GGROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I IVÁN SADOFSCHI COSTAwith an appendix by KEVIN IVÁN PITERMAN
Abstract.
In this series of two articles, we prove that every action of a finite group G on afinite and contractible -complex has a fixed point. The proof goes by constructing a nontrivialrepresentation of the fundamental group of each of the acyclic -dimensional G -complexesconstructed by Oliver and Segev. In the first part we develop the necessary theory and coverthe cases where G = PSL (2 n ) , G = PSL ( q ) with q ≡ or G = Sz(2 n ) . The cases G = PSL ( q ) with q ≡ are addressed in the second part. Contents
1. Introduction 12. Preliminaries on Lie groups 33. A moduli of representations of
Γ = π ( X , x ) : G
44. Some representation theory 105. The graph X OS ( G ) M k W at W -complexes 22Appendix A. Representation theory of PSL ( q ) and Sz( q ) Introduction
A well known result of Jean-Pierre Serre states that an action of a finite group on a tree hasa fixed point [Ser80]. A natural attempt to generalize Serre’s result would be to replace “tree”by “contractible n -complex”. An example by Edwin E. Floyd and Roger W. Richardson [FR59]implies this generalization does not hold for n ≥ . However, Carles Casacuberta and WarrenDicks conjectured that it holds for n = 2 [CD92]. In the compact case and in the form of aquestion, this was also posed by Michael Aschbacher and Yoav Segev [AS93, Question 3]. In thisseries of two articles, we give a positive answer to the question of Aschbacher–Segev, settlingthe compact case of the Casacuberta–Dicks conjecture. Mathematics Subject Classification.
Key words and phrases.
Group actions, contractible -complexes, moduli of group representations, mappingdegree, finite simple groups.Researcher of CONICET. The author was partially supported by grants PICT-2017-2806, PIP11220170100357CO and UBACyT 20020160100081BA. a r X i v : . [ m a t h . A T ] F e b I. SADOFSCHI COSTA
Theorem A.
Every action of a finite group G on a -dimensional finite and contractible complexhas a fixed point. Moreover, if G is a finite group and X is a -dimensional, fixed point free,finite and acyclic G -complex, then the fundamental group of X admits a nontrivial unitaryrepresentation. In [CD92] the conjecture is proved for solvable groups. The question of which groups actwithout fixed points on a finite acyclic -complex was studied independently by Segev [Seg93],who proved this is not possible for the solvable groups and the alternating groups A n for n ≥ .Using the classification of the finite simple groups, Aschbacher and Segev proved that for manygroups any action on a finite -dimensional acyclic complex has a fixed point [AS93]. Then, BobOliver and Yoav Segev [OS02] gave the complete classification of the groups that act withoutfixed points on an acyclic -complex (see also the exposition by Alejandro Adem at the SéminaireBourbaki [Ade03]). Theorem 1.1 (Oliver–Segev) . For any finite group G , there is an essential fixed point free -dimensional (finite) acyclic G -complex if and only if G is isomorphic to one of the simplegroups PSL (2 k ) for k ≥ , PSL ( q ) for q ≡ ± and q ≥ , or Sz(2 k ) for odd k ≥ .Furthermore, the isotropy subgroups of any such G -complex are all solvable. In [SC20], the author proved the G = A (cid:39) PSL (2 ) (cid:39) PSL (5) case of Theorem A andproposed a path to prove Theorem A, which consists on representing (in a nontrivial way)the fundamental group of each of the acyclic -complexes constructed by Oliver and Segev.Concretely, by the results in [SC20], Theorem A follows from Theorems B and C below. Theorem B.
Let G be one of the groups PSL (2 n ) for n ≥ , PSL (3 n ) for n ≥ odd, PSL ( q ) with q ≡
11 (mod 24) or q ≡
19 (mod 24) , or
Sz( q ) for q = 2 n with n ≥ odd. Then thefundamental group of every -dimensional, fixed point free, finite and acyclic G -complex admitsa nontrivial representation in a unitary group U ( m ) . Theorem C ([PSC21]) . Let G be one of the groups PSL ( q ) with q > and q ≡ or q ≡
13 (mod 24) . Then the fundamental group of every -dimensional, fixed point free, finiteand acyclic G -complex admits a nontrivial representation in a unitary group U ( m ) . The proof of Theorem C appears in the second part of this work [PSC21], which is joint withKevin Piterman.To prove Theorems B and C, we use the method of [SC20] but with a more generic approach.If X is a G -graph we consider the group extension Γ = π ( X , x ) : G . If X is obtainedfrom X by attaching orbits of -cells, a result of Kenneth S. Brown [Bro84] gives an extension Γ / (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) (cid:39) π ( X ) : G , where the w i ∈ ker( φ : Γ → G ) (cid:39) π ( X ) are words correspondingto the orbits of -cells of X . Then obtaining a nontrivial representation of π ( X ) reduces toobtaining a representation of Γ which factors through the quotient Γ → Γ / (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) anddoes not factor through φ .We develop general machinery to obtain a moduli of representations M of Γ from a singlerepresentation ρ : G → G , where G is a Lie group. Each word w ∈ Γ induces a map W : M → G and then the proof reduces to finding a suitable point τ ∈ M . With some hypotheses on ρ : G → G , there is a single point ∈ M which gives a representation that factors through φ . Then, by considering W = ( W , . . . , W k ) : M → G k +1 , the proof reduces to finding a point τ (cid:54) = ∈ M such that W ( τ ) = . When we apply these results to the groups in Theorem 1.1 itturns out that M and G k +1 are orientable manifolds of the same dimension. To complete theproof we show that is a regular point of W and that W has degree . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 3 The groups in Theorem B share a key property: they admit a nontrivial representationwhich restricts to an irreducible representation of the Borel subgroup. However, the groups inTheorem C lack this property. In [PSC21] some modifications to the approach of the first partare introduced in order to extend the proof to these groups.
Acknowledgements.
I am grateful to Kevin Piterman for his help in understanding thestructure and representations of the finite simple groups
PSL ( q ) and Sz( q ) and for many fruit-ful discussions. I am also grateful to Ignacio Darago for answering all of my questions onrepresentation theory and Lie theory. I would like to thank Jonathan Barmak and GabrielMinian for useful comments on an earlier version of this work.2. Preliminaries on Lie groups
Recall that a
Lie group G is a smooth manifold with a group structure such that the multi-plication µ : G × G → G , ( x, y ) (cid:55)→ xy and inversion i : G → G , x (cid:55)→ x − are differentiable. Thegroup U ( m ) of m × m unitary matrices is a compact and connected m -dimensional Lie group.If G is a Lie group, the Lie algebra of G is the tangent space T e G at the identity element e ∈ G .The adjoint representation Ad : G → GL( T e G ) is defined by g (cid:55)→ d e Ψ g where Ψ g : G → G is themap given by h (cid:55)→ ghg − . Every Lie group is parallelizable and hence orientable. Lemma 2.1.
Let G be a Lie group with multiplication µ : G × G → G . Then the differential d ( p,q ) µ : T p G × T q G → T pq G is given by ( x, y ) (cid:55)→ d p R q ( x ) + d q L p ( y ) .Proof. The differential d ( e,e ) µ : T e G × T e G → T e G is given by ( x, y ) (cid:55)→ x + y (this is [Lee13,Chapter 7, Problem 7-2]). The general case follows by writing µ = L p R q ◦ µ ◦ ( L p − × R q − ) . (cid:3) Proposition 2.2.
Let M be a manifold, G be a Lie group and f, g : M → G be differentiablemaps.(i) We have the product rule d p ( f · g ) = d f ( p ) R g ( p ) ◦ d p f + d g ( p ) L f ( p ) ◦ d p g .(ii) If f ( p ) = g ( p ) = e , we have d p ( f · g ) = d p f + d p g .(iii) If g ( p ) = e , we have d p ( f · g · f − ) = d f ( p ) − L f ( p ) ◦ d e R f ( p ) − ◦ d p g .(iv) If f ( p ) = e , we have d p f − = − d p f .(v) If f ( p ) = g ( p ) = e , we have d p [ f, g ] = 0 .Proof. These properties follow easily from Lemma 2.1. (cid:3)
Corollary 2.3.
The adjoint representation is given by
Ad( g ) = dL g ◦ dR g − . We denote the centralizer of H in G by C G ( H ) and the center of G by Z ( G ) . Proposition 2.4 ([Bou06, Chapter III, §9, no. 3, Proposition 8]) . Let H be a finite subgroupof a Lie group G . Then the Lie algebra of the centralizer C G ( H ) is obtained by taking the fixedpoints by H of the adjoint representation of G . That is, we have T e C G ( H ) = ( T e G ) H . Theorem 2.5.
Let H ≤ U ( m ) be a subgroup. Then C U ( m ) ( H ) is connected.Proof. A proof using a simultaneous diagonalization argument is given in [Sta05, Proof of The-orem 3.2]. See also [Gra]. (cid:3)
Proposition 2.6 ([Lee13, Corollary 21.6]) . Every continuous action by a compact Lie group ona manifold is proper.
I. SADOFSCHI COSTA
Theorem 2.7 (Quotient Manifold Theorem) . Suppose G is a Lie group acting smoothly, freely,and properly on a smooth manifold M . Then the orbit space M/G is a topological manifold ofdimension equal to dim M − dim G , and has a unique smooth structure with the property thatthe quotient map π : M → M/G is a smooth submersion.Moreover, if M is orientable and G is connected, then M/G is orientable.Proof.
