Parabolic subgroup of a reflection group

In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} , which is generated by the set S = { ( 1 2 ) , ( 2 3 ) , … , ( n − 1 n ) } {\displaystyle S=\{(1\ 2),(2\ 3),\ldots ,(n-1\ n)\}} of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of S; equivalently, they are the groups of permutations that come from partitioning the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} into parts { 1 , 2 , … , a 1 } {\displaystyle \{1,2,\ldots ,a_{1}\}} , { a 1 + 1 , a 1 + 2 , … , a 1 + a 2 } {\displaystyle \{a_{1}+1,a_{1}+2,\ldots ,a_{1}+a_{2}\}} , etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself.

Source: Wikipedia — Parabolic subgroup of a reflection group (CC BY-SA 4.0)

Parabolic subgroup of a reflection group

In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} , which is generated by the set S = { ( 1 2 ) , ( 2 3 ) , … , ( n − 1 n ) } {\displaystyle S=\{(1\ 2),(2\ 3),\ldots ,(n-1\ n)\}} of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of S; equivalently, they are the groups of permutations that come from partitioning the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} into parts { 1 , 2 , … , a 1 } {\displaystyle \{1,2,\ldots ,a_{1}\}} , { a 1 + 1 , a 1 + 2 , … , a 1 + a 2 } {\displaystyle \{a_{1}+1,a_{1}+2,\ldots ,a_{1}+a_{2}\}} , etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself.

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Source: Wikipedia "Parabolic subgroup of a reflection group" · CC BY-SA 4.0

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