THE NORMALISER ACTION AND STRONGLY MODULAR LATTICES Autor(en): