The complex of p-centric and p-radical subgroups and its reduced Lefschetz module John Maginnis and Silvia Onofrei* Kansas State University The Ohio State University AMS Fall Central Sectional Meeting, University of Akron, Ohio, 20-21 October 2012
Silvia Onofrei (OSU), Properties of Lefschetz modules Subgroup Complexes in a Finite Group G Δ = Δ( C ) Subgroup complex ∙ 0-simplices: C = { Q : Q ≤ G } is a collection of subgroups of the group G , closed under G -conjugation and partially ordered by inclusion ∙ n -simplices: σ = ( Q 0 < Q 1 < ... < Q n ) , Q i ∈ C The group G acts by conjugation on the subgroup complex Δ : G σ = ∩ n ∙ isotropy group of σ : i = 0 N G ( Q i ) Δ Q = Δ( C Q ) with C Q = { P ∈ C ∣ Q ≤ N G ( P ) } ∙ fixed point set of Q : Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 1/10
Silvia Onofrei (OSU), Properties of Lefschetz modules The Reduced Lefschetz Module of a Subgroup Complex in a Finite Group G The reduced Lefschetz virtual module with coefficients in a field k of characteristic p ∣ Δ ∣ ˜ ∑ ( − 1 ) i C i (Δ; k ) ∙ alternating sum of chain groups: L G (Δ; k ) := i = − 1 L G (Δ; k ) = ∑ ˜ ( − 1 ) ∣ σ ∣ Ind G ∙ element of Green ring of kG : G σ k − k σ ∈ Δ / G Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 2/10
Silvia Onofrei (OSU), Properties of Lefschetz modules The Reduced Lefschetz Module of a Subgroup Complex in a Finite Group G The reduced Lefschetz virtual module with coefficients in a field k of characteristic p ∣ Δ ∣ ˜ ∑ ( − 1 ) i C i (Δ; k ) ∙ alternating sum of chain groups: L G (Δ; k ) := i = − 1 L G (Δ; k ) = ∑ ˜ ( − 1 ) ∣ σ ∣ Ind G ∙ element of Green ring of kG : G σ k − k σ ∈ Δ / G Theorem (Robinson, 1988) Let G be a finite group, k a field of characteristic p and Δ a subgroup complex in G. The number of indecomposable summands of ˜ L G (Δ; k ) with vertex Q equals the number of indecomposable summands of ˜ L N G ( Q ) (Δ Q ; k ) with vertex Q. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 2/10
Silvia Onofrei (OSU), Properties of Lefschetz modules The Complex of p -Centric and p -Radical Subgroups A nontrivial p -subgroup Q of G is p -radical if Q = O p ( N G ( Q )) is p -centric if Z ( Q ) ∈ Syl p ( C G ( Q )) Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10
Silvia Onofrei (OSU), Properties of Lefschetz modules The Complex of p -Centric and p -Radical Subgroups A nontrivial p -subgroup Q of G is p -radical if Q = O p ( N G ( Q )) is p -centric if Z ( Q ) ∈ Syl p ( C G ( Q )) D p ( G ) complex ∙ collection of p -centric p -radical subgroups of G Dwyer(1997) ∙ best candidate for a p -local geometry Smith, Yoshiara(1997) ∙ used in cohomology decompositions Dwyer(1998), Grodal(2001) Benson, Smith(2008) ˜ L G ( D p ( G ); k ) ∙ not indecomposable, not projective ∙ vertices are subgroups of non-centric p -radicals Sawabe(2005) Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10
Silvia Onofrei (OSU), Properties of Lefschetz modules The Complex of p -Centric and p -Radical Subgroups A nontrivial p -subgroup Q of G is p -radical if Q = O p ( N G ( Q )) is p -centric if Z ( Q ) ∈ Syl p ( C G ( Q )) D p ( G ) complex ∙ collection of p -centric p -radical subgroups of G Dwyer(1997) ∙ best candidate for a p -local geometry Smith, Yoshiara(1997) ∙ used in cohomology decompositions Dwyer(1998), Grodal(2001) Benson, Smith(2008) ˜ L G ( D p ( G ); k ) ∙ not indecomposable, not projective ∙ vertices are subgroups of non-centric p -radicals Sawabe(2005) If G is a finite simple group of Lie type then ˜ L G ( D p ( G ); k ) ≃ St G the irreducible and projective Steinberg module. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 3/10
Silvia Onofrei (OSU), Properties of Lefschetz modules Terminology and Notation: Groups G is a finite group and p a prime divisor of its order a p -local subgroup is the normalizer of a finite p -subgroup of G a p -central element is an element in the center of a Sylow p -subgroup of G kG is the group algebra with k a field of characteristic p Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 4/10
Silvia Onofrei (OSU), Properties of Lefschetz modules Terminology and Notation: Groups G is a finite group and p a prime divisor of its order a p -local subgroup is the normalizer of a finite p -subgroup of G a p -central element is an element in the center of a Sylow p -subgroup of G kG is the group algebra with k a field of characteristic p G has characteristic p if C G ( O p ( G )) ≤ O p ( G ) 1 G has local characteristic p if all p -local subgroups of G have characteristic p 2 G has parabolic characteristic p if all p -local subgroups which contain a Sylow p -subgroup 3 of G have characteristic p Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 4/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Sketch of proof: Set T := ⟨ t ⟩ . Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Sketch of proof: Set T := ⟨ t ⟩ . If O p ( C G ( T )) contains p -central elements then D p ( G ) T is N G ( T ) -contractible. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Sketch of proof: Set T := ⟨ t ⟩ . If O p ( C G ( T )) contains p -central elements then D p ( G ) T is N G ( T ) -contractible. Thus D ( G ) T is mod- p acyclic. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Sketch of proof: Set T := ⟨ t ⟩ . If O p ( C G ( T )) contains p -central elements then D p ( G ) T is N G ( T ) -contractible. Thus D ( G ) T is mod- p acyclic. .A. Smith theory: D ( G ) Q is mod- p acyclic for any p -subgroup Q > T . And P Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
Silvia Onofrei (OSU), Properties of Lefschetz modules No Vertex of ˜ L G ( D p ( G ); k ) Contains p -Central Elements Proposition (Maginnis, Onofrei, 2009) Assume G is a finite group of parabolic characteristic p. Suppose that t is an element of order p in G such that O p ( C G ( t )) contains p-central elements. Then no vertex of the reduced Lefschetz module ˜ L G ( D p ( G ); k ) contains a conjugate of t. Sketch of proof: Set T := ⟨ t ⟩ . If O p ( C G ( T )) contains p -central elements then D p ( G ) T is N G ( T ) -contractible. Thus D ( G ) T is mod- p acyclic. .A. Smith theory: D ( G ) Q is mod- p acyclic for any p -subgroup Q > T . And P It follows ˜ L N G ( Q ) ( D ( G ) Q ; k ) = 0. Fall Central Sectional Meeting, University of Akron, 20-21 October 2012 5/10
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