On splitting of the normalizer of a maximal torus in groups of Lie type Alexey Galt 07.08.2017
Example 1 Let G = SL 2 ( F p ) be the special linear group of degree 2 over F p . � λ � 0 ∗ Then T = { , λ ∈ F p } is a maximal torus of G . The λ − 1 0 normalizer N G ( T ) is the group of all monomial matrices of G and N G ( T ) /T ≃ Sym 2 . But G contains only one element of � − 1 0 � order two: , and this element lies in T . Hence, 0 − 1 N G ( T ) does not split over T .
Example 2 Let G = GL n ( F p ) be the general linear group of degree n over F p . Then T = D n ( F p ) is a maximal torus of G . The normalizer N G ( T ) is the group of all monomial matrices of G and N G ( T ) /T ≃ Sym n . There is a canonical embedding of Sym n into the group of all monomial matrices of G . If H is an image of Sym n under this embedding, then H is a complement for T in N G ( T ). Since the center Z ( G ) of G is contained in T , then a maximal torus of PGL n ( F p ) also has a complement in their normalizer. Moreover, PGL n ( F p ) ≃ PSL n ( F p ) and the same is true for PSL n ( F p ).
Example 2 Let G = GL n ( F p ) be the general linear group of degree n over F p . Then T = D n ( F p ) is a maximal torus of G . The normalizer N G ( T ) is the group of all monomial matrices of G and N G ( T ) /T ≃ Sym n . There is a canonical embedding of Sym n into the group of all monomial matrices of G . If H is an image of Sym n under this embedding, then H is a complement for T in N G ( T ). Since the center Z ( G ) of G is contained in T , then a maximal torus of PGL n ( F p ) also has a complement in their normalizer. Moreover, PGL n ( F p ) ≃ PSL n ( F p ) and the same is true for PSL n ( F p ).
Problems Let G be a simple connected linear algebraic group over the algebraic closure F p of a finite field of positive characteristic p . Let σ be a Steinberg endomorphism and T a maximal σ -invariant torus of G . It’s well known that all the maximal tori are conjugated in G and the quotient N G ( T ) /T is isomorphic to the Weyl group W of G . The following problem arises. Problem 1 Describe the groups G in which N G ( T ) splits over T .
Problems Let G be a simple connected linear algebraic group over the algebraic closure F p of a finite field of positive characteristic p . Let σ be a Steinberg endomorphism and T a maximal σ -invariant torus of G . It’s well known that all the maximal tori are conjugated in G and the quotient N G ( T ) /T is isomorphic to the Weyl group W of G . The following problem arises. Problem 1 Describe the groups G in which N G ( T ) splits over T .
Problems A similar problem arises in finite groups G of Lie type. Let T = T ∩ G be a maximal torus in a finite group of Lie type G , N ( G, T ) = N G ( T ) ∩ G an algebraic normalizer of G . Notice that N ( G, T ) � N G ( T ), but the equality is not true in general. Problem 2 Describe the groups G and their maximal tori T in which N ( G, T ) splits over T .
Problems A similar problem arises in finite groups G of Lie type. Let T = T ∩ G be a maximal torus in a finite group of Lie type G , N ( G, T ) = N G ( T ) ∩ G an algebraic normalizer of G . Notice that N ( G, T ) � N G ( T ), but the equality is not true in general. Problem 2 Describe the groups G and their maximal tori T in which N ( G, T ) splits over T .
History J.Tits ”Normalisateurs de tores I. Groupes de Coxeter ´ Etendus” // Journal of Algebra, 1966, V.4, 96–116. An answer to Problem 1 for simple Lie groups was given in M. Curtis, A. Wiederhold, B. Williams, ”Normalizers of maximal tori” // Springer, Berlin, 1974, Lecture Notes in Math., V. 418, 31–47.
History J.Tits ”Normalisateurs de tores I. Groupes de Coxeter ´ Etendus” // Journal of Algebra, 1966, V.4, 96–116. An answer to Problem 1 for simple Lie groups was given in M. Curtis, A. Wiederhold, B. Williams, ”Normalizers of maximal tori” // Springer, Berlin, 1974, Lecture Notes in Math., V. 418, 31–47.
