About parabolic subgroup of a Weyl group












2














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 $Isubseteq Delta$.




  1. Does $W_I=W_Jimplies I=J$?


  2. 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







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  • 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 $s=id$.
    – Torsten Schoeneberg
    Dec 29 '18 at 4:51








  • 1




    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 $I=J$.
    – James Cheung
    Dec 29 '18 at 17:55








  • 1




    $s=-id$ 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 $ge 2$.
    – Torsten Schoeneberg
    Jan 2 at 1:57


















2














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 $Isubseteq Delta$.




  1. Does $W_I=W_Jimplies I=J$?


  2. 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







share|cite|improve this question
























  • 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 $s=id$.
    – Torsten Schoeneberg
    Dec 29 '18 at 4:51








  • 1




    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 $I=J$.
    – James Cheung
    Dec 29 '18 at 17:55








  • 1




    $s=-id$ 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 $ge 2$.
    – Torsten Schoeneberg
    Jan 2 at 1:57
















2












2








2


1





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 $Isubseteq Delta$.




  1. Does $W_I=W_Jimplies I=J$?


  2. 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







share|cite|improve this question















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 $Isubseteq Delta$.




  1. Does $W_I=W_Jimplies I=J$?


  2. 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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









Zvi

4,960430




4,960430










asked Dec 29 '18 at 4:44









James Cheung

1284




1284












  • 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 $s=id$.
    – Torsten Schoeneberg
    Dec 29 '18 at 4:51








  • 1




    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 $I=J$.
    – James Cheung
    Dec 29 '18 at 17:55








  • 1




    $s=-id$ 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 $ge 2$.
    – Torsten Schoeneberg
    Jan 2 at 1:57




















  • 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 $s=id$.
    – Torsten Schoeneberg
    Dec 29 '18 at 4:51








  • 1




    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 $I=J$.
    – James Cheung
    Dec 29 '18 at 17:55








  • 1




    $s=-id$ 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 $ge 2$.
    – Torsten Schoeneberg
    Jan 2 at 1:57


















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 $s=id$.
– Torsten Schoeneberg
Dec 29 '18 at 4:51






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 $s=id$.
– Torsten Schoeneberg
Dec 29 '18 at 4:51






1




1




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 $I=J$.
– James Cheung
Dec 29 '18 at 17:55






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 $I=J$.
– James Cheung
Dec 29 '18 at 17:55






1




1




$s=-id$ 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 $ge 2$.
– Torsten Schoeneberg
Jan 2 at 1:57






$s=-id$ 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 $ge 2$.
– Torsten Schoeneberg
Jan 2 at 1:57












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