Sylow branching coefficients and a conjecture of Malle and Navarro
aa r X i v : . [ m a t h . R T ] F e b SYLOW BRANCHING COEFFICIENTS AND ACONJECTURE OF MALLE AND NAVARRO
EUGENIO GIANNELLI, STACEY LAW, JASON LONG, AND CAROLINA VALLEJO
Abstract.
We prove that a finite group G has a normal Sylow p -subgroup P if, and only if, every irreducible character of G appearingin the permutation character ( P ) G with multiplicity coprime to p hasdegree coprime to p . This confirms a prediction by Malle and Navarrofrom 2012. Our proof of the above result depends on a reduction to sim-ple groups and ultimately on a combinatorial analysis of the propertiesof Sylow branching coefficients for symmetric groups. Introduction
One of the main research themes in the representation theory of finite groupsis to determine how much information about the algebraic structure of afinite group G can be discovered using knowledge of its character degrees. Afamous result in this line of investigation is the Itˆo-Michler theorem [Itˆo51,Mic86] asserting that a Sylow p -subgroup P of G is abelian and normalin G if, and only if, the character degree χ (1) is coprime to p for everyirreducible character χ ∈ Irr( G ). Separating the two conditions (abelianand normal) on the Sylow p -subgroup P in the context of character degreeshas been a challenge for the last few decades. While the commutativity of P is characterized by Brauer’s height zero conjecture [MN21], the aim of thisarticle is to study canonical subsets of characters whose degrees characterizethe normality of P in G .In [MN12] Malle and Navarro showed that given a prime p and a Sylow p -subgroup P of G , then P is normal in G if, and only if, every irreducibleconstituent of the permutation character ( P ) G has degree coprime to p . Atthe end of their article they conjecture a refinement of this result, proposingthat the normality of P may be detected by looking at an even smallersubset of irreducible characters of G , namely those irreducible constituentsof ( P ) G appearing with multiplicity coprime to p . In this article we verifyMalle and Navarro’s prediction. Theorem A.
Let G be a finite group, p be a prime and P ∈ Syl p ( G ) . Thefollowing statements are equivalent: (i) P is normal in G . (ii) Every χ ∈ Irr( G ) with [ χ P , P ] not divisible by p has degree coprimeto p . (iii) Every χ ∈ Irr( G ) with [ χ P , P ] not divisible by p does not vanish in P . Mathematics Subject Classification.
Key words and phrases.
Sylow subgroups, Character degrees, Vanishing elements andSylow branching coefficients.
In Section 2, we show that in order to prove Theorem A it is enough toprove the validity of its statement for all finite simple non-abelian groups.Roughly speaking, for every finite non-abelian simple group S we must ex-hibit an irreducible character χ of degree divisible by p with trivial Sylowbranching coefficient coprime to p . If P is a Sylow p -subgroup of S , thenthe trivial Sylow branching coefficient of χ is the multiplicity [ χ P , P ] withwhich χ appears as a constituent of the permutation character ( P ) S . (Werefer the reader to Section 3.1.1 for more information on Sylow branchingcoefficients.) The main obstacle in this context comes from simple alter-nating groups (as already observed in [MN12, p.4] and recently remarkedby the same authors in [MN20]). We prove the statement of Theorem Afor simple alternating groups as a consequence of the following much moregeneral statement, concerning symmetric and alternating groups, S n and A n , at all primes. Theorem B.
Let p be a prime and n ∈ N . Let G ∈ { S n , A n } , let P be aSylow p -subgroup of G and let B be a p -block of G . Then there exists anirreducible character χ of height zero in B such that [ χ P , P ] is not divisibleby p . As well as providing the key ingredient for proving Theorem A (note thatthe character χ provided by Theorem B has degree divisible by p whenever B has non-maximal defect), Theorem B also contributes to the study of Sy-low branching coefficients for symmetric and alternating groups and, moregenerally, to the study of the restriction of characters to Sylow subgroups.These topics have recently been at the centre of investigation for their con-nections to the McKay Conjecture [NTV14, GKNT17, INOT17, GN18]. In[GL18] the authors determine those irreducible characters of S n having non-zero trivial Sylow branching coefficient. Despite this positivity result, verylittle is known about the values of these integers. In this sense, TheoremB represents a first step towards a more precise description of these Sylowbranching coefficients.The key idea behind the proof of Theorem B is the following. For anygiven p -block B of S n we introduce a virtual character V B , obtained asa certain integer combination of the irreducible characters of height zeroin B (see Definition 3.13). Using the language of symmetric functions to-gether with algebraic-combinatorial techniques, we then show that p doesnot divide the multiplicity [( V B ) P , P ], and hence deduce that p does notdivide the Sylow branching coefficient corresponding to one of the heightzero characters occurring in V B .It is worth mentioning that the virtual characters V B , and more gener-ally the family of virtual characters introduced in Section 3 below, seemto have further applications to problems involving signed character sums insymmetric and alternating groups (see [IN08, p.2] and [Nav10, Section 6]);this will be the subject of future investigation.The structure of this article is as follows. In Section 2 we prove Theo-rem A, assuming that its statement holds for simple alternating groups. InSection 3 we investigate symmetric and alternating groups specifically, inparticular showing that Theorem A holds for these classes of groups as aconsequence of the more general Theorem B. A reduction to alternating simple groups
The aim of this section is to prove Theorem A. We mimic and adapt theapproach used in [MN12]: we reduce the problem to showing that every finitenon-abelian simple group S possesses a suitable character lying over thetrivial character of a Sylow p -subgroup of S with multiplicity not divisibleby p and vanishing on some element of the aforementioned Sylow p -subgroup.We follow the notation of [Isa06] and [Nav18] for characters. Let G be afinite group and p be a prime. Recall that χ ∈ Irr( G ) has p -defect zero if the p -part of its degree χ (1) is as large as possible, that is, if p does not divide | G | /χ (1). Note that a non-trivial group with a normal Sylow p -subgroup P does not possess a p -defect zero character unless P = 1. Lemma 2.1.
Suppose that χ ∈ Irr( G ) has p -defect zero and let P ∈ Syl p ( G ) .Then (i) χ ( x ) = 0 for every non-trivial x ∈ P . (ii) χ P = f · ρ P where f is coprime to p and ρ P is the regular characterof P .Proof. Part (i) is [Isa06, Theorem 8.17] (also [Nav18, Theorem 4.6]). By (i),we have that [ χ P , P ] = 1 | P | X x ∈ P χ ( x ) = χ (1) | P | = f is a positive integer. Then χ P = f · ρ P and part (ii) follows. (cid:3) The next lemma follows from a standard argument.
