Essential p -dimension of a normalizer of a maximal torus Mark L. MacDonald University of British Columbia May 2011 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential dimension Let k 0 be a field, and consider a functor F : Fields / k 0 → Sets Definition k The essential dimension of x ∈ F ( k ) is ed k 0 ( x ) := min { tr . deg k 0 ( L ) | x is defined over L } . L The essential dimension of F is k 0 ed k 0 ( F ) := sup { ed k 0 ( x ) |∀ x ∈ F ( k ) } . Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential dimension Let k 0 be a field, and consider a functor F : Fields / k 0 → Sets Definition k The essential dimension of x ∈ F ( k ) is ed k 0 ( x ) := min { tr . deg k 0 ( L ) | x is defined over L } . L The essential dimension of F is k 0 ed k 0 ( F ) := sup { ed k 0 ( x ) |∀ x ∈ F ( k ) } . ed( G ) := ed( H 1 ( − , G )) . Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential dimension Let k 0 be a field, and consider a functor F : Fields / k 0 → Sets Definition k The essential dimension of x ∈ F ( k ) is ed k 0 ( x ) := min { tr . deg k 0 ( L ) | x is defined over L } . L The essential dimension of F is k 0 ed k 0 ( F ) := sup { ed k 0 ( x ) |∀ x ∈ F ( k ) } . ed( G ) := ed( H 1 ( − , G )) . ed( SL n ) = ed( GL n ) = ed( Sp 2 n ) = 0 . Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential p -dimension Definition k ′ Let p be a prime. The essential p-dimension of � � � � x ∈ F ( k ) is � � � � L k ed( x ; p ) := min { ed( x k ′ ) | k ′ / k of finite degree � � � } . � � not divisible by p � � � � The essential p-dimension of F is ed( F ) := sup { ed( x ; p ) |∀ x ∈ F ( k ) } . k 0 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential p -dimension Definition k ′ Let p be a prime. The essential p-dimension of � � � � x ∈ F ( k ) is � � � � L k ed( x ; p ) := min { ed( x k ′ ) | k ′ / k of finite degree � � � } . � � not divisible by p � � � � The essential p-dimension of F is ed( F ) := sup { ed( x ; p ) |∀ x ∈ F ( k ) } . k 0 0 ≤ ed( G ; p ) ≤ ed( G ) . Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Goal of this talk Let G be a connected (almost) simple split linear algebraic group, with split maximal torus T . Then the normalizer fits in an exact sequence 1 → T → N → W → 1 . Goal Find the exact value of ed( N ; p ). Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Let N be an extension of a p -group by a torus; i.e. an algebraic group such that 1 → T → N → F → 1. Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Let N be an extension of a p -group by a torus; i.e. an algebraic group such that 1 → T → N → F → 1. Theorem [L¨ otscher, M-, Meyer, Reichstein, 2010] Assume every finite field extension of k has degree a power of p . min dim( V ) − dim N ≤ ed( N ; p ) ≤ min dim( W ) − dim N , where the minimums are taken over all N -representations defined over k such that V is p -faithful, and W is p -generically free. A representation is said to be p-faithful if its kernel is finite order not divisible by p . If additionally N / ker is generically free, then the representation is said to be p -generically free. Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Discrete data To a split algebraic group we can associate the following discrete data: Root system R Character lattice ˆ T Weyl group representation W → GL( ˆ T ) ˆ T Spin 16 � ������� � � � � � � ˆ ˆ T SO 16 T HSpin 16 � � � � � � � � � � � � � � ˆ T PSO 16 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Discrete data To a split algebraic group we can associate the following discrete data: Root system R Character lattice ˆ T Weyl group representation W → GL( ˆ T ) ˆ ˆ T Spin 16 T SL 9 � ������� � ������� � � � � � ˆ ˆ ˆ = ˆ T SO 16 T HSpin 16 T E 8 = T SL 9 /µ 3 � � � ������� � � � � � � � � � � � ˆ ˆ T PSO 16 T PSL 9 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Representations vs. subsets of the character lattice Assume 1 → T → N → F → 1 with T split and F constant p -group. An N -representation decomposes as V = ⊕ λ ∈ Λ V λ for some F -invariant subset Λ ⊂ ˆ T , V � Λ . V is p -faithful iff Λ is p-generating (i.e. generates a sublattice of ˆ T whose index is not divisible by p ). Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Representations vs. subsets of the character lattice Assume 1 → T → N → F → 1 with T split and F constant p -group. An N -representation decomposes as V = ⊕ λ ∈ Λ V λ for some F -invariant subset Λ ⊂ ˆ T , V � Λ . V is p -faithful iff Λ is p-generating (i.e. generates a sublattice of ˆ T whose index is not divisible by p ). Conversely: If N = N G ( T ) for a split maximal torus in a simple algebraic group, then we can construct an N -representation from an F -invariant subset of ˆ T . Λ � V Λ . Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Symmetric p -rank Definition The symmetric p-rank of an (integral) representation W → GL( L ), is the minimal size of an F -invariant p -generating subset of L , where F ⊂ W is a Sylow p -subgroup. Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Symmetric p -rank Definition The symmetric p-rank of an (integral) representation W → GL( L ), is the minimal size of an F -invariant p -generating subset of L , where F ⊂ W is a Sylow p -subgroup. For example, L = Z , and W = Z / 2 acting by negation. Then the symmetric 2-rank equals 2, by taking Λ = {− 1 , 1 } , or Λ = {− 3 , 3 } , etc. Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Example: E 8 , p = 5 F = W ( E 8 ) (5) ∼ = W ( A 4 ) (5) × W ( A 4 ) (5) . α 2 α 1 α 3 α 4 α 5 α 6 α 7 α 8 α 0 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Example: E 8 , p = 5 F = W ( E 8 ) (5) ∼ = W ( A 4 ) (5) × W ( A 4 ) (5) . ���� ���� α 2 ���� ���� ���� ���� ���� ���� α 1 α 3 α 4 α 5 α 6 α 7 α 8 α 0 1) Consider elements of ˆ T E 8 in the Z -basis { α 1 , · · · , α 8 } . If Λ ⊂ ˆ T E 8 is 5-generating, it must contain an element whose α 5 coefficient is not divisible by 5. 2) Such an element has F -orbit of size 25. 3) Therefore ed( N E 8 ( T ); 5) = 25 − 8 = 17. Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Other exceptional cases φ G p SymRank( φ ; p ) ( K ) 2 A 2 G 2 2 4 n 1 A 2 G 2 3 3 n 2 D 4 F 4 2 16 y 3 D 4 3 9 y F 4 1 E 6 2 16 y E 6 2 E 6 2 E 6 2 32 y 1 E 7 2 64 y E 7 1 E +2 2 E 7 2 40 y 7 1 E 8 E 8 2 128 y 1 E 6 E 6 3 27 y 1 E +3 3 E 6 3 27 y 6 1 E 7 E 7 3 27 y 1 E 8 E 8 3 81 y 1 E 8 E 8 5 25 y Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential p -dimension of Weyl groups W ( R ) H 1 ( k , N ) ։ H 1 ( k , G ) ⇒ ed( G ; p ) ≤ ed( N ; p ) H 1 ( k , N ) ։ H 1 ( k , W ) ⇒ ed( W ; p ) ≤ ed( N ; p ) Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Essential p -dimension of Weyl groups W ( R ) H 1 ( k , N ) ։ H 1 ( k , G ) ⇒ ed( G ; p ) ≤ ed( N ; p ) H 1 ( k , N ) ։ H 1 ( k , W ) ⇒ ed( W ; p ) ≤ ed( N ; p ) ed( W ( R ); p ) p = 0 p = 2 p = 3 p = 5 p = 7 p ≥ 11 ⌊ n +1 A n ?? 2 ⌋ ⌊ ( n + 1) / p ⌋ ⌊ n / p ⌋ B n n n C n n n ⌊ n / p ⌋ D n ( n odd) n − 1 n − 1 ⌊ n / p ⌋ D n ( n even) n n ⌊ n / p ⌋ E 6 ?? 4 3 1 0 0 7 7 3 1 1 0 E 7 E 8 8 8 4 2 1 0 4 4 2 0 0 0 F 4 G 2 2 2 1 0 0 0 Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Summary Convert the essential dimension problem ed( N ; p ) into a problem about Weyl group actions on character lattices. Compute the symmetric p -rank by considering each connected simple split group separately. In “most” cases, the bounds match: min dim V − dim N ≤ ed( N ; p ) ≤ min dim W − dim N , where V is p -faithful, and W is p -gen. free. All known cases are consistent with the conjecture that the upper bound is an equality Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
Appendix: D n lattices φ G p Conditions SymRank( φ ; p ) ( K ) 2 2 r +2 ( s − 1) n = 2 r s , s > 1 y 1 D n PSO 2 n 2 n = 2 r ≥ 4 2 2 r y 2 2 r +2 ( s − 1) n = 2 r s , s > 1 y 2 D n PSp 2 n 2 n = 2 r ≥ 4 2 2 r y 1 D +2 2 n − 1 HSpin 2 n 2 n ≥ 6 y n 1 I n 2 n ≥ 4 2 n n SO 2 n 2 I n Sp 2 n , SO 2 n +1 2 n ≥ 2 2 n n 2 n − 1 n ≥ 5 odd y 1 D +4 2 Spin 2 n 2 n − 1 + 2 r +1 n n ≥ 4 even y 2 D +4 2 n Spin 2 n +1 2 n ≥ 2 y n 3 D 4 F 4 3 9 y Mark L. MacDonald Essential p -dimension of a normalizer of a maximal torus
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