PermutationGroups,Banﬀ2009 AndreaLucchini Onthe(non)contractibilityofthesimplicialcomplexassociatedtothecosetposetofaclassicalgroup

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Motivations and Results Outline of the strategy of the proof

On the (non) contractibility of the simplicial complex associated to the coset poset

of a classical group

Andrea Lucchini

Università di Padova

Permutation Groups, Banff 2009

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Introduction

A conjecture of Kennet Brown

The Main Result: a recent result by Massimiliano Patassini

The simplicial complex associated to the coset poset

Definition (S. Bouc - K. Brown)

Thecoset poset C(G )associated to a finite group G is the poset consisting of the proper (right) cosets Hx, with H < G and x ∈ G , ordered by inclusion.

Hx ⊆ Ky ⇐⇒ H ≤ K and Ky = Kx.

We can apply topological concepts to this poset by using the simplicial complex∆ = ∆(C(G ))whose simplices are the finite chains H₁g₁< H₂g₂ < ... < H_ng_n of elements of C(G ).

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Introduction

Euler characteristic and contractibility of ∆(C(G ))

In particular, we can speak of theEuler characteristic of C(G ) χ(C(G )):=X

m

(−1)^mαm

where α_m is the number of chains in C(G ) of length m and of thereduced Euler characteristic of C(G )

˜

χ(C(G )):= χ(C(G )) − 1.

˜

χ(C(G )) 6= 0 ⇒ the simplicial complex ∆(C(G )) is not contractible.

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Introduction

The Dirichlet polynomial P

_G

(s) of a group G

In 2000, Kennet Brown pointed out a connection between C(G ) and theDirichlet polynomial P_G(s)associated to G , given by

P_G(s) =

∞

X

k=1

a_k(G )

k^s , where a_k(G ) = X

H≤G ,|G :H|=k

µG(H),

and µ_G is the Möbius function of the subgroup lattice of G , defined by µ_G(G ) = 1, µ_G(H) = −P

K >Hµ_G(K ) if H < G . Remark (Bouc, Brown)

˜

χ(C(G )) = −P_G(−1).

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Introduction

Conjecture of Brown

Conjecture (Brown, 2000)

Let G be a finite group. Then P_G(−1) 6= 0, hence the simplicial complex associated to the coset poset of G is non-contractible.

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Introduction

Conjecture of Brown

Conjecture (Brown, 2000)

Let G be a finite group. Then P_G(−1) 6= 0, hence the simplicial complex associated to the coset poset of G is non-contractible.

Evidences for the conjectures

1 P_G(−1) 6= 0 for all the finite groups for which PG(−1) have been computed by GAP.

2 (K. Brown - 2000) True for solvable groups.

3 (M. Patassini - 2008) True for PSL(2, q),²B₂(q), ²G₂(q).

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Introduction

Theorem (Brown, 2000)

If G is a finite soluble group, then P_G(−1) 6= 1.

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Introduction

Theorem (Brown, 2000)

If G is a finite soluble group, then P_G(−1) 6= 1.

Proof

Let1 = N₀ E N1E · · · E Nt = G be a chief series of G . Define

1 c_i := the numbers of complements of N_i/Ni −1 in G /N_{i −1},

2 d_i := the order of N_i/N_{i −1}. Then

P_G(s) = Y

1≤i ≤t

(1 − ci/d_i^s) hence

P_G(−1) = Y

1≤i ≤t

(1 − cid_i) 6= 0.

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Introduction

The Main Result: a recent result by Massimiliano Patassini

Let G be the set of groups X satisfying the following:

X is nearly simple, i.e. X /Z (X ) is almost simple with socle G . G is a classical simple group (i.e. PSL_n(q), PSUn(q),

PSp_n(q), PΩ_n(q), where ∈ {o, +, −}).

X does not contain a non-trivial graph automorphism.

G is not isomorphic to PSL₂(49) or to PSp₆(2).

Theorem (M. Patassini 2009) If X ∈ G, then P_X(−1) 6= 0.

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General strategy

The p-Dirichlet polynomial of G Two types of non-parabolic subgroups H contained in a non-parabolic maximal

H non-parabolic, H intersection of parabolic maximal So, we are done!

Assumptions and useful definitions

For the sake of simplicity, we present an outline of the proof in the case X = G , a classical simple group.

Definition

Letp be the characteristic of G . Thep-Dirichlet polynomialof G is P_G^(p)(s)= X

(p,n)=1

a_n(G ) n^s .

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General strategy

General Strategy

It suffices to prove:

P_G^(p)(−1) 6= 0;

p divides |G : H| and µ_G(H) 6= 0 ⇒ |P_G^(p)(−1)|_p < |G : H|²_p.

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General strategy

General Strategy

P_G^(p)(−1) 6= 0;

p divides |G : H| and µ_G(H) 6= 0 ⇒ |P_G^(p)(−1)|_p < |G : H|²_p.

P_G(−1) = P_G^(p)(−1) +P

H∈Mµ_G(H)|G : H| with M:= {H ≤ G | p divides |G : H| and µ_G(H) 6= 0}

(Isaacs, Hawkes, Özaydin) G perfect ⇒ |G : H| divides µ_G(H)

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General strategy

General Strategy

P_G^(p)(−1) 6= 0;

p divides |G : H| and µ_G(H) 6= 0 ⇒ |P_G^(p)(−1)|p < |G : H|²_p. P_G(−1) = P_G^(p)(−1) +P

H∈MµG(H)|G : H| with M:= {H ≤ G | p divides |G : H| and µ_G(H) 6= 0}

(Isaacs, Hawkes, Özaydin) G perfect ⇒ |G : H| divides µ_G(H)

|P_G^(p)(−1)|p < |G : H|²_p ∀H ∈ M ⇒

|P_G^(p)(−1)|p<

µ_G(H)|G : H|

_p ∀H ∈ M ⇒

|P (−1)| = |P^(p)(−1)| 6= 0 ⇒ P (−1) 6= 0.

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General strategy

The p-part of P

_G^(p)

(−1)

P_G^(p)(s) = X

H≥B

µG(H)

|G : H|^s−1 where B is a Borel subgroup.

Theorem

Let G be an untwisted simple group of Lie type of characteristic p over Fq. Then

|P_G^(p)(−1)|p= (p, 2)q^l

with l the number of nodes of the Dynkin diagram associated to G . A similar result holds for twisted groups: in that case l is the number of the ρ-orbits on the nodes, where ρ is a suitable symmetry of the Dynkin diagram.

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General strategy

Two types of subgroups

Strategy

Prove that if µ_G(H) 6= 0 and p divides |G : H|, then

|P_G^(p)(−1)|p< |G : H|²_p.

If H is such a subgroup, then |H|p < |G |p and H is intersection of maximal subgroups of G .

We have two types of such subgroups H:

(A) H is contained in a non parabolic maximal subgroup M of G ; (B) H is not of type(A), H is an intersection of parabolic maximal

subgroups of G and |H| < |G | .

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General strategy

Subgroups of type (A)

(A) H is contained in a non parabolic maximal subgroup M of G . From what it is known about the maximal subgroups of classical groups, it follows

Proposition

|G : H|²_p> |P_G^(p)(−1)|p, with few exceptions (e.g. PSL₂(p²)).

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General strategy

Subgroups of type (B)

(B) H is an intersection of parabolic maximal subgroups of G and

|H|_p< |G |p. Moreover H is not of type (A).

To the classical group G it is associated a certain form κ defined on a vector space V of dimension n over a field Fq of characteristic p.

So we can speak about totally singular subspace of V (w.r.t.

the form κ).

