alpha$?
abstract-algebra group-theory root-systems coxeter-groups weyl-group
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
$endgroup$
$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
1
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
1
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57
add a comment |
1
1
1
$begingroup$
Let $W$ be a Weyl group/Coxeter group. Let $Phi$ be the associated root system, fix a positive root system $Phi^+$ and let
$Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup of $W$ generated by $Isubseteq Delta$.
Does $W_I=W_Jimplies I=J$?
Does $sin W$ with $sneq 1$, $s^2=1implies s=s_alpha$ for some root $alpha$?
abstract-algebra group-theory root-systems coxeter-groups weyl-group
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
$endgroup$
Let $W$ be a Weyl group/Coxeter group. Let $Phi$ be the associated root system, fix a positive root system $Phi^+$ and let
$Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup of $W$ generated by $Isubseteq Delta$.
Does $W_I=W_Jimplies I=J$?
Does $sin W$ with $sneq 1$, $s^2=1implies s=s_alpha$ for some root $alpha$?
abstract-algebra group-theory root-systems coxeter-groups weyl-group
abstract-algebra group-theory root-systems coxeter-groups weyl-group
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
edited Jan 3 at 14:42
user593746
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
asked Dec 29 '18 at 4:44
James CheungJames Cheung
1184
1184
$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
1
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
1
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57
add a comment |
$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
1
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
1
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57
$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
1
1
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
1
1
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57
add a comment |
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$begingroup$
What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$.
$endgroup$
– Torsten Schoeneberg
Dec 29 '18 at 4:51
1
$begingroup$
My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then ${s_alpha:alphainPhi_I}={s_alpha:alphainPhi_J}$. This implies $Phi_I=Phi_J$ and then $I=J$.
$endgroup$
– James Cheung
Dec 29 '18 at 17:55
1
$begingroup$
$s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $ge 2$.
$endgroup$
– Torsten Schoeneberg
Jan 2 at 1:57