Lemma 2.2.
Let G be a finite group, p be a prime and P ∈ Syl p ( G ) . Let χ ∈ Irr( G ) have degree coprime to p . Then χ ( x ) = 0 for every x ∈ P .Proof. Consider the ring R of algebraic integers of C . Let M be a maximalideal of R containing p R . Let ξ ∈ R be a root of unity of order p a . Then( ξ − p a ≡ M . Since R / M is a field we obtain that ξ ≡ M .Given x ∈ P , write Q = h x i . Since χ Q = λ + · · · + λ χ (1) where λ j ∈ Irr( Q ),then χ ( x ) is a sum of χ (1) roots of p -power order and χ ( x ) ≡ χ (1) mod M . In particular, χ ( x ) = 0 as M ∩ Z = p Z . (cid:3) It is a well-known result of Burnside that every non-linear character of afinite group vanishes on some element ([Nav18, Corollary 4.2]). Hence theconverse of the above lemma holds in p -groups. However, it is not the case ingeneral that the non-vanishing property on Sylow p -subgroups characterisescharacters of degree coprime to p , as shown by SL (5) for p = 2. A keyingredient in the proof of Theorem A will be that symmetric and alternatinggroups satisfy the converse of Lemma 2.2 (see Theorem 3.16 below).We will prove Theorem A of the introduction using the following resulton finite non-abelian simple groups. Theorem 2.3.
Let S be a finite non-abelian simple group of order divisibleby a prime p , and let R ∈ Syl p ( S ) . Then S either possesses a p -defect zero EUGENIO GIANNELLI, STACEY LAW, JASON LONG, AND CAROLINA VALLEJO character or there is some
Aut( S ) -invariant θ ∈ Irr( S ) such that [ θ R , R ] isnot divisible by p and θ ( x ) = 0 for some x ∈ R .Proof. By [GO96, Corollary 2], every finite non-abelian simple group pos-sesses a p -defect zero character unless p = 2 and S is one of the sporadicgroups M , M , M , J , HS, Suz, Ru, Co , Co , BM or an alternatinggroup A n with 7 ≤ n = 2 m + m , 2 m + m + 2 for any integer m ; or p = 3and S is either Suz, Co or A n with 3 n + 1 divisible by some prime q con-gruent to 2 (mod 3) to an odd power. For p = 2, let S be a sporadic groupnot admitting a p -defect zero character. Using [GAP] and the command PermChars(CharacterTable("S"), d)) , we can compute the permutationcharacters ( R ) S for S ∈ { M , M , M , J , HS } , where the second argu-ment d is the degree of the desired permutation character. For S = BM thecharacter ( R ) S was computed by T. Breuer .For S ∈ { Suz , Ru , Co , Co } , one can compute ( R ) S by choosing a maxi-mal subgroup M of S containing R with the [GAP] command Maxes(CharacterTable("S")) , computing θ := ( R ) M with PermChars ,and finally inducing θ to S . (In the case where S = Co , choose Co as a maximal subgroup.) One can proceed in a similar way to obtain( R ) S whenever S ∈ { Suz , Co } and p = 3. (The function PermChars hasseveral strategies to determine candidates for permutation characters, andthe second argument determines which one is chosen. In the case where S = Co and p = 3, one should use PermChars(CharacterTable("S"),rec(torso:=[d])) where d is the degree of the desired permutation char-acter.)Once we have ( R ) S stored in [GAP], one can easily check that there issome Aut( S )-invariant θ ∈ Irr( S ) with [ θ R , R ] coprime to p that vanisheson some p -power order element.It remains to find θ for the alternating groups A n with n ≥ p ∈{ , } . This is given by Theorem 3.17 below. (cid:3) The following technical lemma will be useful for proving Theorem A.
Lemma 2.4.
Let G be a finite group, p be a prime and P ∈ Syl p ( G ) .Suppose that N ⊳ G is such that
P N ⊳ G and Q = P ∩ N > . (a) If τ ∈ Irr(
P N ) lies over P with multiplicity coprime to p , then sodoes any χ ∈ Irr( G ) lying over τ . (b) If N has a p -defect zero character η , then G has an irreducible char-acter χ = G such that p does not divide [ χ P , P ] and χ vanishes onthe non-trivial elements of Q .Proof. (a) By assumption m τ = [ τ P , P ] is coprime to p . Write χ P N = e P ti =1 τ x i by Clifford’s theorem [Isa06, Theorem 6.2] with x i ∈ G . By theFrattini argument G = N N G ( P ) and we can choose x i ∈ N G ( P ) (with x =1). By [Isa06, Corollary 11.29], χ (1) /τ (1) = et divides | G : P N | , and hence e and t are coprime to p . In particular, χ P = e P ti =1 ( τ x i ) P = e P ti =1 ( τ P ) x i and so [ χ P , P ] = etm τ is coprime to p .(b) Note that Q = P ∩ N ∈ Syl p ( N ). In particular, η Q = f · ρ Q byLemma 2.1(b). Notice that ( η P N ) P = ( f · ρ Q ) P = f · ρ P , so that p does not See . divide f = [( η P N ) P , P ]. We can choose τ ∈ Irr(
P N ) lying over η and P ,and such that m τ = [ τ P , P ] is not divisible by p . Let χ ∈ Irr( G ) lie over τ . By part (a), the multiplicity [ χ P , P ] is coprime to p . Moreover, χ N isa multiple of the sum of the N G ( P )-conjugates of η , and hence a sum of p -defect zero characters of N . In particular, by Lemma 2.1(a), χ vanisheson every non-trivial element of Q . (cid:3) Proof of Theorem A. (i) ⇒ (ii): If P ⊳ G and χ ∈ Irr(( P ) G ) then χ can beseen as a character of G/P and hence has degree coprime to p .(ii) ⇒ (iii): This implication follows from Lemma 2.2.(iii) ⇒ (i): Suppose that G is a counterexample to the statement ofminimal order. Of course G > N G ( P ) < G . Let 1 < M ⊳ G . Given χ ∈ Irr(
G/M ) lying above P M/M with multiplicity coprime to p , we canview χ as an irreducible character of G , and then χ P M = m P M +∆ where p does not divide m and [∆ , P M ] = 0. Note that every irreducible constituentof χ P M contains M in its kernel, and hence restricts irreducibly to P . Inparticular, [ χ P , P ] = m is not divisible by p . By assumption, χ does notvanish in P (so in P M/M as a character of
G/M ). By minimality of G , weconclude that P M ⊳ G . Hence O p ( G ) = 1.Let N be a minimal normal subgroup of G and write Q = P ∩ N ∈ Syl p ( N ). By the paragraph above we have that P N ⊳ G . Case 1:
Suppose that p divides the order of N . Since N is not a p -groupbecause O p ( G ) = 1, we have that N is semisimple. Let S ⊳ N be a minimalnormal subgroup of G . Then N = Q ri =1 S g i where { S g i } ri =1 = { S g | g ∈ G } and we may assume g = 1. In fact, N = × ri =1 S g i . Write R = Q ∩ S ∈ Syl p ( S ), and note that R >
1. Since S is a non-abelian simple group, byTheorem 2.3 either (a) S has a p -defect zero character θ or (b) S has anAut( S )-invariant θ ∈ Irr( S ) such that p does not divide [ θ R , R ] and θ ( x ) = 0for some x ∈ R .In case (a), note that η = × ri =1 θ g i ∈ Irr( N ) has p -defect zero. ThenLemma 2.4(b) yields a contradiction. In case (b), write η = × ri =1 θ g i ∈ Irr( N ). Note that η is G -invariant as θ is Aut( S )-invariant. Moreover,[ η Q , Q ] = [ θ R , R ] r is not divisible by p and η ( y ) = 0 where y = Q ri =1 x g i ∈ Q . Since ( η P N ) P = ( η Q ) P contains P with multiplicity [ η Q , Q ] coprimeto p , we can choose τ ∈ Irr(
P N ) lying over η such that p does not divide[ τ P , P ]. Let χ ∈ Irr( G ) lie over τ . By Lemma 2.4(a) we have that p doesnot divide [ χ P , P ]. As η is G -invariant, hence χ N = eη and so χ ( y ) = 0 for y ∈ Q ⊆ P , yielding a contradiction also in this case. Case 2:
We are left to deal with the case where N is a p ′ -group. Take K/N a minimal normal subgroup of
G/N with K ⊆ P N . Write Q = P ∩ K ∈ Syl p ( K ). In particular, K = N Q and Q ∼ = K/N is a p -elementaryabelian group. By the Frattini argument G = K N G ( Q ) = N N G ( Q ), andtherefore it is easy to see that C Q ( N ) ⊳ G . The fact that O p ( G ) = 1 forces C Q ( N ) to be trivial, and consequently the action of Q on N is faithful. By[DPSS09, Lemma 2.8] there is some θ ∈ Irr( N ) with K θ = N . (Note thatthe hypotheses of [DPSS09, Lemma 2.8] are fulfilled as Q acts coprimelyand faithfully on N , Q is abelian and N is characteristically simple.) Let EUGENIO GIANNELLI, STACEY LAW, JASON LONG, AND CAROLINA VALLEJO η = θ K ∈ Irr( K ). Then η has p -defect zero as a character of K and Lemma2.4(b) yields the final contradiction. (cid:3) Sylow branching coefficients of S n and A n The main aim of this section is to prove Theorem B. Using Theorem B, wethen complete the proof of Theorem A by showing that Theorem 2.3 holdsfor alternating groups at the primes 2 and 3. This is done in Theorem 3.17.We start by recording some notation and standard facts that will be usedthroughout this section.3.1.
Preliminaries.
For m a natural number we denote by [ m ] the set { , , . . . , m } . For p a prime, ν p ( m ) denotes the p -adic valuation of m ,i.e. m = p ν p ( m ) t where p ∤ t . For a finite group G and a prime number p , we write Irr p ′ ( G ) = { χ ∈ Irr( G ) : p ∤ χ (1) } for the set of irreduciblecharacters of G of degree coprime to p . For a p -block B of G , let Irr ( B )denote the set of height zero characters in B . Recall that if B has defectgroup D then the height ht( χ ) of an irreducible character χ is given byht( χ ) = ν p (cid:0) χ (1) (cid:1) + ν p ( | D | ) − ν p ( | G | ). As is customary we denote by g G theconjugacy class of the element g in G .We start by recording a group-theoretical result that will be used in theproof of Theorem B. Lemma 3.1.
Let G be a finite group, let p be a prime and let P ∈ Syl p ( G ) .Let g be an element of P . Then ν p (cid:0) | P ∩ g G | (cid:1) = ν p (cid:0) | g G | (cid:1) .Proof. From the definition of induced character [Isa06, Chapter 5], we have( P ) G ( g ) = | C G ( g ) | · | g G ∩ P || P | = | G : P | · | g G ∩ P || g G | . We then observe that ( P ) G ( g ) equals the number of fixed left cosets of P in G under the action of h g i by left multiplication. It follows that ( P ) G ( g )is coprime to p . The statement then follows. (cid:3) Characters and combinatorics of S n . We let P ( n ) denote the set ofpartitions of n . Given λ ∈ P ( n ) (also written λ ⊢ n ) we denote its conjugateby λ ′ . The set Irr( S n ) of ordinary irreducible characters of S n is naturally inbijection with P ( n ). For a partition λ ∈ P ( n ), we denote the correspondingirreducible character of S n by χ λ . Given P n a Sylow p -subgroup of S n and ϕ ∈ Irr( P n ), we use the notation introduced in [GL20] by letting Z λϕ denotethe natural number defined by Z λϕ := [( χ λ ) P n , ϕ ] . These multiplicities are called Sylow branching coefficients for symmetricgroups. In this article we will be particularly interested in the case where ϕ = P n is the trivial character of P n . We will sometimes use the symbol Z λ to denote Z λ Pn , to ease the notation.To each partition λ = ( λ , . . . , λ k ) we may associate a Young diagramgiven by [ λ ] = { ( i, j ) ∈ N × N : 1 ≤ i ≤ k, ≤ j ≤ λ i } . The hook of λ corresponding to the node ( i, j ) is denoted by h i,j ( λ ) and we let | h i,j ( λ ) | denote its size. For any e ∈ N , we denote by C e ( λ ) the e -core of the partition λ . This isobtained from λ by successively removing hooks of size e (also called e -hooks)until there are no further removable e -hooks. We say that λ is an e -corepartition if λ = C e ( λ ). The leg length of a hook is one less than the numberof rows it occupies. The e -weight of λ is given by w e ( λ ) = ( | λ | − | C e ( λ ) | ) /e .We refer the reader to [JK81] or [Ols94] for detailed descriptions of thesecombinatorial objects.We record here some useful facts on the degrees of irreducible charactersof S n . Let p be a prime and let λ ∈ P ( n ). An immediate consequence ofthe hook length formula [Jam79, 20.1] is that χ λ has p -defect zero if andonly if λ is a p -core partition. At the other end of the spectrum, the setof irreducible characters of S n of degree not divisible by p was completelydescribed in [Mac71]. We recall this result in language convenient for ourpurposes. Lemma 3.2.
Let p be a prime and n ∈ N . Let n = P ti =1 a i p n i be its p -adicexpansion, where n > n > · · · > n t ≥ and a i ∈ [ p − for all i ∈ [ t ] .Let λ ∈ P ( n ) and let µ = C p n ( λ ) . Then χ λ ∈ Irr p ′ ( S n ) if and only if µ ∈ P ( n − a p n ) and χ µ ∈ Irr p ′ ( S n − a p n ) . Repeated applications of Lemma 3.2 imply the following statement.
Lemma 3.3.
Let n be a natural number, let p be a prime and let λ ∈ P ( n ) .If | C p ( λ ) | ≥ p then p divides χ λ (1) . The Murnaghan–Nakayama rule [JK81, 2.4.7] allows us to compute thevalues of the irreducible characters of S n . Theorem 3.4 (Murnaghan–Nakayama rule) . Let r, n ∈ N with r < n .Suppose that πρ ∈ S n where ρ is an r -cycle and π is a permutation of theremaining n − r numbers. Then χ λ ( πρ ) = X µ ( − h ( λ \ µ ) χ µ ( π ) where the sum runs over all partitions µ obtained from λ by removing an r -hook, and h ( λ \ µ ) denotes the leg length of the hook removed. The cycle types of elements in S n are naturally parametrised by thepartitions of n . Since the order of cycles is irrelevant, when we refer tocycle types we may sometimes use compositions rather than partitions of n .We further remark that if σ ∈ S n has cycle type given by the composition α = (1 m m . . . ), then we say that σ contains exactly m i i -cycles, for all i ∈ N . Moreover, its centraliser has size | C S n ( σ ) | = Q i ∈ N i m i · m i !.3.1.2. Characters and blocks of S n and A n . Let p be a prime. It is wellknown that the p -blocks of S n are parametrised by p -core partitions [JK81,6.1.21]. In this article we will denote the p -block corresponding to the p -core γ by B ( γ, w ), where w is the natural number such that n = | γ | + pw . Asexplained in [JK81, 6.2.39], defect groups of B ( γ, w ) are Sylow p -subgroupsof S pw . Moreover, the set Irr ( B ( γ, w )) can be described as follows. Lemma 3.5.
Let n be a natural number and let p be a prime. Let γ be a p -core partition such that n = | γ | + pw , for some w ∈ N . Let pw = P ti =1 a i p n i EUGENIO GIANNELLI, STACEY LAW, JASON LONG, AND CAROLINA VALLEJO be its p -adic expansion, where n > n > · · · > n t ≥ and a i ∈ [ p − , forall i ∈ [ t ] . Given λ ∈ P ( n ) and µ = C p n ( λ ) , we have that χ λ ∈ Irr ( B ( γ, w )) if and only if χ µ ∈ Irr ( B ( γ, w − a p n − )) . Proof.
This follows from [Ols76, Lemma 3.1]. (cid:3)
In other words, Lemma 3.5 tells us that χ λ ∈ Irr ( B ( γ, w )) if and only ifthere exists a sequence of partitions λ = λ , λ , . . . , λ a = µ such that λ i +1 is obtained by removing a p n -hook from λ i , and such that λ a labels anirreducible character of height zero in B ( γ, w − a p n − ). When p = 2, wehave the following. Lemma 3.6.
Let n be a natural number and let λ ∈ P ( n ) be such that λ = C ( λ ) . If χ λ is an irreducible character of height zero in its -block,then λ = λ ′ .Proof. Let χ λ ∈ Irr( B ( γ, w )), for some 2-core γ and some w ∈ N . Let2 w = 2 n + 2 n + · · · + 2 n t be the binary expansion of 2 w where t ∈ N and n > n > · · · > n t ≥
1. By Lemma 3.5, λ has a unique 2 n -hook. Assumefor a contradiction that λ = λ ′ . Then | h i,i ( λ ) | is odd for all i ∈ N . Hencethe unique 2 n -hook of λ is off the main diagonal, i.e. it is h i,j ( λ ) for some i = j . This contradicts the assumption that λ = λ ′ . (cid:3) We now briefly recall a description of the irreducible characters and blocksof alternating groups, and refer the reader to [Ols90, Section 4] for furtherdetail. Let λ ∈ P ( n ). If λ = λ ′ then ( χ λ ) A n = ( χ λ ′ ) A n is an irreduciblecharacter of A n . On the other hand, if λ = λ ′ then ( χ λ ) A n = ϕ + λ + ϕ − λ with ϕ ± λ ∈ Irr( A n ). All of the irreducible characters of A n are of one of thesetwo forms. Turning to blocks, we let B ( γ, w ) be a p -block of S n and wefirst suppose that p is odd. If w > B ( γ, w ) covers a unique blockˆ B of A n . Moreover, B ( γ, w ) and B ( γ ′ , w ) are the only blocks covering ˆ B .If w = 0 then B ( γ,
0) covers a unique block of A n , unless γ = γ ′ . In thelatter case B ( γ,
0) covers two blocks of A n respectively containing the twoirreducible constituents of ( χ γ ) A n . Finally, if B ( γ, w ) covers ˆ B then theirdefect groups are isomorphic. On the other hand, if p = 2 then γ = γ ′ . Inparticular we have that B ( γ, w ) covers a unique block of A n if and only if w >
0. Moreover, if D is a defect group of B ( γ, w ), then D ∩ A n is a defectgroup of any 2-block of A n covered by B ( γ, w ).3.2. Virtual characters of S n . As mentioned in the introduction, thefollowing definition will play a central role in the proof of Theorem B.
Definition 3.7.
Let λ be any partition and let e ∈ N . Set n := | λ | + e . Welet V λ [ e ] be the virtual character of S n defined as follows: V λ [ e ] := X α ( − h ( α \ λ ) χ α where α runs over all partitions of n obtained from λ by adding an e -hook.As before, h ( α \ λ ) denotes the leg length of the e -hook added. Example 3.8.
Let λ = (3 ,
1) and e = 3. We observe that λ has exactly threeaddable 3-hooks. In particular, we have that V (3 , [3] = χ (6 , − χ (3 , , + χ (3 , ) . ♦ We describe the values taken by the virtual characters just introduced.
Theorem 3.9.
Let λ be any partition and let e ∈ N . Let n = | λ | + e and let σ ∈ S n . Suppose that the disjoint cycle decomposition of σ contains exactly k e -cycles. Then V λ [ e ]( σ ) = ( ke · χ λ ( τ ) if k > , k = 0 , where τ ∈ S n − e has cycle type equal to that of σ except with one fewer e -cycle. We observe that Theorem 3.9 extends [Jam79, Theorem 21.7]. The re-sult may be known to experts in the field, but we could not find it in theliterature. To prove Theorem 3.9, we use results from [Sta99], translatingbetween the language of symmetric polynomials and class functions. Webriefly summarise here the relevant notation.For a partition µ , we let s µ denote the corresponding Schur function.For e ∈ N , p e denotes the power sum symmetric function P i x ei in indeter-minates x i . If µ = ( µ , µ , . . . , µ k ), then p µ is defined to be the product p µ p µ · · · p µ k . The Frobenius characteristic map ch is a ring isomorphismbetween the algebra of class functions of finite symmetric groups and thering of symmetric functions (for more detail, see [Sta99, § f is aclass function of S n , then ch( f ) = X µ ⊢ n z − µ f ( µ ) p µ , (3.1)where z µ = | C S n ( ω ) | with ω ∈ S n an element of cycle type µ . In particular,ch( χ λ ) = s λ , for any partition λ . Moreover, for all class functions f of S m and g of S n we have that ch satisfies ch( f ◦ g ) = ch( f ) · ch( g ). Here f ◦ g denotes the induced class function ( f × g ) S m + n S m × S n . Proof of Theorem 3.9.
From [Sta99, Theorem 7.17.1], we have that s λ · p e = X α ( − h ( α \ λ ) s α where the sum runs over all partitions α obtained from λ by adding an e -hook. It is easy to see from (3.1) that if ch( f ) = p e , then f is the classfunction of S e given by f ( ω ) = ( e if the cycle type of ω is ( e ) , . It follows thatch( χ λ ◦ f ) = ch( χ λ ) · ch( f ) = s λ · p e = X α ( − h ( α \ λ ) ch( χ α ) . Since ch is bijective and linear, we have that χ λ ◦ f = V λ [ e ] . Thus it remainsto prove that ( χ λ ◦ f )( σ ) = ( ke · χ λ ( τ ) if k > , k = 0 , where τ ∈ S n − e has cycle type equal to that of σ except with one fewer e -cycle. Since χ λ ◦ f = ( χ λ × f ) S n + e , this follows directly from [Isa06, (5.1)]and the definition of f given above. (cid:3) We now extend Definition 3.7 by allowing the addition of multiple hooks.
Definition 3.10.
Let λ be any partition. For e, f ∈ N , define V λ [ e, f ] := X α ( − h ( α \ λ ) X β ( − h ( β \ α ) χ β . Here α runs over all partitions obtained from λ by adding an e -hook. Foreach fixed such α , we have β running over partitions obtained from α byadding an f -hook.We define V λ [ e , e , . . . , e u ] analogously for any u ∈ N ∪ { } and anysequence of natural numbers e , . . . , e u . If u = 0 , we set V λ [ e , . . . , e u ] := χ λ . Observe that V λ [ e, f ] = P α ( − h ( α \ λ ) V α [ f ]. It is then easy to seethat Theorem 3.9 implies V λ [ e, f ] = V λ [ f, e ]. Similarly, we have that V λ [ e , . . . , e u ] = V λ [ e ρ (1) , . . . , e ρ ( u ) ] for any ρ ∈ S u . Example 3.11.
Following on from Example 3.8, we can compute V (3 , [3 , S : V (3 , [3 ,
3] = ( − · V (6 , [3] + ( − · V (3 , , [3] + ( − · V (3 , ) [3]= χ (9 , + χ (6 , − χ (6 , , + 2 χ (6 , ) + χ (4 , , + 2 χ (3 , , − χ (3 , , , ) − χ (3 , , , , + χ (3 , ) . ♦ We now turn to the study of the restriction to Sylow p -subgroups ofthe virtual characters introduced in Definition 3.10. As mentioned at thebeginning of Section 3, we will use the symbol P n to denote a fixed Sylow p -subgroup of S n . Theorem 3.12.
Let p be a prime, let n ∈ N and let γ be a p -core partitionsuch that | γ | ≤ n and p | n − | γ | . Let u ∈ N ∪ { } and t , . . . , t u ∈ N be suchthat n − | γ | = p t + · · · + p t u . Then p ∤ (cid:2) V γ [ p t , . . . , p t u ] P n , P n (cid:3) . Proof. If | γ | = n then u = 0 and V γ [ p t , . . . , p t u ] = χ γ is a p -defect zerocharacter. Then p does not divide [( χ γ ) P n , P n ] by Lemma 2.1.We now assume that | γ | < n and u >
0. To ease the notation we let e i = p t i , for all i ∈ [ u ]. We recall that χ γ is zero on every non-trivialelement of P | γ | since γ is a p -core partition. By repeated applications ofTheorem 3.9, we find that V γ [ e , . . . , e u ] is zero on every element of P n except those with cycle type ( e , e , . . . , e u , | γ | ). Let σ ∈ P n have cycletype ( e , e , . . . , e u , | γ | ). Let T σ be the class function of P n taking value 1 on those elements with the same cycle type as σ and 0 otherwise. Then V γ [ e , . . . , e u ] P n = zT σ for some z ∈ N . Thus (cid:2) V γ [ e , . . . , e u ] P n , P n (cid:3) = [ zT σ , P n ] = 1 | P n | X τ ∈ P n zT σ ( τ ) = z | P n | · | P n ∩ σ S n | . On the other hand, from Theorem 3.9 we also have that V γ [ e , . . . , e u ]( σ ) = χ γ (1) · Y i ≥ ( i a i · a i !)where a i = |{ j ∈ [ u ] : e j = i }| . Hence z = zT σ ( σ ) = V γ [ e , . . . , e u ]( σ ) = χ γ (1) · | C S n ( σ ) || γ | ! . Therefore, to conclude the proof it suffices to show that ν p (cid:0) χ γ (1) (cid:1) − ν p ( | γ | !) + ν p (cid:0) | C S n ( σ ) | (cid:1) − ν p ( | S n | ) + ν p (cid:0) | P n ∩ σ S n | (cid:1) = 0 , (3.2)where we have used that ν p ( | P n | ) = ν p ( | S n | ). Since γ is a p -core, we have ν p (cid:0) χ γ (1) (cid:1) = ν p ( | γ | !). Thus (3.2) follows from the Orbit–Stabiliser theorem(giving | S n | = | σ S n | · | C S n ( σ ) | ) and Lemma 3.1. (cid:3) Starting with Definition 3.7, in this section we introduced a family ofvirtual characters of S n and we have studied their properties. The mainingredient of our proof of Theorem B for symmetric groups (Corollary 3.14below) is a specific member of this family. We highlight this specific virtualcharacter in the following definition. Definition 3.13.