The first part is [Lee13, Theorem 21.10]. For the second part we fix an orientation on M and G . Since G is connected, the translations L g , R g : G → G and g : M → M are homotopic tothe identity map and thus preserve the orientation. A tedious but straightforward computationwith the charts constructed in the proof of [Lee13, Theorem 21.10] allows to extract an orientedatlas, showing that M/G is orientable. (cid:3) A moduli of representations of
Γ = π ( X , x ) : G By G -complex we mean a G -CW complex. That is, a CW complex with a continuous G -actionthat is admissible (i.e. the action permutes the open cells of X , and maps a cell to itself onlyvia the identity). See [OS02, Appendix A] for more details. A graph is a -dimensional CWcomplex. By G -graph we always mean a -dimensional G -complex.If X is a connected G -graph, there is a group extension → π ( X , v ) i −→ Γ φ −→ G → which is most easily defined by lifting the action of G to the universal cover (cid:101) X of X . Inthis section we construct a moduli M of representations of the group extension Γ and studyits properties (note that we are using the word moduli in a rather informal way, meaning ageometric object whose points correspond to certain representations of Γ ). The starting pointto construct M is a result in Bass–Serre theory due to K.S. Brown [Bro84]. Theorem 3.1 (Brown) . Let X be obtained from a G -graph X by attaching m orbits of -cellsalong (the orbits of ) the closed edge paths ω , . . . , ω m based at a vertex v . Then there is a groupextension → π ( X, v ) i −→ Γ / (cid:104)(cid:104) i ( ω ) , . . . , i ( ω m ) (cid:105)(cid:105) φ −→ G → , where the maps i and φ are given by factoring through the quotient. In order to describe Brown’s construction of Γ and the maps i and φ we need some choices.By admissibility of the action, the group G acts on the set of oriented edges. If e is an orientededge, the same -cell with the opposite orientation is denoted by e − . Each oriented edge e hasa source s ( e ) and a target t ( e ) . For each -cell of X we choose a preferred orientation in sucha way that these orientations are preserved by G . This determines a set P of oriented edges.We choose a tree of representatives for X /G . That is, a tree T ⊂ X such that the vertex set V of T is a set of representatives of X (0)1 /G . Such tree always exists and the -cells of T areinequivalent modulo G . We give an orientation to the -cells of T so that they are elements of P . We also choose a set of representatives E of P/G in such a way that s ( e ) ∈ V for every e ∈ E and such that each oriented edge of T is in E . If e is an oriented edge, the unique element of V that is equivalent to t ( e ) modulo G will be denoted by w ( e ) . For every e ∈ E we fix an element g e ∈ G such that t ( e ) = g e · w ( e ) . If e ∈ T , we specifically choose g e = 1 . Then Γ = F ( x e : e ∈ E ) ∗ ∗ v ∈ V G v (cid:104)(cid:104) R (cid:105)(cid:105) , ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 5 where F ( x e : e ∈ E ) is the free group with basis { x e : e ∈ E } and (cid:104)(cid:104) R (cid:105)(cid:105) denotes the normalsubgroup generated by the set R of relations of the following two types:(i) x e = 1 if e ∈ T , and(ii) x − e ι s ( e ) ( g ) x e = ι w ( e ) ( g − e gg e ) for every e ∈ E and g ∈ G e ,where ι v : G v (cid:44) → F ( x e : e ∈ E ) ∗ ∗ v ∈ V G v denotes the canonical inclusion.Let φ : Γ → G be the map induced by the coproduct of the inclusions G v → G and the map F ( x e : e ∈ E ) → G defined by x e (cid:55)→ g e . Let N = ker( φ ) = i ( π ( X , v )) . Let i v : G v (cid:44) → Γ bethe canonical inclusion. We will not give the description of i here, instead we refer to [Bro84] orto [SC20, Section 4].In the following proposition we use a morphism ρ : G → G to construct a moduli of repre-sentations of Γ in the Lie group G . This extends the construction in [SC20, Theorem 5.4]. Theorem 3.2.
Let X be a G -graph with the necessary choices to apply Theorem 3.1. Take avertex v ∈ V as the root of T and assume the orientation P is taken so that every edge of T isoriented away from v . Let Γ be the group given by Brown’s result and consider a representation ρ : G → G of G in a Lie group G . Let M = (cid:89) e ∈ E C G ( ρ ( G e )) . Suppose τ = ( τ e ) e ∈ E ∈ M . For v ∈ V , we define τ v = τ e s τ e s − · · · τ e τ e where ( e , e , . . . , e s ) is the unique path from v to v byedges in T (with this definition τ v = ). Then we have a representation ρ τ : Γ → G given by ρ τ ( i v ( g )) = τ − v ρ ( g ) τ v for v ∈ V,ρ τ ( x e ) = τ − s ( e ) τ − e ρ ( g e ) τ w ( e ) for e ∈ E. We thus have a moduli of representations ρ : M → hom(Γ , G ) τ (cid:55)→ ρ τ Moreover, each word w ∈ Γ induces a differentiable map W : M → G given by τ (cid:55)→ ρ τ ( w ) .Proof. If e ∈ T then τ w ( e ) = τ t ( e ) = τ e τ s ( e ) and g e = 1 . Therefore ρ τ ( x e ) = and relations oftype (i) are satisfied. Now if e ∈ E , g ∈ G e we have ρ τ ( x e ) − ρ τ ( i s ( e ) ( g )) ρ τ ( x e ) = τ − w ( e ) ρ ( g e ) − τ e τ s ( e ) · τ − s ( e ) ρ ( g ) τ s ( e ) · τ − s ( e ) τ − e ρ ( g e ) τ w ( e ) = τ − w ( e ) ρ ( g e ) − τ e ρ ( g ) τ − e ρ ( g e ) τ w ( e ) = τ − w ( e ) ρ ( g e ) − ρ ( g ) ρ ( g e ) τ w ( e ) = ρ τ ( i w ( e ) ( g − e gg e )) and thus the type (ii) relations x − e i s ( e ) ( g ) x e = i w ( e ) ( g − e gg e ) also hold. (cid:3) Different points of M may correspond to equal representations of Γ . The quotient M intro-duced in the following result allows us to deal with this issue. Theorem 3.3.
Let H = { ( α v ) v ∈ V : α v = } ⊆ (cid:89) v ∈ V C G ( ρ ( G v )) . Assume H is compact.(i) There is a free right action M (cid:120) H given by ( τ · α ) e = ρ ( g e ) α − w ( e ) ρ ( g e ) − · τ e · α s ( e ) (ii) Moreover ρ τ = ρ τ (cid:48) if and only if τ, τ (cid:48) lie in the same orbit of the action of H . I. SADOFSCHI COSTA (iii) The quotient M = M / H is a smooth manifold, the map p : M → M is a smoothsubmersion and dim( M ) = dim( M ) − dim( H ) .(iv) If H is connected then M is orientable.(v) We have an induced map ρ : M → hom(Γ , G ) . Each word w ∈ Γ induces a differentiablemap W : M → G such that W = W ◦ p .Proof. (i) Since G s ( e ) ⊇ G e ⊆ G t ( e ) , the good definition follows from ρ ( g e ) α − w ( e ) ρ ( g e ) − ∈ C G ( ρ ( G t ( e ) )) which holds since t ( e ) = g e · w ( e ) . If ( τ · α ) e = τ e for all e ∈ T , by induction(traversing the tree T starting from the root v ) it follows that α v = 1 for all v ∈ V . Then theaction is free.(ii) Let τ ∈ M , α ∈ H . If e ∈ T then ( τ α ) e = α − t ( e ) τ e α s ( e ) . If v ∈ V , ( τ α ) v = α − v τ v . Then ρ τα ( i v ( g )) = ( τ α ) − v ρ ( g )( τ α ) v = τ − v α v ρ ( g ) α − v τ v = τ − v ρ ( g ) τ v = ρ τ ( i v ( g )) . Moreover, for e ∈ E we have ρ τα ( x e ) = ( τ α ) − s ( e ) ( τ α ) − e ρ ( g e )( τ α ) w ( e ) = ( α − s ( e ) τ s ( e ) ) − ( ρ ( g e ) α − w ( e ) ρ ( g e ) − τ e α s ( e ) ) − ρ ( g e )( α − w ( e ) τ w ( e ) )= τ − s ( e ) τ − e ρ ( g e ) τ w ( e ) = ρ τ ( x e ) . Then ρ τ = ρ τα . For the other implication, if τ, τ (cid:48) ∈ M satisfy ρ τ = ρ τ (cid:48) , by defining α v = τ v ( τ (cid:48) v ) − we obtain a point α = ( α v ) v ∈ V ∈ H and τ α = τ (cid:48) .(iii) By Proposition 2.6 the action is proper. Then by Theorem 2.7, the quotient M = M / H has a (unique) smooth manifold structure such that p : M → M is a submersion and dim( M ) =dim( M ) − dim( H ) .(iv) This follows from the second part of Theorem 2.7.(v) This follows by passing to the quotient. (cid:3) Corollary 3.4. If G = U ( m ) then M and M are connected and orientable.Proof. This is immediate from Theorem 2.5. (cid:3)
A representation ρ : Γ → G is said to be universal if N ⊆ ker( ρ ) (or equivalently, if ρ factorsthrough φ ). Under suitable hypotheses, = p ( ) is the only point in M which corresponds toa universal representation: Proposition 3.5.
Suppose that G is finite and that each element of G fixes a vertex in X . Let G ⊆ GL m ( C ) and assume the restriction ρ | G v : G v → G is an irreducible representation of G v . Then { } = { τ ∈ M : ρ τ is universal } .Proof. First note that ρ = ρ = ρ ◦ φ is universal. Now consider τ ∈ M such that ρ τ isuniversal. By passing to the quotient we have a representation (cid:101) ρ τ : G → G such that ρ τ = (cid:101) ρ τ ◦ φ .Now note that, since each element of G fixes a vertex of X , from the definition of ρ τ it followsthat the representations ρ and (cid:101) ρ τ have the same character and are therefore isomorphic. Hence,we can take α ∈ GL m ( C ) such that for all g ∈ G we have α (cid:101) ρ τ ( g ) α − = ρ ( g ) . Now since for every ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 7 g ∈ G v we have (cid:101) ρ τ ( g ) = ρ ( g ) , and since ρ | G v is irreducible, by Schur’s lemma (Theorem A.1.2)it follows that α is a scalar matrix and therefore (cid:101) ρ τ = ρ . Then ρ τ = ρ and therefore by part(ii) of Theorem 3.3, p ( τ ) = p ( ) in M . (cid:3) Remark . If ρ | G v : G v → G is not irreducible, we could still consider the quotient M of M by the action of C G ( ρ ( G v )) . In this case, the points in M correspond to characters (notrepresentations) of Γ and the image of the induced map W is only defined up to conjugation by C G ( ρ ( G v )) . Note that the quotient of G by the conjugation action of C G ( ρ ( G v )) is not, ingeneral, a manifold.The following result relates a closed edge path ω ∈ X to the differential at of the map M → G induced by the word i ( ω ) ∈ Γ . Theorem 3.7.
Let X be a G -graph (with the necessary choices to form M ). Consider a closededge path ω = ( a e ε , . . . , a n e ε n n ) in X , based at v , with e i ∈ E , a i ∈ G and ε i ∈ { , − } .Let w = i ( ω ) ∈ N = ker( φ ) . Let W : M → G be the induced differentiable map. Let =( ) e ∈ E ∈ M and consider the inclusion j e : C G ( ρ ( G e )) (cid:44) → G . Then, with the identification T M (cid:39) (cid:77) e ∈ E T C G ( ρ ( G e )) we have d W = − n (cid:88) i =1 ε i · d ρ ( a i ) − L ρ ( a i ) ◦ d R ρ ( a i ) − ◦ d j e i . Proof.
By the definition of i : π ( X ) → Γ (see [Bro84] or [SC20, Section 4]) we can write w = i v ( h ) · x ε e · i v ( h ) · x ε e · . . . · i v n − ( h n ) · x ε n e n · i v ( g g · · · g n ) − so that for each i we have g i = h i g ε i e i and a i = (cid:40) g · · · g i − h i if ε i = 1 g · · · g i − h i g − e i if ε i = − . Then W ( τ ) = (cid:32) n (cid:89) i =1 ( τ − v i − ρ ( h i ) τ v i − )( τ − s ( e i ) τ − e i ρ ( g e i ) τ w ( e i ) ) ε i (cid:33) τ − v ρ ( g g · · · g n ) − τ v = (cid:32) n (cid:89) i =1 ρ ( h i )( τ − e i ρ ( g e i )) ε i (cid:33) ρ ( g g · · · g n ) − . In the last equality we used that τ v = 1 and that s ( e i ) and w ( e i ) are (in some order whichdepends on ε i ) v i − and v i . We have P i ( ) = ρ ( a i ) where P i is the prefix of W ending justbefore the occurrence of τ − ε i e i . Note that, since W ( ) = , if S i is the suffix of W starting justafter the occurrence of τ − ε i e i , we have S i ( ) = ρ ( a i ) − . To conclude, we apply the product ruleProposition 2.2. (cid:3) Lemma 3.8 (cf. [SC20, Lemma 6.7]) . Let Γ be a group, G be a Lie group, M be a differentiablemanifold, and ρ : M → hom(Γ , G ) be a function such that for each w ∈ Γ the mapping W : M → G defined by W ( z ) = ρ ( z )( w ) is differentiable. Let N (cid:47) Γ be a normal subgroup and suppose that p ∈ M is such that ρ ( p )( w ) = for each w ∈ N . Then for any elements w , . . . , w k ∈ N and x , . . . , x k ∈ (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ [ N, N ] we have rk d p W ≥ rk d p X , where W = ( W , . . . , W k ) and X = ( X , . . . , X k ) are the induced maps M → G k +1 . I. SADOFSCHI COSTA
Proof.
For each j = 0 , . . . , k we consider numbers a j , (cid:96) j ∈ N , elements u j, , . . . , u j,a j ,v j, , . . . , v j,a j ∈ N , elements p j, , . . . p j,(cid:96) j ∈ Γ , indices α j, , . . . , α j,(cid:96) j ∈ { , . . . , k } and signs ε j, , . . . , ε j,(cid:96) j ∈ { , − } such that x j = (cid:96) j (cid:89) s =1 p j,s w ε j,s α j,s p − j,s a j (cid:89) i =1 [ u j,i , v j,i ] . Then the induced maps M → G satisfy X j = (cid:96) j (cid:89) s =1 P j,s W ε j,s α j,s P − j,s a j (cid:89) i =1 [ U j,i , V j,i ] and using Proposition 2.2 we obtain d p X j = (cid:96) j (cid:88) s =1 ε j,s · d P j,s ( p ) − L P j,s ( p ) ◦ d e R P j,s ( p ) − ◦ d p W α j,s . To conclude, note that we have shown there is an R -linear endomorphism A of T ( G k +1 ) suchthat d p X = A ◦ d p W . (cid:3) We now prove some results that will be used later to obtain homotopies between maps
M → G .We obtain these homotopies from homotopies M × I → G k +1 that are H -equivariant. Lemma 3.9.
Let w, w (cid:48) ∈ Γ and let g ∈ G v for some v ∈ V . Then the maps M → G inducedby the words ww (cid:48) and wi v ( g ) w (cid:48) are homotopic. Moreover, if C G ( C G ( ρ ( G v ))) is connected thenthe same holds for the induced maps M → G .Proof. Let
W, W (cid:48) : M → G be the maps induced by w and w (cid:48) respectively. Let γ : I → G be apath with γ (0) = , γ (1) = ρ ( g ) . The following map H : M × I → G ( τ, t ) (cid:55)→ W ( τ ) τ − v γ ( t ) τ v W (cid:48) ( τ ) is a homotopy between the maps M → G induced by ww (cid:48) and wi v ( g ) w (cid:48) . Moreover, since ρ ( g ) ∈ C G ( C G ( ρ ( G v ))) , we can take γ ( I ) ⊆ C G ( C G ( ρ ( G v ))) if the latter is connected and inthis case the following computation H ( τ α, t ) = W ( τ α )( τ α ) − v γ ( t )( τ α ) v W (cid:48) ( τ α )= W ( τ )( τ α ) − v γ ( t )( τ α ) v W (cid:48) ( τ )= W ( τ )( α − v τ v ) − γ ( t )( α − v τ v ) W (cid:48) ( τ )= W ( τ ) τ − v α v γ ( t ) α − v τ v W (cid:48) ( τ )= W ( τ ) τ − v γ ( t ) τ v W (cid:48) ( τ )= H ( τ, t ) shows the homotopy H is H -equivariant, giving a homotopy between the induced maps M → G . (cid:3) In the following two propositions we use the notation (cid:89) i = (cid:96) b i = b (cid:96) b (cid:96) − b (cid:96) − · · · b b . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 9 Proposition 3.10.
Let η ∈ E − T and let ( e , . . . , e k ) and ( e (cid:48) , . . . , e (cid:48) (cid:96) ) be the unique paths in T from v to s ( η ) and w ( η ) respectively (see Figure 1). Suppose that γ , . . . , γ k , β , . . . , β (cid:96) : I → G are paths such that: • For i = 1 , . . . , k and for every t ∈ I , γ i ( t ) commutes with C G ( ρ ( G t ( e i ) )) . • For i = 1 , . . . , (cid:96) and for every t ∈ I , β i ( t ) commutes with C G ( ρ ( G t ( e (cid:48) i ) )) .Then there is an H -equivariant homotopy F : M × I → G defined by F ( τ, t ) = γ ( t ) (cid:32) k (cid:89) i =1 τ − e i γ i ( t ) (cid:33) τ − η ρ ( g η ) (cid:32) (cid:89) i = (cid:96) β i ( t ) τ e (cid:48) i (cid:33) β ( t ) Moreover, if γ i (0) = for i = 0 , . . . , k and β i (0) = for i = 0 , . . . , (cid:96) then F = X η where X η isthe map induced by x η . e e (cid:48) e · · · e k e (cid:48) · · · e (cid:48) (cid:96) − e (cid:48) (cid:96) ηv t ( η ) t ( e (cid:48) (cid:96) ) = w ( η ) Figure 1.
The paths in Propositions 3.10 and 3.11. Note that t ( e (cid:48) (cid:96) ) = w ( η ) = g − η · t ( η ) . Also note that we may have k = 0 or (cid:96) = 0 . Proof.
The following computation shows that F is H -equivariant. F ( τ α, t ) = γ ( t ) (cid:32) k (cid:89) i =1 ( τ α ) − e i γ i ( t ) (cid:33) ( τ α ) − η ρ ( g η ) (cid:32) (cid:89) i = (cid:96) β i ( t )( τ α ) e (cid:48) i (cid:33) β ( t )= γ ( t ) (cid:32) k (cid:89) i =1 α − s ( e i ) τ − e i α t ( e i ) γ i ( t ) (cid:33) α − s ( η ) τ − η ρ ( g η ) α w ( η ) ρ ( g η ) − · ρ ( g η ) (cid:32) (cid:89) i = (cid:96) β i ( t ) α − t ( e (cid:48) i ) τ e (cid:48) i α s ( e (cid:48) i ) (cid:33) β ( t )= γ ( t ) (cid:32) k (cid:89) i =1 τ − e i γ i ( t ) (cid:33) τ − η ρ ( g η ) (cid:32) (cid:89) i = (cid:96) β i ( t ) τ e (cid:48) i (cid:33) β ( t )= F ( τ, t ) . For the second part, note that X η ( τ ) = ρ τ ( x η )= τ − s ( η ) τ − η ρ ( g η ) τ w ( η ) = (cid:32) k (cid:89) i =1 τ − e i (cid:33) τ − η ρ ( g η ) (cid:89) i = (cid:96) τ e (cid:48) i . (cid:3) Proposition 3.11.
Suppose that C G ( C G ( ρ ( G v ))) is connected for each v ∈ V . Let η ∈ E − T and let ( e , . . . , e k ) and ( e (cid:48) , . . . , e (cid:48) (cid:96) ) be the unique paths in T from v to s ( η ) and w ( η ) respectively.Let A e ∈ G be elements defined for every e ∈ E . Suppose that C , . . . , C k , B , . . . , B (cid:96) ∈ G satisfy: • A − e i C i A e i +1 commutes with C G ( ρ ( G t ( e i ) )) for i = 1 , . . . , k − . • A − e k C k A η commutes with C G ( ρ ( G t ( e k ) )) . • A − e (cid:48) i +1 B i A e (cid:48) i commutes with C G ( ρ ( G t ( e (cid:48) i ) )) for i = 1 , . . . , (cid:96) − . • ρ ( g η ) − A − η ρ ( g η ) B (cid:96) A e (cid:48) (cid:96) commutes with C G ( ρ ( G w ( η ) )) .Then there is an H -equivariant homotopy between the map X η : M → G induced by x η andthe map Z : M → G defined by Z ( τ ) = (cid:32) k (cid:89) i =1 A e i τ − e i A − e i C i (cid:33) A η τ − η A − η ρ ( g η ) (cid:32) (cid:89) i = (cid:96) B i A e (cid:48) i τ e (cid:48) i A − e (cid:48) i (cid:33) . Proof.
Since the centralizers C G ( C G ( ρ ( G v ))) are connected, we can take paths: • γ −→ A e , in G . • γ i −→ A − e i C i A e i +1 such that γ i ( I ) commutes with C G ( ρ ( G t ( e i ) )) for i = 1 , . . . , k − . • γ k −→ A − e k C k A η such that γ k ( I ) commutes with C G ( ρ ( G t ( e k ) )) . • β (cid:96) −→ ρ ( g η ) − A − η ρ ( g η ) B (cid:96) A e (cid:48) (cid:96) such that β l ( I ) commutes with C G ( ρ ( G w ( η ) )) . • β i −→ A − e (cid:48) i +1 B i A e (cid:48) i such that β i ( I ) commutes with C G ( ρ ( G t ( e (cid:48) i ) )) for i = 1 , . . . , (cid:96) − . • β −→ A − e (cid:48) , in G .The result now follows from Proposition 3.10. (cid:3) Some representation theory
In this section we prove Lemma 4.7, which is useful to apply Proposition 3.11. We also proveLemma 4.5, a consequence of Schur’s lemma which is used later to compute the dimension ofcentralizers in U ( n ) . We start by recalling the following classical results Theorem 4.1 ([EGH +
11, Theorem 4.6.2]) . Every representation ρ : G → GL n ( C ) of a finitegroup G is isomorphic to a unitary representation (cid:101) ρ : G → U ( n ) . Theorem 4.2.
Let G be a finite group. If two unitary representations of G are isomorphic thenthere is a unitary isomorphism between them.Proof. When the representations are irreducible this is [Dor71, Lemma 33.1]. For a proof in thegeneral case see [Was]. (cid:3) If A, A (cid:48) are matrices then A ⊕ A (cid:48) denotes the block diagonal matrix (cid:32) A A (cid:48) (cid:33) . If ρ, ρ (cid:48) arerepresentations of a group G then ρ ⊕ ρ (cid:48) denotes the representation such that ( ρ ⊕ ρ (cid:48) )( g ) = ρ ( g ) ⊕ ρ (cid:48) ( g ) for all g ∈ G . We denote the n × n identity matrix by I n .It is easy to verify that block scalar matrices commute with scalar block matrices: Proposition 4.3.
Let X ∈ M n ( C ) and λ ∈ M k ( C ) be two matrices. Let A = X ⊕ . . . ⊕ X ∈ M kn ( C ) and let B ∈ M kn ( C ) = M k ( M n ( C )) be the matrix defined by B i,j = λ i,j I n . Then A and B commute. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 11 Remark . Let ρ , . . . , ρ k be pairwise non-isomorphic irreducible representations of a finitegroup G and let n , . . . , n k be natural numbers. Consider the representation ρ = (cid:76) ki =1 ρ n i i ,where ρ n i i denotes the sum ρ i ⊕ . . . ⊕ ρ i of n i copies of ρ i . Let d i be the degree of ρ i and let n = (cid:80) ki =1 d i n i be the degree of ρ . Then, by Schur’s lemma (Theorem A.1.2), we have C U ( n ) ( ρ ( G )) = k (cid:89) i =1 C U ( d i n i ) ( ρ n i i ) , where the product on the right is included in U ( n ) as block diagonal matrices. Again by Schur’slemma, we have an isomorphism U ( n i ) (cid:39) −−→ C U ( d i n i ) ( ρ n i i ) which is given by A (cid:55)→ (cid:101) A , where (cid:101) A ∈ M n i d i ( C ) = M n i ( M d i ( C )) is the scalar block matrix defined by (cid:101) A s,t = A s,t I d i , which is infact unitary. Then C U ( n ) ( ρ ( G )) (cid:39) (cid:81) ki =1 U ( n i ) and in particular we have dim C U ( n ) ( ρ ( G )) = (cid:80) ki =1 n i . Lemma 4.5.
Let G be a finite group and let ρ : G → U ( n ) be a unitary representation withcharacter χ . Then dim C U ( n ) ( ρ ( G )) = (cid:104) χ, χ (cid:105) G .Proof. If ρ is isomorphic to (cid:76) ki =1 ρ n i i , where ρ , . . . , ρ k are pairwise non-isomorphic irreduciblerepresentations of G , from the orthogonality relations and Remark 4.4 we obtain (cid:104) χ, χ (cid:105) G = (cid:80) ki =1 n i = dim C U ( n ) ( ρ ( G )) . (cid:3) Recall that
Res GH χ denotes the restriction of a character χ of G to a subgroup H ≤ G . Definition 4.6.
Let G be a finite group and let H ⊆ G be a subgroup. Let Θ be a subset ofthe irreducible characters of H . If ρ is a representation of G with character χ we define d ( ρ, Θ) = (cid:42)
Res GH χ, (cid:88) θ ∈ Θ θ (1) θ (cid:43) H. Note that d ( ρ, Θ) is the sum of the dimensions of the irreducible factors of ρ | H whose charactersbelong to Θ . Lemma 4.7.
Let H , H be two subgroups of a finite group G and let ρ : G → U ( n ) be arepresentation. Suppose that n = n (cid:48) + n (cid:48)(cid:48) and A , A ∈ U ( n ) are matrices such that, for i = 1 , ,the representations (cid:101) ρ i : H i → U ( n ) given by (cid:101) ρ i ( g ) = A i ρ ( g ) A − i satisfy • There is a number k i ∈ N and irreducible representations (cid:101) ρ i,s : H i → U ( m i,s ) with s =1 , . . . , k i such that we have a block diagonal decomposition (cid:101) ρ i = (cid:101) ρ i, ⊕ . . . ⊕ (cid:101) ρ i,k i . • If (cid:101) ρ i,s and (cid:101) ρ i,s (cid:48) are isomorphic then (cid:101) ρ i,s = (cid:101) ρ i,s (cid:48) . • There exists l i ∈ N such that n (cid:48) = m i, + . . . + m i,l i and n (cid:48)(cid:48) = m i,l i +1 + . . . + m i,k i . • If s, s (cid:48) are numbers such that ≤ s ≤ l i and l i + 1 ≤ s (cid:48) ≤ k i then (cid:101) ρ i,s and (cid:101) ρ i,s (cid:48) are notisomorphic.Suppose that for each irreducible factor ρ (cid:48) of ρ we have d ( ρ (cid:48) , Θ ) = d ( ρ (cid:48) , Θ ) , where Θ i is the setgiven by the characters of the representations (cid:101) ρ i,s with ≤ s ≤ l i .Then there is a matrix C ∈ U ( n (cid:48) ) × U ( n (cid:48)(cid:48) ) such that A − CA commutes with C U ( n ) ( ρ ( G )) .Proof. By Theorem 4.1 and Theorem 4.2, we can take T ∈ U ( n ) and irreducible representations ρ j : G → U ( m j ) with j = 1 , . . . , k such that T ρT − = ρ ⊕ . . . ⊕ ρ k . Moreover, we can do this so that whenever ρ j and ρ j (cid:48) are isomorphic we have ρ j = ρ j (cid:48) . For each i = 1 , we take matrices D i, , . . . , D i,k with D i,j ∈ U ( m j ) such that D i,j ρ j D − i,j : H i → U ( m j ) is block diagonal, with each block equal to some (cid:101) ρ i,s and such that the irreducible blocks withcharacters in Θ i appear consecutively at the start of the diagonal. These blocks have a total sizeof d ( ρ j , Θ ) = d ( ρ j , Θ ) (see Figure 2). We choose the D i,j so that ρ j = ρ j (cid:48) implies D i,j = D i,j (cid:48) .Let D i = D i, ⊕ . . . ⊕ D i,k . Note that by Proposition 4.3 and Remark 4.4, D i commutes with C U ( n ) ( T ρ ( G ) T − ) . D ,j ρ j D − ,j d ( ρ j , Θ ) = d ( ρ j , Θ ) D ,j ρ j D − ,j Figure 2.
The matrices D i,j ρ j D − i,j for i = 1 , . Each shaded block is equal tosome (cid:101) ρ i,s . Later in the proof of Lemma 4.7, the condition d ( ρ j , Θ ) = d ( ρ j , Θ ) allows to take the same permutation matrix σ for i = 1 , .Now note that we can take a permutation matrix σ ∈ U ( n ) such that for i = 1 , therepresentation ( σD i T ) ρ | H i ( σD i T ) − : H i → U ( n ) is block diagonal with irreducible blocks and such that the blocks with characters in Θ i appearconsecutively at the start of the diagonal.Now for i = 1 , we take a permutation matrix σ i ∈ U ( n (cid:48) ) × U ( n (cid:48)(cid:48) ) such that A i ρ | H i A − i = ( σ i σD i T ) ρ | H i ( σ i σD i T ) − . Let C i = A i ( σ i σD i T ) − . Then C i ∈ C U ( n ) ( A i ρ ( H i ) A − i ) ⊆ U ( n (cid:48) ) × U ( n (cid:48)(cid:48) ) and we have A i = C i σ i σD i T . Finally, letting C = C σ σ − C − ∈ U ( n (cid:48) ) × U ( n (cid:48)(cid:48) ) the matrix A − CA = T − D − D T commutes with C U ( n ) ( ρ ( G )) , because D i commutes with C U ( n ) ( T ρ ( G ) T − ) for i = 1 , . (cid:3) In what follows, C n denotes a cyclic group of order n and D n denotes a dihedral group oforder n . Proposition 4.8.
Let n be an odd number and consider subgroups H = (cid:104) g (cid:105) (cid:39) C n and H (cid:39) C of a group G (cid:39) D n . Let µ k : H → C be the character given by µ k ( g j ) = ξ kj , where ξ = e π i n and let H be the trivial character of H .(i) Letting Θ = { µ , . . . , µ ( n − / } and Θ = { H } , we have d ( ρ, Θ ) = d ( ρ, Θ ) for anynontrivial irreducible representation ρ of G .(ii) Letting Θ = { µ , µ , . . . , µ ( n − / } and Θ = { H } , we have d ( ρ, Θ ) = d ( ρ, Θ ) forany irreducible representation ρ of G other than the nontrivial degree representation. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 13 Proof.
The irreducible characters of D n are given in Proposition A.3.1. For a representation ρ with character ψ we have Res GH ψ = µ and Res GH ψ = 1 H . For a representation ρ withcharacter ψ we have Res GH ψ = µ and Res GH ψ = ν , where ν : H → C denotes the nontrivialirreducible character of H . For a representation ρ with character χ i we have Res GH χ i = µ i + µ − i and Res GH χ i = 1 H + ν . (cid:3) The graph X OS ( G ) We recall here the examples obtained by Oliver and Segev for each of the groups in Theo-rem 1.1.
Proposition 5.1 ([OS02, Example 3.4]) . Set G = PSL ( q ) , where q = 2 k and k ≥ . Thenthere is a -dimensional acyclic fixed point free G -complex X , all of whose isotropy subgroupsare solvable. More precisely X can be constructed to have three orbits of vertices with isotropysubgroups isomorphic to B = F q (cid:111) C q − , D q − , and D q +1) ; three orbits of edges with isotropysubgroups isomorphic to C q − , C and C ; and one free orbit of -cells. Proposition 5.2 ([OS02, Example 3.5]) . Assume that G = PSL ( q ) , where q = p k ≥ and q ≡ ± . Then there is a -dimensional acyclic fixed point free G -complex X , all ofwhose isotropy subgroups are solvable. More precisely, X can be constructed to have four orbitsof vertices with isotropy subgroups isomorphic to B = F q (cid:111) C ( q − / , D q − , D q +1 , and A ; fourorbits of edges with isotropy subgroups isomorphic to C ( q − / , C , C and C ; and one free orbitof -cells. Proposition 5.3 ([OS02, Example 3.7]) . Set q = 2 k +1 for any k ≥ . Then there is a -dimensional acyclic fixed point free Sz( q ) -complex X , all of whose isotropy subgroups are solvable.More precisely, X can be constructed to have four orbits of vertices with isotropy subgroupsisomorphic to M ( q, θ ) , D q − , C q + √ q +1 (cid:111) C , C q −√ q +1 (cid:111) C ; four orbits of edges with isotropysubgroups isomorphic to C q − , C , C and C ; and one free orbit of -cells. The
Oliver–Segev graph X OS ( G ) is the -skeleton of any -dimensional fixed point free acyclic G -complex of the type constructed in Propositions 5.1 to 5.3. The graph X OS ( G ) is unique up to G -homotopy equivalence (see [SC20, Proposition 3.10]). For any k ≥ , we also consider the G -graph X OS + k ( G ) obtained from X OS ( G ) by attaching k free orbits of -cells. The G -homotopytype of X OS + k ( G ) does not depend on the particular way these free orbits are attached (againby [SC20, Proposition 3.10]). Lemma 5.4.
Let G be a finite group and let X be a G -graph. Let u, v, w be vertices of X and let e, e (cid:48) be edges such that e has endpoints { u, v } and e (cid:48) has endpoints { v, w } . Suppose that G e (cid:48) ⊆ G e .Consider the G -graph (cid:102) X obtained from X by removing the orbit of e (cid:48) and attaching an orbitof edges of type G/G e (cid:48) with endpoints { u, w } . Then X and (cid:102) X are G -homotopy equivalent. ee (cid:48) v uwX ee (cid:48) e (cid:48)(cid:48) v uw e e (cid:48)(cid:48) v uw (cid:101) X Proof.
We can obtain (cid:102) X from X by doing an equivariant elementary expansion of type G/G e (cid:48) followed by an equivariant elementary collapse of type G/G e (cid:48) (see [Ill74] for the precise defini-tion of equivariant expansion and equivariant collapse). These modifications are G -homotopyequivalences. (cid:3) For each of the groups G in Theorem 1.1, we describe a feasible way to connect the orbits inthe graph X OS ( G ) . Recall that, by [OS02, Theorem 4.1], the graph X OS ( G ) H is a tree for eachsolvable subgroup H ≤ G . When considering the groups Sz( q ) we use the notation r = √ q . Proposition 5.5.
For each of the groups G in Theorem 1.1, we can construct X OS ( G ) so thatthe orbits are connected as in Figure 3.Proof. In all cases, the orbit types are given by Propositions 5.1 to 5.3. For the groups G =PSL ( q ) , a possible way to connect the orbits is described in [OS02, Section 3]. For each of thesegroups, the structure in Figure 3 coincides with this one up to an application of Lemma 5.4.For G = Sz( q ) we give more detail here. First note that, since q − (cid:45) q ± r + 1) , the orbitof type C q − has to connect B to D q − . Now the two orbits of type C must connect B , C q + r +1 (cid:111) C and C q − r +1 (cid:111) C (in some way). The orbit C must connect D q − to one ofthe other three orbits of vertices. Note that, in any case, we can repeatedly use Lemma 5.4 toobtain the desired structure. (cid:3) G q G v G v G v G v PSL ( q ) 2 n B = F q (cid:111) C q − D q − D q +1) - PSL ( q ) q ≡ B = F q (cid:111) C ( q − / D q − D q +1 A Sz( q ) 2 n B = M ( q, θ ) D q − C q + r +1 (cid:111) C C q − r +1 (cid:111) C Table 1.
Stabilizers of vertices for the graph X OS ( G ) G q G η G η G η G η G η (cid:48) i PSL ( q ) 2 n C q − C C - ( q ) q ≡ C ( q − / C C × C C q ) 2 n C q − C C C Table 2.
Stabilizers of edges for the graph X OS ( G ) Now, for each of the groups G in Theorem B, we fix our choices regarding X OS + k ( G ) in orderto apply Brown’s result to it. In each case the stabilizers are given in Tables 1 and 2. Ourchoices in each case are the following. • For G = PSL (2 n ) we take V = { v , v , v } , E = { η , η , η , η (cid:48) , . . . , η (cid:48) k } , T = { η , η } ,with v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k . • For G = PSL ( q ) with q = 3 n or q ≡
19 (mod 24) we take V = { v , v , v , v } , E = { η , η , η , η , η (cid:48) , . . . , η (cid:48) k } , T = { η , η , η } , with v η −→ v , v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k . • For G = PSL ( q ) with q ≡
11 (mod 24) we take V = { v , v , v , v } , E = { η , η , η , η ,η (cid:48) , . . . , η (cid:48) k } , T = { η , η , η } , with v η −→ v , v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 15 BD q − D q +1) C q − C C G = PSL (2 n ) . B A D q − D q +1 C q − C C C G = PSL (3 n ) , odd n . B A D q − D q +1 C q − C C C G = PSL ( q ) , q ≡
19 (mod 24) . B A D q − D q +1 C q − C C C G = PSL ( q ) , q ≡
11 (mod 24) . B A D q − D q +1 C q − C C C G = PSL ( q ) , q ≡
13 (mod 24) . B D q +1 D q − A C q − C C C G = PSL ( q ) , q ≡ . B C q − r +1 (cid:111) C D q − C q + r +1 (cid:111) C C C q − C C G = Sz( q ) , q = 2 n . Figure 3.
One of the possible ways to construct X OS ( G ) in each case. • For G = Sz( q ) we take V = { v , v , v , v } , E = { η , η , η , η , η (cid:48) , . . . , η (cid:48) k } , T = { η , η , η } ,with v η −→ v , v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k .Note that in all cases the stabilizer of v is a Borel subgroup of G . In what follows Γ k is thegroup obtained by applying Brown’s result to the action of G on X OS + k ( G ) with these choices.For each of the groups G in Theorem B, we consider a closed edge path ξ in X OS ( G ) such thatattaching a free orbit of -cells along this path gives an acyclic -complex. We define x = i ( ξ ) ,where i : π ( X OS ( G ) , v ) → Γ is the inclusion given by Brown’s theorem. We set x i = x η (cid:48) i for i = 1 , . . . , k . Let ˜ η be the unique edge of X OS ( G ) which lies in E − T . We define y = x ˜ η and y i = x η (cid:48) i for i = 1 , . . . , k .We conclude this section with a lemma needed in Section 8. We explain here some notationwhich is only needed in this proof. If x = (cid:80) g ∈ G x g g ∈ Z [ G ] then we define x = (cid:80) g ∈ G x g g − .We have x + y = x + y and x · y = y · x . If H ≤ G is a subgroup we define N ( H ) = (cid:80) h ∈ H h . Lemma 5.6.
Let G be one of the groups in Theorem 1.1. Let E be a set of representatives of theorbits of edges in X OS ( G ) . Let X be an acyclic -complex obtained from X OS ( G ) by attachinga free orbit of -cells along (the orbit of ) a closed edge path ξ = ( a e ε , . . . , a n e ε n n ) with e i ∈ E , a i ∈ G and ε i ∈ {− , } . Let G e be the stabilizer of e . Then it is possible to choose, for each e ∈ E an element x e ∈ Z [ G ] such that n (cid:88) i =1 ε i a i N ( G e i ) x e i . Therefore for any representation V of G we have V = (cid:88) e ∈ E s e V G e , where s e = (cid:88) i ∈ I e ε i a i and I e = { i : e i = e } .Proof. We consider the cellular chain complex of X (which is a complex of left Z [ G ] -modules).Let α be the -cell attached along ξ . We have isomorphisms C ( X ) (cid:39) Z [ G ] and C ( X ) (cid:39) (cid:77) e ∈ E Z [ G/G e ] given by α (cid:55)→ and e (cid:55)→ · G e respectively. With these identifications, thedifferential d : C ( X ) → C ( X ) is given by d (1) = n (cid:88) i =1 ε i a i G e i = (cid:88) e ∈ E s e G e . Now the differential d : C ( X ; Z ) → C ( X ; Z ) identifies with the map d : (cid:77) e ∈ E Z [ G/G e ] → Z [ G ]1 · G e (cid:55)→ N ( G e ) s e . Since X is acyclic, the differential d is surjective and there are elements y e ∈ Z [ G ] suchthat (cid:80) e ∈ E y e N ( G e ) s e . Finally, since N ( H ) = N ( H ) and letting x e = y e we have (cid:80) e ∈ E s e N ( G e ) x e . (cid:3) Representations and centralizers
The results in this section provide, for each of the groups G in Theorem B, a suitable irre-ducible representation ρ of G in G = U ( m ) . The values of m are recorded in Table 3. G q m
PSL ( q ) 2 n q − ( q ) 3 n with n odd q − PSL ( q ) q ≡ or
19 (mod 24) q − Sz( q ) 2 n with n odd r ( q − Table 3.
The degree m of ρ in each case.For i = 0 , and each of the groups G in Theorem B, we fix a generator ˆ g i of G η i . In thefollowing propositions I n denotes the n × n identity matrix. Proposition 6.1.
Let G = PSL ( q ) with q = 2 n and n ≥ . Let G = U ( q − . There is anirreducible representation ρ : G → G satisfying the following properties:(i) There is a matrix A η ∈ G such that A η ρ (ˆ g ) A − η = diag( ξ, ξ , . . . , ξ q − ) where ξ = e πiq − . Then C G ( ρ ( G η )) = A − η U (1) q − A η has dimension q − . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 17 (ii) There is a matrix A η ∈ G such that we have A η ρ (ˆ g ) A − η = I q − ⊕ − I q and therefore C G ( ρ ( G η )) = A − η (cid:16) U (cid:16) q − (cid:17) × U (cid:16) q (cid:17)(cid:17) A η has dimension (cid:0) q − (cid:1) + (cid:0) q (cid:1) .(iii) The centralizer C G ( ρ ( G η )) has dimension (cid:0) q − (cid:1) + (cid:0) q (cid:1) .(iv) The restriction of ρ to the Borel subgroup G v is irreducible.(v) The centralizer C G ( ρ ( G v )) has dimension q .(vi) The centralizer C G ( ρ ( G v )) has dimension q .(vii) The trivial representation G v does not occur in the restriction ρ | G v .Proof. For G = PSL ( q ) and q = 2 n we take ρ realizing the degree q − character θ of Theo-rem A.4.1. By Theorem 4.1, we can take ρ to be unitary. By Lemma A.1.1 and Lemma 4.5we can prove parts (i) to (vii) by computing inner products of the restrictions of θ . Theserestrictions are computed using Proposition A.4.2. (cid:3) Proposition 6.2.
Let G = PSL ( q ) where q ≡ and q > . Let r = (cid:112) q/p andlet G = U (cid:16) q − (cid:17) . There is an irreducible representation ρ : G → G satisfying the followingproperties:(i) There is a matrix A η ∈ G such that A η ρ (ˆ g ) A − η = diag(1 , ξ, ξ , . . . , ξ q − − ) where ξ = e π i ( q − / . Then C G ( ρ ( G η )) = A − η U (1) q − A η has dimension q − .(ii) There is a matrix A η ∈ G such that A η ρ (ˆ g ) A − η = I q +14 ⊕ − I q − and therefore C G ( ρ ( G η )) = A − η (cid:18) U (cid:18) q + 14 (cid:19) × U (cid:18) q − (cid:19)(cid:19) A η has dimension ( q +14 ) + ( q − ) .(iii) The centralizer C G ( ρ ( G η )) has dimension ( q +58 ) + 3( q − ) .(iv) The dimension of C G ( ρ ( G η )) is given by dim C G ( ρ ( G η )) = ( q − ) + ( q +3 r ) + ( q − r ) if q ≡ q − ) if q ≡ q − ) + 2( q +16 ) if q ≡ . (v) The restriction of ρ to the Borel subgroup G v is irreducible.(vi) The centralizer C G ( ρ ( G v )) has dimension q +14 .(vii) The centralizer C G ( ρ ( G v )) has dimension q +14 . (viii) The dimension of C G ( ρ ( G v )) is given by dim C G ( ρ ( G v )) = q +6 q +2148 if q ≡ q − q +1348 if q ≡ q − q +4548 if q ≡ . (ix) The nontrivial degree representation of G v (cid:39) D q − does not occur in the restriction ρ | G v .Proof. For G = PSL ( q ) with q ≡ we take ρ by factoring a representation realizingthe degree q − character η of Theorem A.5.1 through the quotient SL ( q ) → PSL ( q ) . ByTheorem 4.1, we can take ρ to be unitary. By Lemma A.1.1 and Lemma 4.5 we can prove parts(i) to (ix) by computing inner products of the restrictions of η . These restrictions are computedusing Proposition A.5.3. (cid:3) Proposition 6.3.
Let G = Sz( q ) with q = 2 n and n ≥ odd. Let r = √ q and let G = U (cid:16) r ( q − (cid:17) . There is an irreducible representation ρ : G → G satisfying the following properties:(i) There is a matrix A η ∈ G such that A η ρ (ˆ g ) A − η = ξI r/ ⊕ ξ I r/ ⊕ . . . ⊕ ξ q − I r/ ⊕ ξ q − I r/ . where ξ = e πiq − . Then C G ( ρ ( G η )) = A − η U ( r/ q − A η has dimension q ( q − .(ii) There is a matrix A η ∈ G such that A η ρ (ˆ g ) A − η = I r ( q − / ⊕ − I rq/ and then C G ( ρ ( G η )) = A − η ( U ( r ( q − / × U ( rq/ A η has dimension q ( q − q +2)4 .(iii) For i = 2 , the centralizer C G ( ρ ( G η i )) has dimension q ( q − q +4)8 .(iv) The restriction of ρ to the Borel subgroup G v is irreducible.(v) The centralizer C G ( ρ ( G v )) has dimension q .(vi) The centralizer C G ( ρ ( G v )) has dimension ( q − qr + 2 q + 2 r ) .(vii) The centralizer C G ( ρ ( G v )) has dimension ( q + qr + 2 q − r ) .(viii) The trivial representation G v does not occur in the restriction ρ | G v .Proof. For G = Sz( q ) we take ρ realizing the degree r ( q − character W of Proposition A.6.3.By Theorem 4.1, we can take ρ to be unitary. By Lemma A.1.1 and Lemma 4.5 we can proveparts (i) to (viii) by computing inner products of the restrictions of W . These restrictions arecomputed using Proposition A.6.4. (cid:3) The dimension of M k From now on, let M k be the moduli of representations of Γ k obtained from the representation ρ given by Propositions 6.1 to 6.3 using Theorem 3.2. Let M k be the corresponding quotientobtained using Theorem 3.3. Note that M k = M × G k and that M k = M × G k . FromCorollary 3.4 we know that M k and M k are connected and orientable. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 19 Proposition 7.1.
For each of the groups G in Theorem B, the dimension of M k is equal to thedimension of G k +1 .Proof. This reduces to the case of k = 0 and by part (iii) of Theorem 3.3 we only have to verifythat (cid:88) e ∈ E dim C G ( ρ ( G e )) − (cid:88) v ∈ V,v (cid:54) = v dim C G ( ρ ( G v )) = dim G . In each case, this follows from a computation with Propositions 6.1 to 6.3. (cid:3)
The following result, whose proof is due to Piterman, gives an alternative proof of Proposi-tion 7.1 which sheds light on why dim M k = dim G k +1 . Lemma 7.2 (Piterman) . Let X be an acyclic -dimensional G -complex and let ϕ, ψ be twocharacters of G . Let V , E , F be representatives of the orbits of vertices, edges and -cells in X .Then (cid:104) ϕ, ψ (cid:105) G + (cid:88) e ∈ E (cid:104) Res GG e ϕ, Res GG e ψ (cid:105) G e = (cid:88) v ∈ V (cid:104) Res GG v ϕ, Res GG v ψ (cid:105) G v + (cid:88) f ∈ F (cid:104) Res GG f ϕ, Res GG f ψ (cid:105) G f . Proof.
Since X is acyclic, (cid:101) C − ( X ; C ) ⊕ (cid:101) C ( X ; C ) (cid:39) (cid:101) C ( X ; C ) ⊕ (cid:101) C ( X ; C ) as G -modules. Then,letting α H be the character of the G -module C [ G/H ] we have α G + (cid:88) e ∈ E α G e = (cid:88) v ∈ V α G v + (cid:88) f ∈ F α G f and now the result follows from Frobenius reciprocity: (cid:104) Res GH ϕ, Res GH ψ (cid:105) H = (cid:104) ϕ, Ind GH Res GH ψ (cid:105) G = (cid:104) ϕ, α H ψ (cid:105) G (cid:3) Let χ be the character of ρ . Since (cid:104) χ, χ (cid:105) G = 1 and (cid:104) Res GG v χ, Res GG v χ (cid:105) G v = 1 , applyingLemma 7.2 to χ (and using Lemma 4.5) we obtain another proof of Proposition 7.1.8. The differential of W at For i = 0 , . . . , k we consider the map X i : M → G induced by x i . Proposition 8.1.
Let X = ( X , . . . , X k ) : M k → G k +1 . Then is a regular point of X .Proof. The proof reduces to the case of k = 0 . Consider the representation Ad ◦ ρ : G → GL( T G ) which (by Corollary 2.3) is given by g · v = d ρ ( g ) − L ρ ( g ) ◦ d R ρ ( g ) − ( v ) . By Proposition 2.4 wehave T C G ( ρ ( H )) = ( T G ) H . Then by Lemma 5.6 we have T G = (cid:80) e ∈ E s e · T C G ( ρ ( G e )) .Now the result follows from Theorem 3.7. (cid:3) Proposition 8.2. If w , . . . , w k ∈ N satisfy N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] , then is a regularpoint of W = ( W , . . . , W k ) : M k → G k +1 .Proof. This is follows from Lemma 3.8 and Proposition 8.1. (cid:3)
Now since W ◦ p = W we have: Corollary 8.3. If w , . . . , w k ∈ N satisfy N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] , then is a regular pointof W = ( W , . . . , W k ) : M k → G k +1 . The degree of W In this section we prove the degree of W is . We start by recalling the definition and someproperties of the degree (see e.g. [Lee13, Chapter 17] for a detailed exposition). Let M and M (cid:48) be oriented m -manifolds. The degree deg( f ) of a smooth map f : M → M (cid:48) is the unique integer k such that (cid:90) M f ∗ ( ω ) = k (cid:90) M (cid:48) ω for every smooth m -form ω on M (cid:48) . If x ∈ M is a regular point of f , then d x f : T x M → T f ( x ) M (cid:48) is an isomorphism between oriented vector spaces and we can consider its sign sg d x f . If y ∈ M (cid:48) is a regular value of f we have deg( f ) = (cid:88) x ∈ f − ( y ) sg d x f. In particular if f is not surjective then deg( f ) = 0 . Homotopic maps have the same degree.If N and N (cid:48) are oriented n -manifolds and g : N → N (cid:48) is a smooth map then deg( f × g ) =deg( f ) deg( g ) . If M (cid:48)(cid:48) is an oriented m -manifold and h : M (cid:48) → M (cid:48)(cid:48) is smooth then deg( h ◦ f ) =deg( h ) deg( f ) .For i = 0 , . . . , k we consider the map Y i : M → G induced by y i . Table 4 gives the value of Y in the different cases that we consider G q Y ( τ )PSL ( q ) 2 n τ − η τ − η τ − η ρ ( g η )PSL ( q ) 3 n τ − η τ − η τ − η τ − η ρ ( g η )PSL ( q ) q ≡
19 (mod 24) τ − η τ − η τ − η τ − η ρ ( g η )PSL ( q ) q ≡
11 (mod 24) τ − η τ − η τ − η τ − η ρ ( g η ) τ η τ η Sz( q ) 2 n τ − η τ − η τ − η τ − η ρ ( g η ) Table 4.
The map Y , for each of the groups G in Theorem B.The proof goes by proving that Y : M → G has degree . We do this by homotoping themap Y to a map Z : M → G which is not surjective. This homotopy needs to be H -equivariant.In order to define the map Z , we need to choose a matrix C ∈ U ( m (cid:48) ) × U ( m (cid:48)(cid:48) ) with certainproperty. The numbers m (cid:48) and m (cid:48)(cid:48) satisfy m = m (cid:48) + m (cid:48)(cid:48) and are defined in Table 5. Let A η and A η be the matrices in Propositions 6.1 to 6.3. By Lemma 4.7 and Proposition 4.8, in eachcase we can take a matrix (cid:101) C ∈ U ( m ) × U ( m ) such that A − η (cid:101) CA η commutes with C G ( G v ) .We set C = A − η (cid:101) CA η ∈ C G ( G η ) . G q m (cid:48) m (cid:48)(cid:48) PSL ( q ) 2 n q/ − q/ ( q ) 3 n ( q + 1) / q − / ( q ) q ≡
19 (mod 24) ( q + 1) / q − / ( q ) q ≡
11 (mod 24) ( q + 1) / q − / q ) 2 n r ( q − / rq/ Table 5.
The numbers m (cid:48) and m (cid:48)(cid:48) . Proposition 9.1.
In all cases the map Z : M → G of Table 6 is not surjective. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 21 G q Z ( τ ) T ( τ )PSL ( q ) 2 n ( A − η A η ) τ − η ( A − η A η ) − C · τ − η τ − η ρ ( g η ) τ − η τ − η ρ ( g η )PSL ( q ) 3 n ( A − η A η ) τ − η ( A − η A η ) − C · τ − η τ − η τ − η ρ ( g η ) τ − η τ − η τ − η ρ ( g η )PSL ( q ) q ≡
19 (mod 24) ( A − η A η ) τ − η ( A − η A η ) − C · τ − η τ − η τ − η ρ ( g η ) τ − η τ − η τ − η ρ ( g η )PSL ( q ) q ≡
11 (mod 24) ( A − η A η ) τ − η ( A − η A η ) − C · τ − η τ − η τ − η ρ ( g η ) τ η · C − ( A − η A η ) τ η ( A − η A η ) − τ − η τ − η τ − η ρ ( g η ) τ η Sz( q ) 2 n ( A − η A η ) τ − η ( A − η A η ) − C · τ − η τ − η τ − η ρ ( g η ) τ − η τ − η τ − η ρ ( g η ) Table 6.
The definition of the maps Z and T for each of the groups G inTheorem B. Proof.
In each case, we consider the manifold M = (cid:81) i> C G ( ρ ( G η i )) and the Lie group H = (cid:81) i> C G ( ρ ( G v i )) . The action M (cid:120) H restricts to a free action of H ≤ H on the factor M of M . Now consider the map T : M → G defined in Table 6. Note that T factors through thequotient M → M/H giving a map T : M/H → G . By Theorem 2.7, M/H is a manifold and wehave dim
M/H = dim M − dim H = dim M − dim H + dim C G ( ρ ( G v )) − dim C G ( ρ ( G η )) < dim M − dim H = dim G . Therefore T is not surjective (this is a corollary to Sard’s Theorem, see [Lee13, Corollary 6.11]).Now note that A η C G ( ρ ( G η )) A − η ⊆ U ( m ) × U ( m ) = A η C G ( ρ ( G η )) A − η and then A − η A η C G ( ρ ( G η ))( A − η A η ) − ⊆ C G ( ρ ( G η )) . Moreover since (cid:101) C ∈ U ( m (cid:48) ) × U ( m (cid:48)(cid:48) ) we have C = A − η (cid:101) CA η ∈ C G ( ρ ( G η )) . To conclude, note that the image of Z equals the imageof T . (cid:3) Proposition 9.2.
For each of the groups G in Theorem B, the degree of Y : M → G is .Proof. By Proposition 3.11 (and Theorem 2.5) the map Y : M → G is H -equivariantly homo-topic to the map Z defined in Table 6. Passing to the quotient we see that there is a homotopybetween the maps Y , Z : M → G . By Proposition 9.1, Z is not surjective and therefore Z isnot surjective. We conclude the degree of Y is . (cid:3) Remark . In the power of and the Suzuki cases, with some more work we can use Proposi-tion 3.11 to deform Y into a map Z with image ( U ( m (cid:48) ) × U ( m (cid:48)(cid:48) )) · ρ ( g ˜ η ) . This can be achievedby extending Propositions 6.1 and 6.3 and inserting additional matrices in the definition of Z (which again can be obtained using Lemma 4.7). Corollary 9.4.
The degree of Y = ( Y , . . . , Y k ) : M k → G k +1 is . Proof.
We have M k = M × G k . Now, by Proposition 9.2, the map Y : M × G k → G k +1 hasdegree since it is the product of the maps Y : M → G and the identity maps Y i : G → G for i = 1 , . . . , k . (cid:3) Proposition 9.5.
Let w , . . . , w k ∈ Γ k and let W = ( W , . . . , W k ) : M k → G k +1 . Then deg( W ) = 0 ∈ Z .Proof. First note that, by Lemma 3.9 (and Theorem 2.5), we only need to address the casewhen the w i are words in the generators y , . . . , y k . Now consider the map Y = ( Y , . . . , Y k ) and consider the map (cid:102) W : G k +1 → G k +1 induced by the words w , . . . , w k ∈ F ( y , . . . , y k ) ,which makes the following diagram commute M k G k +1 G k +1 YW (cid:102) W By Corollary 9.4 Y has degree and since deg( W ) = deg( (cid:102) W ) · deg( Y ) we are done. (cid:3) Group actions on contractible -complexes We now prove the main results of this article.
Theorem 10.1.
Let G be one of the groups in Theorem B. Let w , . . . , w k ∈ N . If N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] then there is a point τ ∈ M k such that(i) ρ τ ( w i ) = 1 for i = 0 , . . . , k , and(ii) ρ τ is not universal.Proof. By Proposition 9.5 the degree of W is . By Corollary 8.3, is a regular point of W . Therefore, there must exist a point τ ∈ W − ( ) with τ (cid:54) = . To conclude note that byProposition 3.5, τ is not universal. (cid:3) Theorem 10.2 ([SC20, Theorem 3.6]) . Let G be one of the groups in Theorem 1.1. Let X be a fixed point free -dimensional finite acyclic G -complex. Then there is a fixed point free -dimensional finite acyclic G -complex X (cid:48) obtained from the G -graph X OS ( G ) by attaching k ≥ free orbits of -cells and k + 1 free orbits of -cells and an epimorphism π ( X ) → π ( X (cid:48) ) .Proof of Theorem B. By Theorem 10.2 it is enough to prove the result when X is obtained from X OS ( G ) by attaching k ≥ free orbits of -cells and k + 1 free orbits of -cells. By Theorem 3.1,there are words w , . . . , w k ∈ N such that π ( X ) (cid:39) N (cid:104)(cid:104) w ,...,w k (cid:105)(cid:105) Γ k and since H ( X ) = 0 wehave N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] . Now passing to the quotient the representation ρ τ given byTheorem 10.1 we obtain a nontrivial representation π ( X ) → U ( m ) . (cid:3) Theorem 10.3 ([SC20, Theorem 3.8]) . Let P ( G ) be the following proposition: “there is a non-trivial representation in U ( m ) of the fundamental group of every acyclic G -complex obtainedfrom X OS ( G ) by attaching k ≥ free orbits of -cells and ( k + 1) free orbits of -cells”. To proveTheorem A it is enough to prove P ( G ) for each of the following groups G : • PSL (2 p ) for p prime; • PSL (3 p ) for an odd prime p ; • PSL ( q ) for a prime q > such that q ≡ ± and q ≡ ± ; • Sz(2 p ) for p an odd prime. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 23 Remark . Note that in [SC20], Theorem 10.3 is proved for representations in
SO( m ) . How-ever the same proof gives the unitary version: since the Gerstenhaber–Rothaus theorem holdsfor any compact connected Lie group [GR62], we have that [SC20, Proposition A.3] also holdsfor unitary representations. Also note that the second part of Theorem A is not addressed inthe original statement of Theorem 10.3. Nevertheless, this also is immediate from the originalproof of Theorem 10.3. Proof of Theorem A.
By Theorem 10.3, the result now follows from Theorem B and Theorem C. (cid:3)
Appendix A. Representation theory of
PSL ( q ) and Sz( q ) by Kevin Piterman
A.1.
Summary on representation theory for finite groups.
In this subsection, we give abrief summary on representation theory for finite groups. The main references are [Dor71] and[Ser77].Let G be a finite group. We will work with the field C of complex numbers. Denote by x thecomplex conjugate of an element x ∈ C .By Maschke’s theorem, the group algebra C [ G ] is semisimple. Hence every C [ G ] -module splitsas a direct sum of irreducible (or simple) C [ G ] -modules (that is, with no nontrivial and proper C [ G ] -submodule). Recall also that a C [ G ] -module V is the same as a representation of G ona C -vector space V , which is group homomorphism ρ : G → Aut C ( V ) . Here Aut C ( V ) denotesthe group of C -linear automorphisms of V . We make no distinction and we denote such arepresentation by ρ or V . The degree of the representation ρ is the dimension of V as C -vectorspace. The character of ρ is the function χ ρ : G → C such that χ ρ ( g ) equals the trace of ρ ( g ) , for g ∈ G . Note that characters are constant on conjugacy classes, and that the degree of ρ equalsthe value of χ ρ at ∈ G . From the theory of characters, the representation ρ is completelydetermined by χ ρ . This means that two C [ G ] -modules V, W give rise to the same character ifand only if V and W are isomorphic. A character χ of G is irreducible if it is the character ofan irreducible C [ G ] -module.Let Irr( G ) denote the set of irreducible characters of G and let C ( G ) denote the set of functions α : G → C which are constant on the conjugacy classes of G . Then C ( G ) is a C -vector spacewhose dimension is equal to the number of conjugacy classes of G . There is an inner product in C ( G ) defined as follow. If α, β ∈ C ( G ) , then (cid:104) α, β (cid:105) G := 1 | G | (cid:88) g ∈ G α ( g ) β ( g ) . By Schur’s orthogonality relations,
Irr( G ) is an orthonormal basis for this inner product, fromwhich it follows that | Irr( G ) | equals the number of conjugacy classes of G . In particular, thecharacter of every representation ρ : G → Aut C ( V ) can be uniquely written as a linear combina-tion of irreducible characters, with non-negative integer coefficients. Let Irr( G ) = { χ , . . . , χ r } and denote by V i the irreducible module associated to χ i .If ρ is a representation that splits as a direct sum V = (cid:76) ri =1 V n i i with n i ≥ , then thecharacter of V is the linear combination χ = (cid:80) ri =1 n i χ i . Since Irr( G ) forms an orthonormalbasis, the coefficients n i for an arbitrary representation ρ can be computed by taking the inner products n i = (cid:104) χ ρ , χ i (cid:105) . In particular, we have the following useful lemma to detect irreduciblecharacters. Lemma A.1.1.
Let G be a finite group and χ : G → C be a character. Then (cid:104) χ, χ (cid:105) = 1 if andonly if χ is irreducible. Below we state Schur’s lemma. Recall from the Frobenius theorem that C is the only divisionalgebra (of finite dimension) over C . Theorem A.1.2 (Schur’s Lemma) . Let
V, W be two irreducible C [ G ] -modules. Then every G -linear map between V and W is either the zero map or an isomorphism. Therefore the followingassertions hold: • If V and W are not isomorphic, then the zero map is the unique G -linear map betweenthem. • The endomorphism algebra
End C [ G ] ( V ) is a division algebra over C . Hence, the elementsof End C [ G ] ( V ) are the scalar multiples of the identity map. Note that an element x ∈ Aut C ( V ) satisfies xρ ( g ) = ρ ( g ) x for all g ∈ G if and only if x preserves the action of G . That is, C Aut C ( V ) ( G ) is exactly the group of C [ G ] -automorphisms of V . This centralizer can be computed by using Schur’s lemma.A.2. Characters of cyclic groups.
The following proposition computes the irreducible char-acters of cyclic groups.
Proposition A.2.1.
Let C n be a cyclic group of order n generated by an element g . Theirreducible characters for C n are µ k for k = 1 , . . . , n , and they are defined as follows. Let µ = e π i n . Then the irreducible character µ k has degree and is given by g i (cid:55)→ µ ki . A.3.
Characters of dihedral group D n ( n odd). Recall that the dihedral group of order n , denoted by D n , is the group generated by an involution s and an element r of order n suchthat srs = r − . Proposition A.3.1 ([Ser77, Section 5.3]) . Let n be an odd number and let µ = e π i n . Then thecharacter of table of D n is r k sr k ψ ψ − χ i µ ik + µ − ik Here, i = 1 , . . . , n − . A.4.
Linear groups in even characteristic.
We review the conjugacy classes and irreduciblecharacters of the groups SL ( q ) and PSL ( q ) .Recall that PSL ( q ) is the quotient of SL ( q ) by its center, which is Z (SL ( q )) = { , − } if q is odd, and Z (SL ( q )) = 1 if q is even. By composing with the quotient map SL ( q ) → PSL ( q ) ,we see that every character of PSL ( q ) arises from a character of SL ( q ) whose kernel contains Z (SL ( q )) (recall that the kernel of a character of a group G is the kernel of the associatedgroup homomorphism G → Aut C ( V ) ). Furthermore, it is known that the irreducible characters χ of PSL ( q ) are exactly those arising from the irreducible characters ˜ χ of SL ( q ) containing Z (SL ( q )) in their kernel. ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 25 We recall below the conjugacy classes and character tables for linear groups defined over afield of even characteristic. For any x ∈ G , let ( x ) denote the conjugacy class of x . Theorem A.4.1 ([Dor71, Theorem 38.2]) . Let F be a finite field of q = 2 f elements, and let ν be a generator of the cyclic group F ∗ = F − . Denote (cid:32) (cid:33) , c = (cid:32) (cid:33) , a = (cid:32) ν ν − (cid:33) in G = SL ( F ) .Then | a | = q − , | c | = 2 , G contains an element b of order q + 1 and there are exactly q + 1 conjugacy classes (1) , ( c ) , ( a ) , ( a ) , . . . , ( a q − ) , ( b ) , ( b ) , . . . , ( b q ) , such that x c a l b m | ( x ) | q − q ( q + 1) q ( q − for ≤ l ≤ q − , ≤ m ≤ q .Let ρ ∈ C be a primitive ( q − -th root of unity, σ ∈ C a primitive ( q + 1) -th root of . Thenthe character table of G over C is c a l b m G ψ q − χ i q + 1 1 ρ il + ρ − il θ j q − − − ( σ jm + σ − jm ) for ≤ i ≤ q − , ≤ j ≤ q , ≤ l ≤ q − , ≤ m ≤ q . The following proposition can be deduced by using Theorem A.4.1 and a structure descriptionof the groups listed in the first column.
Proposition A.4.2.
Let G = SL (2 n ) . The following table shows the size of ( x ) ∩ L , for eachconjugacy class ( x ) of G . L c a l b m B q − q D q − q − D q +1) q + 1 0 2 C q − C Here ≤ l ≤ q − and ≤ m ≤ q . A.5.
Linear groups in odd characteristic.
We provide here similar results for the lineargroups in odd characteristic.
Theorem A.5.1 ([Dor71, Theorem 38.1]) . Let F be the finite field of q = p n elements, p oddprime, and let ν be a generator of the cyclic group F ∗ = F − . Denote Γ = (cid:32) (cid:33) , z = (cid:32) − − (cid:33) , c = (cid:32) (cid:33) , d = (cid:32) ν (cid:33) , a = (cid:32) ν ν − (cid:33) , in G = SL ( F ) .Then | a | = q − , G has an element b of order q + 1 and there are exactly q + 4 conjugacyclasses: (1) , ( z ) , ( c ) , ( d ) , ( zc ) , ( a ) , ( a ) , . . . , ( a q − ) , ( b ) , ( b ) , . . . , ( b q − ) , such that x z c d zc zd a l b m | ( x ) | ( q − ( q − ( q − ( q − q ( q + 1) q ( q − with ≤ l ≤ q − , ≤ m ≤ q − .Denote (cid:15) = ( − q − . Let ρ ∈ C be a primitive ( q − -th root of unity, σ ∈ C a primitive ( q + 1) -th root of unity. Then the complex character table of G is z c d a l b m G ψ q q − χ i q + 1 ( − i ( q + 1) 1 1 ρ il + ρ − il θ j q − − j ( q − − − − ( σ jm + σ − jm ) ξ ( q + 1) (cid:15) ( q + 1) (1 + √ (cid:15)q ) (1 − √ (cid:15)q ) ( − l ξ ( q + 1) (cid:15) ( q + 1) (1 − √ (cid:15)q ) (1 + √ (cid:15)q ) ( − l η ( q − − (cid:15) ( q − ( − √ (cid:15)q ) ( − − √ (cid:15)q ) 0 ( − m +1 η ( q − − (cid:15) ( q − ( − − √ (cid:15)q ) ( − √ (cid:15)q ) 0 ( − m +1 for ≤ i ≤ q − , ≤ j ≤ q − , ≤ l ≤ q − , ≤ m ≤ q − .The values for the columns of the classes ( zc ) and ( zd ) can be obtained from the relations χ ( zc ) = χ ( z ) χ (1) χ ( c ) , χ ( zd ) = χ ( z ) χ (1) χ ( d ) , for all irreducible character χ of G .Remark A.5.2 . By [Dor71, p.234], if q ≡ , then c − ∈ ( d ) , ( zc ) − ∈ ( zd ) , d − ∈ ( c ) and ( zd ) − ∈ ( zc ) .Note also that the unique element of order in SL ( q ) is z , and that every element of order in SL ( q ) is conjugate to b q +14 if q ≡ . By the above assertion on the inverse of theelements, we see that there is a unique class of involutions in the quotient PSL ( q ) , correspondingto the involution b q +14 .The following proposition can be deduced from Theorem A.5.1, Remark A.5.2 and a structuredescription of the groups listed in the first column.We denote by x the image of an element x ∈ SL ( q ) in the quotient PSL ( q ) . Recall that if q = 3 n , n odd, then q ≡ . Denote by δ x,y the Kronecker delta function, which is if x = y and otherwise. Proposition A.5.3.
Let G = PSL ( q ) with q ≡ . The following table shows the sizeof ( x ) ∩ L , for each conjugacy class ( x ) of G . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES I 27 L c d a l b m b q +14 B q − q − q A : q ≡ q ≡ δ l, q − q ≡ δ l, q +16 D q − q − D q +1 q +32 C C × C C q − C : q ≡ q ≡ δ l, q − q ≡ δ l, q +16 Here ≤ l ≤ q − and ≤ m ≤ q − . A.6.
The Suzuki groups.
In this section we recall the basic facts on the conjugacy classesand irreducible characters of the Suzuki groups G = Sz( q ) , where q is an odd power of . Let r := √ q . The following results are due to Suzuki [Suz62]. Proposition A.6.1 (Maximal subgroups) . The solvable maximal subgroups of
Sz( q ) are:(1) B , the Borel subgroup, of order q ( q − and normalizer of a Sylow -subgroup.(2) D q − , dihedral of order q − and normalizer of a cyclic subgroup of order q − .(3) C + (cid:39) C q + r +1 (cid:111) C , normalizer of a cyclic subgroup of order q + r + 1 .(4) C − (cid:39) C q − r +1 (cid:111) C , normalizer of a cyclic subgroup of order q − r + 1 . Proposition A.6.2 (Conjugacy classes) . Let G = Sz( q ) , with q = 2 n and n odd. There are q + 3 conjugacy classes of elements in G , and they are given as follows. Let σ, ρ, π , π , π ∈ G be elements of order , , q − , q + r + 1 and q − r + 1 respectively. Then the conjugacy classesare (1) , ( σ ) , ( ρ ) , ( ρ − ) , ( π a ) ( ≤ a ≤ q − ), ( π b ) ( ≤ b ≤ q + r ) and ( π c ) ( ≤ c ≤ q − r ). In Proposition A.6.3 we reproduce the character table of the Suzuki group
Sz( q ) , followingthe notation and results from [Suz62, Section 17]. Proposition A.6.3 (Character table) . Let G = Sz( q ) , with q = 2 n and n odd. Let (cid:15) be aprimitive ( q − -th root of unity. For ≤ i ≤ q − , define the function (cid:15) i in the powers of π asfollows: (cid:15) i ( π a ) = (cid:15) ia + (cid:15) − ia . Let (cid:15) be a primitive ( q + r + 1) -th root of unity. For ≤ j ≤ q + r , define the function (cid:15) j in thepowers of π as follows: (cid:15) j ( π b ) = (cid:15) jb + (cid:15) jbq + (cid:15) − jb + (cid:15) − jbq . Let (cid:15) be a primitive ( q − r + 1) -th root of unity. For ≤ k ≤ q − r , define the function (cid:15) k in thepowers of π as follows: (cid:15) k ( π c ) = (cid:15) kc + (cid:15) kcq + (cid:15) − kc + (cid:15) − kcq . The following table gives the irreducible characters for G . σ ρ ρ − π a π b π c G X q − − X i q + 1 1 1 1 (cid:15) i ( π a ) 0 0 Y j ( q − q − r + 1) r − − − − (cid:15) j ( π b ) 0 Z k ( q − q + r + 1) − r − − − − (cid:15) k ( π c ) W r q − − r/ r i / − r i / − W r q − − r/ − r i / r i / − Here ≤ a, i ≤ q − , ≤ b, j ≤ q + r and ≤ c, k ≤ q − r . In the following proposition, we compute the size of ( x ) ∩ L for each conjugacy class ( x ) of theSuzuki group G = Sz( q ) . The proof of this proposition follows from the structure descriptionof each one of the groups appearing in the first column. See [Suz62] for more details on thesedescriptions. Proposition A.6.4.
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Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires. Buenos Aires,Argentina.
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