Geometric actions of classical groups Raffaele Rainone School of - PowerPoint PPT Presentation

Geometric actions of classical groups Raffaele Rainone School of Mathematics University of Southampton Groups St. Andrews August 2013 Raffaele Rainone Geometric actions of classical groups

Algebraic groups Let k be an algebraically closed field of characteristic p � 0. An algebraic group G is an affine algebraic variety, defined over k , with a group structure such that µ : G × G → G ι : G → G x − 1 ( x , y ) �→ xy x �→ are morphisms of varieties. Example The prototype is the special linear group SL n ( k ) = { A ∈ M n ( k ) | det( A ) = 1 } Raffaele Rainone Geometric actions of classical groups

Actions of algebraic groups Let G be an algebraic group and Ω a variety (over k ). An action of G on Ω is a morphism of varieties (with the usual properties) G × Ω → Ω ( x , ω ) �→ x .ω We can define orbits and stabilisers as usual: orbits are locally closed subsets of Ω, and we can define dim G . x = dim G . x for ω ∈ Ω, G ω � G is closed Raffaele Rainone Geometric actions of classical groups

Actions of algebraic groups Let G be an algebraic group and Ω a variety (over k ). An action of G on Ω is a morphism of varieties (with the usual properties) G × Ω → Ω ( x , ω ) �→ x .ω We can define orbits and stabilisers as usual: orbits are locally closed subsets of Ω, and we can define dim G . x = dim G . x for ω ∈ Ω, G ω � G is closed Lemma Let H � G be a closed subgroup. Then (i) G / H is a (quasi-projective) variety (ii) there is a natural (transitive) action G × G / H → G / H Raffaele Rainone Geometric actions of classical groups

Fixed point spaces Let G be an algebraic group acting on a variety Ω. For any x ∈ G , the fixed point space C Ω ( x ) = { ω ∈ Ω : x .ω = ω } ⊆ Ω is closed. Proposition Let Ω = G / H. Then, for x ∈ G, if x G ∩ H = ∅ � 0 dim C Ω ( x ) = dim Ω − dim x G + dim( x G ∩ H ) otherwise General aim : given x ∈ G of prime order, derive bounds on f Ω ( x ) = dim C Ω ( x ) dim Ω Raffaele Rainone Geometric actions of classical groups

Classical groups Let V be an n -dimensional k -vector space. GL( V ) = invertible linear maps V → V Sp( V ) = { x ∈ GL( V ) : β ( x . u , x . v ) = β ( u , v ) } O( V ) = { x ∈ GL( V ) : Q ( x . u ) = Q ( u ) } where: β is a symplectic form on V Q is a non-degenerate quadratic form on V . We write Cl ( V ) for GL( V ) , Sp( V ) , O( V ) Similarly Cl n for GL n , Sp n , O n Raffaele Rainone Geometric actions of classical groups

Subgroup structure: geometric subgroups Let G = Cl ( V ) be a classical group. We define 5 families of positive-dimension subgroups that arise naturally from the underlying geometry of V C 1 stabilisers of subspaces U ⊂ V C 2 stabilisers of direct sum decompositions V = V 1 ⊕ . . . ⊕ V t C 3 stabilisers of totally singular decompositions V = U ⊕ W , when G = Sp( V ) or O( V ) C 4 stabilisers of tensor product decompositions V = V 1 ⊗ . . . ⊗ V t C 5 stabiliser of non-degenerate forms on V Set C ( G ) = � C i . Raffaele Rainone Geometric actions of classical groups

Subgroup structure Example C 2 Let G = GL n . Assume V = V 1 ⊕ . . . ⊕ V t where dim V i = n / t . Then H = GL n / t ≀ S t , and H ◦ = GL n / t × . . . × GL n / t C 3 Let G = Sp n . Assume V = U ⊕ W where U , W are maximal totally singular subspaces. Then H = GL n / 2 . 2 and � ∼ �� A H ◦ = � : A ∈ GL n / 2 = GL n / 2 A − t Raffaele Rainone Geometric actions of classical groups

Subgroup structure Example C 2 Let G = GL n . Assume V = V 1 ⊕ . . . ⊕ V t where dim V i = n / t . Then H = GL n / t ≀ S t , and H ◦ = GL n / t × . . . × GL n / t C 3 Let G = Sp n . Assume V = U ⊕ W where U , W are maximal totally singular subspaces. Then H = GL n / 2 . 2 and � ∼ �� A H ◦ = � : A ∈ GL n / 2 = GL n / 2 A − t Theorem (Liebeck - Seitz, 1998) Let G = SL ( V ) , Sp ( V ) or SO ( V ) and H � G closed and positive dimensional. Then either H is contained in a member of C ( G ) , or H ◦ is simple and acts irreducibly on V . Raffaele Rainone Geometric actions of classical groups

Aim G = Cl ( V ) classical algebraic group H � G closed geometric subgroup Ω = G / H Main aim Derive bounds on f Ω ( x ) = dim C Ω ( x ) dim Ω for all x ∈ G of prime order. Further aims sharpness, characterisazions? “Local bounds”: how does the action of x on V influence f Ω ( x )? Raffaele Rainone Geometric actions of classical groups

Background Let G be a simple algebraic group, H � G closed. Set Ω = G / H . Theorem (Lawther, Liebeck, Seitz (2002)) If G exceptional then, for x ∈ G of prime order, f Ω ( x ) � δ ( G , H , x ) Theorem (Burness, 2003) Either there exists an involution x ∈ G f Ω ( x ) = dim C Ω ( x ) � 1 2 − ǫ dim Ω for a small ǫ � 0 , or ( G , Ω) is in a short list of known exceptions. Raffaele Rainone Geometric actions of classical groups

