Multilinear tools through filters on groups Joshua Maglione jmaglione@math.uni-bielefeld.de Universit¨ at Bielefeld Fakult¨ at f¨ ur Mathematik
What is intrinsic to a group? Main question: What structure is intrinsic to a group G ? A group G given with the shapes: � 1 � I 2 � I 3 � � � A B C , , . 0 I 12 0 I 6 0 I 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
What is intrinsic to a group? Main question: What structure is intrinsic to a group G ? A group G given with the shapes: � 1 � I 2 � I 3 � � � A B C , , . 0 I 12 0 I 6 0 I 4 A group given with the shapes: 1 A C I 2 X Z I 2 B I 3 Y I 8 I 4 What about scalars? J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
1 A C I 2 X Z G = I 2 B H = I 3 Y I 8 I 4 Verbal subgroups produce a series: 1 A C 1 C 1 I 2 B > I 2 > I 2 , I 8 I 8 I 8 I 2 X Z I 2 Z I 2 I 3 Y > I 3 > I 3 . I 4 I 4 I 4 In both groups, γ 1 /γ 2 ∼ γ 2 ∼ = K 16 , = K 8 . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
In some cases, the shape is intrinsic � I a X Z X ∈ M ab ( K ) � � For a field K , let H abc ( K ) = I b Y Y ∈ M bc ( K ) . � � I c Z ∈ M ac ( K ) � Theorem (J.B. Wilson 2017) For groups H abc ( K ) , the integers a, b, c are isomorphism invariants, and they can be computed in polynomial time. Now: special case of a larger body of work with U. First, J.B. Wilson. Idea: All that information found in algebras associated to [ , ] : γ 1 /γ 2 × γ 1 /γ 2 γ 2 , [ , ] : K ab + bc × K ab + bc K ac . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Larger examples are refinable 1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ = 1 ∗ ∗ ∗ G = 1 ∗ ∗ 1 ∗ 1 E.g. γ 0 = γ 1 = G and γ s +1 = [ γ s , γ 1 ], for s ≥ 1. γ 0 γ 1 γ 2 γ 3 γ 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Larger examples are refinable 1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ = 1 ∗ ∗ ∗ G = 1 ∗ ∗ 1 ∗ 1 E.g. γ 0 = γ 1 = G and γ s +1 = [ γ s , γ 1 ], for s ≥ 1. γ 0 γ 1 γ 2 γ 3 γ 4 L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Filters produce refinable graded Lie algebras → 2 G into the normal subgroups with � N d , � � A filter is a function φ : [ φ s , φ t ] ≤ φ s + t and s � t implies φ s ≥ φ t . Theorem (J.B. Wilson 2013) If φ : N d → 2 G is a filter, then � L ( φ ) = φ s / � φ s + t | t � = 0 � s � =0 is an N d -graded Lie ring. Each graded ideal lifts to a filter refinement. J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Efficient refinements for filters Theorem (M. 2017) If φ : N d → 2 G is a totally ordered filter and H ⊳ G refines φ , then there exists an efficient algorithm (polynomial time in log | G | ) that constructs a filter from φ including H . Filter Not a filter Filter Provides structure that connects N d to subgroups of G that can be updated. Refine Generate Allows for efficient recursion. Repeat J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Survey of 500,000,000 groups of order 2 10 40 30 Percent 20 10 0 2 3 4 5 6 7 8 9 10 Length after refinement Filters uncover new characteristic structure [M.-Wilson] . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refining the algebra to get smaller steps φ (1) : N → 2 G φ (2) : N 2 → 2 G Refine � � φ (1) � φ (2) � � � L = L s L = L s s � =0 s � =0 Refine φ (3) : N 3 → 2 G φ : N d → 2 G Refine � � � φ (3) � L = L s L ( φ ) = L s s � =0 s � =0 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K 2 1 0 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K 2 1 0 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Refinement improves even well-known examples L ( γ ) = K 5 ⊕ K 4 ⊕ K 3 ⊕ K 2 ⊕ K L ( φ ) = K 3 ⊕ K 2 ⊕ K 2 ⊕ K 2 ⊕ K ⊕ K 2 ⊕ K 2 ⊕ K 2 1 0 0 1 2 3 4 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Use module and ring theory to start refinement Suppose ◦ : U × V W is a bilinear map of K -vector spaces. Some algebras associated to ◦ are L ◦ = { ( X, Z ) | ( Xu ) ◦ v = Z ( u ◦ v ) } , M ◦ = { ( X, Y ) | ( uX ) ◦ v = u ◦ ( Y v ) } , R ◦ = { ( Y, Z ) | u ◦ ( vY ) = ( u ◦ v ) Z } , Cent( ◦ ) = { ( X, Y, Z ) | ( uX ) ◦ v = u ◦ ( vY ) = ( u ◦ v ) Z } , Der( ◦ ) = { ( X, Y, Z ) | ( uX ) ◦ v + u ◦ ( vY ) = ( u ◦ v ) Z } . Ongoing work with Brooksbank and Wilson using representation theory of Lie algebras in the context of isomorphism problems. Multilinear Algebra package for Magma on GitHub [M.-Wilson] . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Looking for structure in new places Theorem (Brooksbank-M.-Wilson, 2017) There exists a polynomial-time algorithm to test isomorphism of groups of exponent p with central commutator subgroup isomorphic to ( Z /p Z ) 2 . 70 Grp Iso Used by Brooksbank, 60 Lin Alg O’Brien, and Wilson to 50 Minutes efficiently search for local 40 structure. 30 20 Implemented in Magma . 10 0 5 5 5 50 5 100 5 150 5 200 5 256 | G | J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Study the group through L ( φ ) Two fundamental problems arise in partially-ordered case: Let G be nilpotent, and γ : N → 2 G the lower central series. Set φ : N 2 → 2 G such that for s = ( s 1 , s 2 ) ∈ N 2 , φ s = γ s 1 . The associated Lie algebra is trivial � L ( φ ) = φ s / � φ s + t | t � = 0 � = 0 . s � =0 J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Study the group through L ( φ ) Two fundamental problems arise in partially-ordered case: Let G be nilpotent, and γ : N → 2 G the lower central series. Set φ : N 2 → 2 G such that for s = ( s 1 , s 2 ) ∈ N 2 , φ s = γ s 1 . The associated Lie algebra is trivial � L ( φ ) = φ s / � φ s + t | t � = 0 � = 0 . s � =0 Theorem (M. 2018) If G is nilpotent and φ : N d → 2 G is a filter, then there exists a filter θ : N d → 2 G such that im ( φ ) ⊆ im ( θ ) and there is a surjection L ( θ ) → G . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
The other problem 1 a c : a, b, c ∈ K Let K be a field of order q and G = 1 b . 1 There are q + 1 distinct subgroups G ′ < H < G . There is a filter φ : N q +1 → 2 G , such that dim L ( φ ) = q + 2 . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
A bijection between G and L ( φ ) is recovered A filter φ : N d → 2 G is compatible if there exists X ⊂ G : 1 G = �X� , 2 for all s ∈ N d , � φ s ∩ X� = φ s , 3 H �→ H ∩ X is a complete lattice embedding from im( φ ) to 2 X , 4 for all x ∈ X , there exists a unique s ∈ N d such that x ∈ φ s \� φ s + t | t � = 0 � . Theorem (M. 2018) Suppose G is nilpotent and polycyclic. If φ : N d → 2 G is compatible, then there exists a bijection between the set of bases for L ( φ ) and the polycyclic generating sets for G . J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
Filters provide different location for structure Filters refine many examples of groups: 97% in survey of 2 10 . Algebras associated to bilinear maps L s × L t L s + t . Structure from entire N d -graded L ( φ ). Developed for isomorphism, but are general tools. J. Maglione (Bielefeld) Multilinear tools through filters jmaglione@math.uni-bielefeld.de
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