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General strategy

Totally singular subspaces of V and parabolic maximal subgroups

We have the following 1-1 correspondence:

Stab_G :







non-trivial totally singular subspaces of V







→

parabolic maximal subgroups of G

Given a subgroup H of type(B):

Stab_G :L_H =







W non-trivial totally singular s.t. Stab_G(W ) ≥ H







→







parabolic maximal subgroups of G

containing H







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General strategy

The property P

Definition

We say that L_H fulfills the property P if there exists W ∈ L_H such that for each U ∈ L_H, we have W ≤ U or W ≥ U.

Theorem

Let H be a subgroup of type(B).

If L_H fulfills the property P, then µ_G(H) = 0.

If L_H does not fulfill the property P, then

|G : H|²_p> |P_G^(p)(−1)|_p.

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General strategy

Example: L

_H

fulfills the property P

Let G = PSL₄(q), q odd, let V = he₁, ..., e₄i.

W₁= he1i, W₂= he2i.

H = M1∩ M₂, M_i = StabG(Wi) for i = 1, 2.

Then

H is of type (B).

|G : H|_p= q and |P_G^(p)(−1)|p= q³, so|G : H|²_p ≯ |P_G^(p)(−1)|p

L_H = {W₁, W₂, W₁+ W₂};

L_H fulfills the property P.

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General strategy

Example: L

_H

fulfills the property P

Let G = PSL₄(q), q odd, let V = he1, ..., e4i.

H = M₁∩ M₂, M_i = Stab_G(W_i) for i = 1, 2.

Let M₃ = StabG(W1+ W2). The lattice generated by the maximal subgroups over H is

G

|

M₁ M₃ M₂

M₁∩ M₃ M₃∩ M₂

H

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General strategy

Example: L

_H

fulfills the property P

Let G = PSL₄(q), q odd, let V = he1, ..., e4i.

H = M₁∩ M₂, M_i = StabG(Wi) for i = 1, 2.

Let M₃ = StabG(W1+ W2). The corresponding values of the Möbius function are:

1

|

−1 − 1 − 1

1 1

0

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General strategy

L

_H

fulfills the property P

In general, the idea is: find a "redundant" element W in L_H, i.e for each M ⊆ L_H s.t. W ∈ M,

\

U∈M

Stab_G(U) = H ⇒ \

U∈M−{W }

Stab_G(U) = H.

In the previous case, the subspace W₁+ W2 is redundant.

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General strategy

Summarizing

(A). H is contained in a maximal subgroup M of G and

|M|_p< |G |p

(B). H is not of type (A)and H is an intersection of parabolic maximal subgroups of G and |H|p< |G |p.

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General strategy

Summarizing

|M|_p< |G |p ⇒ |G : H|²_p> |P_G^(p)(−1)|p

(B). H is not of type (A)and H is an intersection of parabolic maximal subgroups of G and |H|_p< |G |p.

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General strategy

Summarizing

|M|_p< |G |p ⇒ |G : H|²_p> |P_G^(p)(−1)|p

(B). H is not of type (A)and H is an intersection of parabolic maximal subgroups of G and |H|_p< |G |p. ⇒ if µ_G(H) 6= 0, then |G : H|²_p> |P_G^(p)(−1)|p.

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General strategy

Summarizing

|M|_p< |G |p ⇒ |G : H|²_p> |P_G^(p)(−1)|p

(B). H is not of type (A)and H is an intersection of parabolic maximal subgroups of G and |H|_p< |G |p. ⇒ if µ_G(H) 6= 0, then |G : H|²_p> |P_G^(p)(−1)|p.

Thus we conclude thatP_G(−1) 6= 0.

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General strategy

Corollary

If X = Gⁿ, with G a classical simple group, then P_X(−1) 6= 0.

Proof

P_X(s) = Y

1≤i ≤n−1

P_G(s) −i |Aut(G )|

|G |^s

⇓ P_X(−1) = Y

1≤i ≤n−1

(PG(−1) − i |Aut(G )||G |)

⇓ ( since |P_G^(p)(−1)|p< |G |²_p)

|P_X^(p)(−1)|_p = Y

1≤i ≤n−1

|P_G^(p)(−1)|_p6= 0