Let γ be a p -core partition and let n = | γ | + wp for someinteger w > . Let B = B ( γ, w ) be the p -block of S n labelled by γ . Let wp = P i ≥ a i p i with a i ∈ { , , . . . , p − } for each i . We denote by V B thevirtual character of S n defined as V B = V γ [ e , . . . , e u ] , where e , . . . , e u are natural numbers such that |{ j ∈ [ u ] : e j = p i }| = a i forall i ≥ , and u = P i a i . In other words, the numbers e , . . . , e u in Definition 3.13 are the various p i appearing in the p -adic expansion wp = P i ≥ a i p i , counted with multiplicity.We remark that every irreducible character χ ∈ Irr( S n ) appearing withnon-zero coefficient in V B belongs to Irr ( B ). This follows directly fromLemma 3.5. Corollary 3.14.
Theorem B holds when G is a finite symmetric group.Proof. Let G = S n and suppose B = B ( γ, w ) for some p -core γ and w ≥ w = 0 then χ γ ∈ Irr ( B ) since γ is a p -core partition. By Lemma 2.1, p ∤ [( χ γ ) P n , P n ].Now suppose w >
0. Consider the virtual character V B = V γ [ e , . . . , e u ],introduced in Definition 3.13, and note that e i > e i is a power of p for every i ∈ [ u ]. It follows from Theorem 3.12 that p ∤ [( V B ) P n , P n ].Hence, there exists an irreducible character χ occurring in V B that satisfies p ∤ [ χ P n , P n ]. As remarked after Definition 3.13 we know that χ ∈ Irr ( B ),as desired. (cid:3) We are now ready to treat the case of alternating groups, which willconclude the proof of Theorem B.
Corollary 3.15.
Theorem B holds when G is a finite alternating group.Proof. Let G = A n and suppose B is a p -block of A n . Let ¯ B = B ( γ, w ) bea block of S n covering B , and let ¯ D and D = ¯ D ∩ A n be defect groups of ¯ B and B respectively. Let ¯ P ∈ Syl p ( S n ) and P := ¯ P ∩ A n ∈ Syl p ( A n ).First assume p is odd. Then P = ¯ P and D = ¯ D . By Corollary 3.14,there exists χ λ ∈ Irr ( ¯ B ) such that p ∤ [( χ λ ) ¯ P , ¯ P ]. If λ = λ ′ then ϕ :=( χ λ ) A n ∈ B . Since χ λ ∈ Irr ( ¯ B ) we have that ϕ ∈ Irr ( B ), as | ¯ D | = | D | and p is odd. Moreover, p ∤ [ ϕ P , P ] = [( χ λ ) ¯ P , ¯ P ]. On the other hand, if λ = λ ′ then ( χ λ ) A n = ϕ + λ + ϕ − λ and at least one of ϕ + λ and ϕ − λ belongs to B .Note ϕ + λ (1) = ϕ − λ (1) = χ λ (1). Since p is odd we have that both ϕ + λ and ϕ − λ are height zero characters in their block. Moreover, p ∤ [( ϕ + λ ) P , P ] =[( ϕ − λ ) P , P ] = [( χ λ ) ¯ P , ¯ P ].Now assume p = 2. First suppose w >
0. Let 2 w = 2 n + 2 n + · · · + 2 n t bethe binary expansion of 2 w where t ∈ N and n > · · · > n t ≥
1. Let V ¯ B = V γ [2 n , . . . , n t ]. By Lemma 3.5, every irreducible character χ λ ∈ Irr( S n )occurring in the linear combination V ¯ B belongs to Irr ( ¯ B ). Moreover, λ = λ ′ for all such characters χ λ by Lemma 3.6. Since γ is self-conjugate, if χ λ occurs in V ¯ B then so does χ λ ′ . In particular, we can write V ¯ B as asum of terms of the form ± ( χ λ + χ λ ′ ) or ( χ λ − χ λ ′ ) for various λ . Since2 ∤ [( V ¯ B ) ¯ P , ¯ P ] by Theorem 3.9, we deduce that 2 ∤ [( χ λ ) ¯ P , ¯ P ]+ [( χ λ ′ ) ¯ P , ¯ P ]for some χ λ ∈ Irr ( ¯ B ). Let ϕ := ( χ λ ) A n . Since | D | = | ¯ D | , we deduce that ϕ ∈ Irr ( B ). Moreover,[ ϕ P , P ] = [( χ λ ) ¯ P , ¯ P ] + [( χ λ ′ ) ¯ P , ¯ P ] . We conclude that 2 ∤ [ ϕ P , P ], as desired.Finally, if p = 2 and w = 0, then D = 1 and B contains a uniqueirreducible character ϕ that is at the same time of p -defect zero and ofheight zero in B . By Lemma 2.1 we have that 2 ∤ [ ϕ P , P ]. The proof isconcluded. (cid:3) As promised at the start of Section 3, we use Theorem B to prove The-orem 3.17 and thereby complete the proof of Theorem A. In order to dothis, we first show that irreducible characters in symmetric and alternat-ing groups of degree coprime to p are characterised by the non-vanishingproperty on Sylow p -subgroups. Theorem 3.16.