Algebraic groups The answer for Problem 1 is in the following table: Group Conditions of existence of a complement SL n ( F p ) p = 2 or n is odd PSL n ( F p ) No conditions Sp 2 n ( F p ) p = 2 PSp 2 n ( F p ) p = 2 or n � 2 SO 2 n +1 ( F p ) No conditions SO 2 n ( F p ) No conditions PSO 2 n ( F p ) No conditions G 2 ( F p ) No conditions F 4 ( F p ) p = 2 E k ( F p ) p = 2
Algebraic groups The answer for Problem 1 is in the following table: Group Conditions of existence of a complement SL n ( F p ) p = 2 or n is odd PSL n ( F p ) No conditions Sp 2 n ( F p ) p = 2 PSp 2 n ( F p ) p = 2 or n � 2 SO 2 n +1 ( F p ) No conditions SO 2 n ( F p ) No conditions PSO 2 n ( F p ) No conditions G 2 ( F p ) No conditions F 4 ( F p ) p = 2 E k ( F p ) p = 2
Let O p ′ ( G σ ) � G � G σ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G . Let W be a Weyl group of G , π a natural homomorphism from N = N G ( T ) into W . Two elements w 1 , w 2 are called σ -conjugate if w 1 = ( w − 1 ) σ w 2 w for some element w of W . Proposition There is a bijection between the G -classes of σ -stable maximal tori of G and the σ -conjugacy classes of W . Define C W,σ ( w ) = { x ∈ W | ( x − 1 ) σ wx = w } . Proposition Let g σ g − 1 ∈ N and π ( g σ g − 1 ) = w . Then g )) σ / ( T g ) σ ≃ C W,σ ( w ) . ( N G ( T
Let O p ′ ( G σ ) � G � G σ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G . Let W be a Weyl group of G , π a natural homomorphism from N = N G ( T ) into W . Two elements w 1 , w 2 are called σ -conjugate if w 1 = ( w − 1 ) σ w 2 w for some element w of W . Proposition There is a bijection between the G -classes of σ -stable maximal tori of G and the σ -conjugacy classes of W . Define C W,σ ( w ) = { x ∈ W | ( x − 1 ) σ wx = w } . Proposition Let g σ g − 1 ∈ N and π ( g σ g − 1 ) = w . Then g )) σ / ( T g ) σ ≃ C W,σ ( w ) . ( N G ( T
Let O p ′ ( G σ ) � G � G σ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G . Let W be a Weyl group of G , π a natural homomorphism from N = N G ( T ) into W . Two elements w 1 , w 2 are called σ -conjugate if w 1 = ( w − 1 ) σ w 2 w for some element w of W . Proposition There is a bijection between the G -classes of σ -stable maximal tori of G and the σ -conjugacy classes of W . Define C W,σ ( w ) = { x ∈ W | ( x − 1 ) σ wx = w } . Proposition Let g σ g − 1 ∈ N and π ( g σ g − 1 ) = w . Then g )) σ / ( T g ) σ ≃ C W,σ ( w ) . ( N G ( T
Let O p ′ ( G σ ) � G � G σ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G . Let W be a Weyl group of G , π a natural homomorphism from N = N G ( T ) into W . Two elements w 1 , w 2 are called σ -conjugate if w 1 = ( w − 1 ) σ w 2 w for some element w of W . Proposition There is a bijection between the G -classes of σ -stable maximal tori of G and the σ -conjugacy classes of W . Define C W,σ ( w ) = { x ∈ W | ( x − 1 ) σ wx = w } . Proposition Let g σ g − 1 ∈ N and π ( g σ g − 1 ) = w . Then g )) σ / ( T g ) σ ≃ C W,σ ( w ) . ( N G ( T
Linear groups In case of linear group W ≃ Sym n and the σ -conjugacy classes C W,σ ( w ) of W are coincide with ordinary conjugacy classes of symmetric group. Each such class corresponds to the cycle-type ( n 1 )( n 2 ) . . . ( n m ). Let { n 1 , . . . , n m } be a partition of n . We assume that n 1 = . . . = n l 1 < . . . < n l 1 + ... + l r − 1 +1 = . . . = n l 1 + ... + l r and a 1 = n l 1 l 1 , a 2 = n l 1 + l 2 l 2 , . . . , a r = n l 1 + ... + l r l r . Theorem Let T be a maximal torus of G = SL n ( q ) with the cycle-type ( n 1 )( n 2 ) . . . ( n m ). Then T has a complement in N if and only if q is even or a i is odd for some 1 � i � r .
Linear groups In case of linear group W ≃ Sym n and the σ -conjugacy classes C W,σ ( w ) of W are coincide with ordinary conjugacy classes of symmetric group. Each such class corresponds to the cycle-type ( n 1 )( n 2 ) . . . ( n m ). Let { n 1 , . . . , n m } be a partition of n . We assume that n 1 = . . . = n l 1 < . . . < n l 1 + ... + l r − 1 +1 = . . . = n l 1 + ... + l r and a 1 = n l 1 l 1 , a 2 = n l 1 + l 2 l 2 , . . . , a r = n l 1 + ... + l r l r . Theorem Let T be a maximal torus of G = SL n ( q ) with the cycle-type ( n 1 )( n 2 ) . . . ( n m ). Then T has a complement in N if and only if q is even or a i is odd for some 1 � i � r .
Symplectic and orthogonal groups Let n = n ′ + n ′′ , { n 1 , . . . , n k } and { n k +1 , . . . , n m } be partitions of n ′ and n ′′ , respectively. A set {− n 1 , . . . , − n k , n k +1 , . . . , n m } will be called a cycle-type and denoted by ( n 1 ) . . . ( n k )( n k +1 ) . . . ( n m ). As above we assume that n 1 = . . . = n l 1 < . . . < n l 1 + ... + l r − 1 +1 = . . . = n l 1 + ... + l r Let a 1 = n l 1 l 1 , a 2 = n l 1 + l 2 l 2 , . . . , a r = n l 1 + ... + l r l r .
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