Matrix Groups GAP examples Matrix groups in GAP Schreier-Sims - PowerPoint PPT Presentation

Matrix Groups Max Neunhöffer Introduction Matrix Groups GAP examples Matrix groups in GAP Schreier-Sims Max Neunhöffer Problems Group algebras SLPs Constructive recognition The problem Troubles University of St Andrews GAC 2010, Allahabad

Matrix Groups Introduction Max Neunhöffer Let F be a field. Set Introduction � M ∈ F d × d | M is invertible � GL d ( F ) := . GAP examples Matrix groups in This is a group under matrix multiplication. GAP Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Introduction Max Neunhöffer Let F be a field. Set Introduction � M ∈ F d × d | M is invertible � GL d ( F ) := . GAP examples Matrix groups in This is a group under matrix multiplication. GAP We are here particularly interested in the case that Schreier-Sims F = F q is the finite field with q = p f elements. Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Introduction Max Neunhöffer Let F be a field. Set Introduction � M ∈ F d × d | M is invertible � GL d ( F ) := . GAP examples Matrix groups in This is a group under matrix multiplication. GAP We are here particularly interested in the case that Schreier-Sims F = F q is the finite field with q = p f elements. Problems Group algebras Definition (Matrix group, projective group) SLPs Constructive A matrix group is a subgroup of some GL d ( F ) . recognition The problem Troubles

Matrix Groups Introduction Max Neunhöffer Let F be a field. Set Introduction � M ∈ F d × d | M is invertible � GL d ( F ) := . GAP examples Matrix groups in This is a group under matrix multiplication. GAP We are here particularly interested in the case that Schreier-Sims F = F q is the finite field with q = p f elements. Problems Group algebras Definition (Matrix group, projective group) SLPs Constructive A matrix group is a subgroup of some GL d ( F ) . recognition The problem We call two matrices M , N ∈ GL d ( F ) equivalent, if one is Troubles a scalar multiple of the other and denote the equivalence class of M by [ M ] . Then PGL d ( F ) := { [ M ] | M ∈ GL d ( F ) } is a group with the well-defined multiplication [ M ] · [ N ] := [ MN ] . This is called the projective group.

Matrix Groups GAP examples Max Neunhöffer Introduction GAP examples Matrix groups in GAP Schreier-Sims Problems Group algebras see other window SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction GAP examples Matrix groups in GAP Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims { W ≤ V | dim F q W = k } for some 1 ≤ k < d . Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims { W ≤ V | dim F q W = k } for some 1 ≤ k < d . Problems Group algebras If vG ⊆ V is an orbit, then we have a group SLPs homomorphism Constructive recognition ψ : G → Σ vG , g �→ ( vG → vG , vh �→ vhg ) . The problem Troubles

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims { W ≤ V | dim F q W = k } for some 1 ≤ k < d . Problems Group algebras If vG ⊆ V is an orbit, then we have a group SLPs homomorphism Constructive recognition ψ : G → Σ vG , g �→ ( vG → vG , vh �→ vhg ) . The problem Troubles Lemma If vG contains a basis of V, then ψ is injective.

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims { W ≤ V | dim F q W = k } for some 1 ≤ k < d . Problems Group algebras If vG ⊆ V is an orbit, then we have a group SLPs homomorphism Constructive recognition ψ : G → Σ vG , g �→ ( vG → vG , vh �→ vhg ) . The problem Troubles Lemma If vG contains a basis of V, then ψ is injective. In this case, we can explicitly compute the image of g ∈ G by acting,

Matrix Groups Matrix groups in GAP(currently!) Max Neunhöffer GAP handles matrix groups via permutation groups: Introduction Let G ≤ GL d ( F q ) . Then G acts linearly on GAP examples V := F 1 × d , Matrix groups in q GAP V modulo scalars (projective action), Schreier-Sims { W ≤ V | dim F q W = k } for some 1 ≤ k < d . Problems Group algebras If vG ⊆ V is an orbit, then we have a group SLPs homomorphism Constructive recognition ψ : G → Σ vG , g �→ ( vG → vG , vh �→ vhg ) . The problem Troubles Lemma If vG contains a basis of V, then ψ is injective. In this case, we can explicitly compute the image of g ∈ G by acting, explicitly compute the preimage of a permutation by reading off the images of the basis vectors in vG .

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in GAP Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in Matrix groups act on lots of stuff, just pick an orbit! GAP Schreier-Sims Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in Matrix groups act on lots of stuff, just pick an orbit! GAP Schreier-Sims One problem is to find a short orbit! Problems Group algebras SLPs Constructive recognition The problem Troubles

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in Matrix groups act on lots of stuff, just pick an orbit! GAP Schreier-Sims One problem is to find a short orbit! Problems Various heuristics are used, for example try vectors in Group algebras intersections of eigenspaces of random elements, but in SLPs general, this is difficult. Constructive recognition The problem Troubles

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in Matrix groups act on lots of stuff, just pick an orbit! GAP Schreier-Sims One problem is to find a short orbit! Problems Various heuristics are used, for example try vectors in Group algebras intersections of eigenspaces of random elements, but in SLPs general, this is difficult. Constructive recognition One usually uses projective action and action on vectors The problem Troubles alternatingly.

Matrix Groups Matrix Schreier-Sims Max Neunhöffer In principle one can use the Schreier-Sims procedure to Introduction compute a stabiliser chain for matrix groups as well. GAP examples Matrix groups in Matrix groups act on lots of stuff, just pick an orbit! GAP Schreier-Sims One problem is to find a short orbit! Problems Various heuristics are used, for example try vectors in Group algebras intersections of eigenspaces of random elements, but in SLPs general, this is difficult. Constructive recognition One usually uses projective action and action on vectors The problem Troubles alternatingly. Magma has lots of algorithms for matrix groups using stabiliser chains.