Let G be a finite symmetric or alternating group, p be aprime and P ∈ Syl p ( G ) . Then χ ∈ Irr( G ) has degree coprime to p if andonly if χ ( x ) = 0 for every x ∈ P .Proof. Let χ ∈ Irr( G ). By Lemma 2.2, we need to prove that if χ ( x ) = 0for every x ∈ P then p ∤ χ (1).Let n ∈ N with p -adic expansion n = P ti =1 a i p n i , where n > n > · · · >n t ≥ a i ∈ [ p −
1] for all i ∈ [ t ]. Since the theorem holds trivially for n < p , we assume from now on that n ≥ p . We call an element g ∈ S n a p -adic element if in the disjoint cycle decomposition of g there are a i cycles of length p n i for each i ∈ [ t ]. We proceed by splitting the proof into twocases according to G = S n or G = A n . (i) G = S n : Given χ ∈ Irr( S n ) with p | χ (1), we claim that χ ( g ) = 0 forany p -adic element g ∈ P . We show that the above claim holds by inductionon t , the p -adic length of n . Let χ = χ λ for some λ ⊢ n .If t = 1, then n = ap k for some a ∈ [ p −
1] and k ∈ N . In this setting g hascycle type ( p k , p k , . . . , p k ) (i.e. g is the product of a cycles of length p k ). Since p divides χ λ (1), by Lemma 3.2 we have that | C p k ( λ ) | >
0. Equivalently,the p k -weight of λ is strictly smaller than a . Hence it is not possible tosuccessively remove a p k -hooks from λ . Using the Murnaghan–Nakayamarule, we conclude that χ λ ( g ) = 0.Let us now assume that t ≥ t −
1. In thissetting we have that g = ρπ , where ρ is the product of a cycles of length p n and π is a p -adic element of S n − a p n . Clearly, the p n -weight w p n ( λ )of λ is smaller than or equal to a . If w p n ( λ ) < a then χ λ ( g ) = 0 bythe Murnaghan–Nakayama rule. Otherwise, w p n ( λ ) = a and ν := C p n ( λ )is a partition of n − a p n . Since p divides χ λ (1), Lemma 3.2 implies that p | χ ν (1). The inductive hypothesis now guarantees that χ ν ( π ) = 0. Thisconcludes the proof of our claim, as another application of the Murnaghan–Nakayama rule shows that there exists k ∈ Z such that χ λ ( g ) = k · χ ν ( π ) = 0.We can now conclude that if χ λ ( x ) = 0 for all x ∈ P then χ λ ∈ Irr p ′ ( S n ),because we can always find a p -adic element of S n lying in P . (ii) G = A n : If p is odd then P ∈ Syl p ( A n ) is also a Sylow p -subgroup of S n ,and hence P contains a p -adic element g . Consider ϕ ∈ Irr( A n ) with p | ϕ (1),and let λ ⊢ n be such that ϕ is an irreducible constituent of ( χ λ ) A n . If λ = λ ′ ,then ϕ = ( χ λ ) A n and hence ϕ ( g ) = χ λ ( g ) = 0, as p divides χ λ (1). If λ = λ ′ then p divides χ λ (1) = 2 ϕ (1). Let δ := ( | h , ( λ ) | , | h , ( λ ) | , . . . , | h ℓ,ℓ ( λ ) | ) bethe partition of n given by the diagonal hook lengths of λ . We observe that δ cannot be equal to the cycle type of g . This is clear whenever a i ≥ i ∈ [ t ], because | h j,j ( λ ) | > | h j +1 ,j +1 ( λ ) | for all j ∈ [ ℓ − a i = 1 for all i ∈ [ t ], then Lemma 3.2 shows that we cannothave | h i,i ( λ ) | = p n i for all i ∈ [ t ], because p divides χ λ (1). It follows that ϕ ( g ) = χ λ ( g ) = 0, by [JK81, 2.5.13].If p = 2, then P is a subgroup of index 2 of a Sylow 2-subgroup of S n .Let ϕ ∈ Irr( A n ) with 2 | ϕ (1) and let λ ⊢ n be such that ϕ is an irreducibleconstituent of ( χ λ ) A n . We observe that χ λ (1) is even. From Lemma 3.2 wededuce that there exists s ∈ [ t ] such that | C s ( λ ) | > n − (2 n + · · · + 2 n s ).Let r = min { s ∈ [ t ] : | C s ( λ ) | > n − (2 n + · · · + 2 n s ) } . We now pick g ∈ S n of cycle type (2 n , n , . . . , n t ) if t is even , (2 n , n − , n − , n , . . . , n t ) if t is odd and r = 1 , (2 n − , n − , n , n , . . . , n t ) if t is odd and r > . We observe that such a g is an even permutation and therefore can befound in P . Moreover, we have that χ λ ( g ) = 0 by the Murnaghan–Nakayama rule. It follows that if λ = λ ′ then ϕ ( g ) = χ λ ( g ) = 0. Finally, if λ = λ ′ thenclearly | h i,i ( λ ) | is odd for all i ∈ [ t ], hence we cannot have h i,i ( λ ) = 2 n i forall i ∈ [ t ]. Therefore ϕ ( g ) = χ λ ( g ) = 0, by [JK81, 2.5.13]. (cid:3) Theorem 3.17.
Let p ∈ { , } and let n ≥ be a natural number. Let R be a Sylow p -subgroup of A n . Then A n either possesses a p -defect zerocharacter or there is some Aut( A n ) -invariant θ ∈ Irr( A n ) such that [ θ R , R ] is not divisible by p and θ ( x ) = 0 for some x ∈ R .Proof. From Theorem 3.16 we know that for θ ∈ Irr( A n ) the condition θ ( x ) = 0 for some x ∈ R is equivalent to having that p divides θ (1). We alsorecall that for every n = 6, we have that θ is Aut( A n )-invariant if and onlyif θ = ( χ λ ) A n for some partition λ of n such that λ = λ ′ .We distinguish two cases, depending on the value of p ∈ { , } . (i) p = 2 : If n = 6 then both constituents of ( χ (3 , , ) A are 2-defect zeroirreducible characters of A . Suppose now that n = 6. Let γ be the following2-core partition: γ = ( (2 ,
1) if n is odd , (3 , ,
1) if n is even . Let 2 w = n − | γ | and let 2 w = 2 n + 2 n + · · · + 2 n t be its binary ex-pansion, where t ∈ N and n > n > · · · > n t ≥
1. Let B := B ( γ, w )be the corresponding 2-block of S n . By Lemma 3.6, if χ λ ∈ Irr ( B ) then λ = λ ′ . Considering as in the proof of Corollary 3.15 the virtual character V γ [2 n , n , . . . , n t ], we deduce that there exists λ ∈ P ( n ) labelling a char-acter in Irr ( B ) such that the sum of Sylow branching coefficients Z λ + Z λ ′ is odd. Since | γ | ≥ χ λ (1) is even, by Lemma 3.3. Weconclude that θ = ( χ λ ) A n is an irreducible character of A n with the desiredproperties, since [ θ R , R ] = Z λ + Z λ ′ . (ii) p = 3 : Direct verification shows that the statement holds for n ∈{ , , } . More precisely, the alternating groups A and A admit 3-defectzero irreducible characters. On the other hand, θ = ( χ (8 , , ) A is such that3 | θ (1) and 3 ∤ [ θ R , R ] where R ∈ Syl ( A ).Now suppose that n / ∈ { , , } . Let γ be the following 3-core partition: γ = (4 ,
2) if n ≡ , (3 ,
1) if n ≡ , (6 , , , ,
1) if n ≡ . Let B be the 3-block of S n labelled by γ . Since γ = γ ′ we have that everyirreducible character in Irr ( B ) is labelled by a non-self-conjugate partition.By Theorem B there exists χ λ ∈ Irr ( B ) such that Z λ is coprime to 3. Since | γ | ≥ χ λ (1), by Lemma 3.3. We conclude that θ = ( χ λ ) A n is an irreducible character of A n with the desired properties. (cid:3) Acknowledgments.