Background Further motivation arises from finite permutation group. Let Ω be a finite set and G � Sym(Ω). For x ∈ G , the fixed point ratio is defined fpr Ω ( x ) = | C Ω ( x ) | | Ω | If G is transitive with point stabiliser H then fpr Ω ( x ) = | x G ∩ H | | x G | Raffaele Rainone Geometric actions of classical groups

Background Further motivation arises from finite permutation group. Let Ω be a finite set and G � Sym(Ω). For x ∈ G , the fixed point ratio is defined fpr Ω ( x ) = | C Ω ( x ) | | Ω | If G is transitive with point stabiliser H then fpr Ω ( x ) = | x G ∩ H | | x G | Bounds on fpr have been studied and applied to a variety of problems, e.g. base sizes monodromy groups of covering of Riemann surfaces (random) generation of simple groups Raffaele Rainone Geometric actions of classical groups

Fixed point spaces Let G = Cl ( V ), H � G closed and Ω = G / H . Recall, for x ∈ H fixed, dim C Ω ( x ) = dim Ω − dim x G + dim( x G ∩ H ) To compute dim C Ω ( x ) we need: (i) information on the centraliser C G ( x ), so dim x G = dim G − dim C G ( x ) (ii) informations on the fusion of H -classes in G , so we can compute dim( x G ∩ H ). Raffaele Rainone Geometric actions of classical groups

Conjugacy classes I For x ∈ GL n we have x = x s x u = x u x s . Raffaele Rainone Geometric actions of classical groups

Conjugacy classes I For x ∈ GL n we have x = x s x u = x u x s . Up to conjugation, x u = [ J a n n , . . . , J a 1 x s = [ λ 1 I a 1 , λ 2 I a 2 , . . . , λ n I a n ] , 1 ] Raffaele Rainone Geometric actions of classical groups

Conjugacy classes I For x ∈ GL n we have x = x s x u = x u x s . Up to conjugation, x u = [ J a n n , . . . , J a 1 x s = [ λ 1 I a 1 , λ 2 I a 2 , . . . , λ n I a n ] , 1 ] Fact Let s , s ′ and u , u ′ in G = Cl ( V ). Then s ∼ G s ′ , u ∼ G u ′ if, and only if, they are GL( V )-conjugate (unless p = 2 and u , u ′ are unipotent). Raffaele Rainone Geometric actions of classical groups

Conjugacy classes I For x ∈ GL n we have x = x s x u = x u x s . Up to conjugation, x u = [ J a n n , . . . , J a 1 x s = [ λ 1 I a 1 , λ 2 I a 2 , . . . , λ n I a n ] , 1 ] Fact Let s , s ′ and u , u ′ in G = Cl ( V ). Then s ∼ G s ′ , u ∼ G u ′ if, and only if, they are GL( V )-conjugate (unless p = 2 and u , u ′ are unipotent). It is well known how to compute dim x G for unipotent and semisimple elements. For example if G = GL n : n s = n 2 − � dim x G a 2 i i =1 n u = n 2 − 2 � � ia 2 dim x G ia i a j − i 1 ≤ i < j ≤ n i =1 Raffaele Rainone Geometric actions of classical groups

Conjugacy classes II Recall: dim C Ω ( x ) = dim Ω − dim x G + dim( x G ∩ H ). Raffaele Rainone Geometric actions of classical groups

Conjugacy classes II Recall: dim C Ω ( x ) = dim Ω − dim x G + dim( x G ∩ H ). In general it is hard to compute dim( x G ∩ H ), but the following result is useful: Theorem (Guralnick, 2007) If H ◦ is reductive then x G ∩ H = x H 1 ∪ . . . ∪ x H m for some m. Thus dim( x G ∩ H ) = max i { dim x H i } . Raffaele Rainone Geometric actions of classical groups

Example Let G = GL 18 , H = GL 6 ≀ S 3 and p = 3. Set Ω = G / H , thus dim Ω = 18 2 − 3 · 6 2 = 216. Let 1 ] , dim x G = 174 x = [ J 2 3 , J 3 2 , J 6 Then x G ∩ H = x G ∩ H ◦ and x G ∩ H = � 4 i =1 x H where i x 1 = [ J 2 3 | J 2 2 , J 2 1 | J 2 , J 4 1 ] , x 2 = [ J 2 3 | J 3 2 | J 6 1 ] , x 3 = [ J 3 , J 2 , J 1 | J 3 , J 2 , J 1 | J 2 , J 4 1 ] , x 4 = [ J 3 , J 2 , J 1 | J 3 , J 3 1 | J 2 2 , J 2 1 ] and dim x H 1 = 46 , dim x H 2 = 42 , dim x H 3 = 54 , dim x H 4 = 52 Thus dim( x G ∩ H ) = 54. Therefore f Ω ( x ) = 4 9 > 1 3 . Raffaele Rainone Geometric actions of classical groups

Main result: Global bounds Recall that H ∈ C 2 ∪ C 3 is a stabiliser of a decomposition V = V 1 ⊕ . . . ⊕ V t . Theorem (R., 2013) Let G = Cl n and H ∈ C 2 ∪ C 3 . Set Ω = G / H and fix x ∈ H of prime order r. Then 1 r − ǫ � f Ω ( x ) = dim C Ω ( x ) � 1 − 1 dim Ω n where  0 r = p    1 p � = r > n ǫ = r  rt 2  p � = r � n  4 n 2 ( t − 1) Raffaele Rainone Geometric actions of classical groups