Part of this work was done while the fourth authorwas visiting the first at the University of Florence supported by the Span-ish National Research Council through the “Ayuda extraordinaria a Cen-tros de Excelencia Severo Ochoa” (20205CEX001). The third author was supported by ERC Consolidator Grant 647678. The fourth author is par-tially supported by the Spanish Ministerio de Ciencia e Innovaci´on PID2019-103854GB-I00 and FEDER funds. We thank Gabriel Navarro and ThomasBreuer for helping us checking Theorem 2.3 for some sporadic groups. More-over, we would like to thank Gunter Malle for comments on a previousversion. References [DPSS09]
S. Dolfi, E. Pacifici, L. Sanus and P. Spiga,
On the orders of zeros ofirreducible characters,
J. Algebra (2009), 345–352.[GAP] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0;2020, .[GKNT17]
E. Giannelli, A. Kleshchev, G. Navarro and P.H. Tiep,
Restriction ofodd degree characters and natural correspondences,
Int. Math. Res. Not. (2017),no. 20, 6089–6118.[GL18]
E. Giannelli and S. Law,
On permutation characters and Sylow p -subgroupsof S n , J. Algebra (2018), 409–428.[GL20]
E. Giannelli and S. Law,
Sylow branching coefficients for symmetric groups,to appear in
J. London Math. Soc. (2) 00 (2020), 1–32, DOI:10.1112/jlms.12389.[GN18]
E. Giannelli and G. Navarro,
Restricting irreducible characters to Sylow p -subgroups, Proc. Amer. Math. Soc. (2018), no. 5, 1963–1976.[GO96]
A. Granville and K. Ono,
Defect zero p -blocks for finite simple groups, Trans.Amer. Math. Soc. (1996), 331–347.[Isa06]
I. M. Isaacs,
Character theory of finite groups , AMS-Chelsea, Providence, 2006.[IN08]
I. M. Isaacs and G. Navarro,
Character sums and double cosets,
J. Algebra (2008) 3749–3764.[INOT17]
I. M. Isaacs, G. Navarro, J. B. Olsson and P. H. Tiep,
Character restric-tion and multiplicities in symmetric groups,
J. Algebra (2017), 271–282.[Itˆo51]
N. Itˆo,
Some studies of group characters,
Nagoya Math. J. (1951), 17–28.[Jam79] G. James,
The representation theory of the symmetric groups , Lecture Notes inMathematics, vol. 682, Springer, Berlin, 1978.[JK81]
G. James and A. Kerber,
The representation theory of the symmetric group ,Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Pub-lishing Co., Reading, Mass., 1981.[Mac71]
I. G. MacDonald,
On the degrees of the irreducible representations of sym-metric groups,
Bull. London Math. Soc. (1971), 189–192.[MN12] G. Malle and G. Navarro,
Characterizing normal Sylow p -subgroups by char-acter degrees, J. Algebra (2012), 402–406.[MN20]
G. Malle and G. Navarro,
On principal indecomposable degrees and Sylowsubgroups,
Arch. Math. , in press 2020.[MN21]
G. Malle and G. Navarro,
Brauer’s Height Zero Conjecture for PrincipalBlocks, arXiv:2102.08270 .[Mic86]
G. Michler,
Brauer’s conjectures and the classification of finite simple groups,
Representation theory, II (Ottawa, Ont., 1984) , 129–142, Lecture Notes in Math.,Springer, Berlin, 1986.[Nav10]
G. Navarro,
Problems in Character Theory, in
Character theory of finite groups ,Contemp. Math. , Amer. Math. Soc., Providence (2010), 97–125.[Nav18]
G. Navarro,
Character theory and the McKay conjecture , Cambridge UniversityPress, 2018.[NTV14]
G. Navarro, P. H. Tiep and C. Vallejo,
McKay natural correspondenceson characters,
Algebra Number Theory no. 8 (2014), 1839–1856.[Ols76] J. Olsson,
McKay numbers and heights of characters,
Math. Scand. (1976),25–42.[Ols90] J.B. Olsson , p -blocks of symmetric and alternating groups and their coveringgroups, J. Algebra (1990), no. 1, 188–213. [Ols94]
J. B. Olsson,
Combinatorics and representations of finite groups , Vorlesungenaus dem Fachbereich Mathematik der Universit¨at Essen, Heft 20, 1994.[Sta99]
R. P. Stanley,
Enumerative combinatorics , Volume 2, Cambridge UniversityPress, 1999.(E. Giannelli)
Dipartimento di Matematica e Informatica U. Dini, Viale Mor-gagni 67/a, Firenze, Italy
Email address : [email protected] (S. Law) Department of Pure Mathematics and Mathematical Statistics, Uni-versity of Cambridge, Cambridge CB3 0WB, UK
Email address : [email protected] (J. Long) Mathematical Institute, University of Oxford, Andrew Wiles Build-ing, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Email address : [email protected] (C. Vallejo) Departamento de Matem´aticas, Edificio Sabatini, UniversidadCarlos III de Madrid, Avenida Universidad 30, 28911, Legan´es. Madrid, Spain
Email address ::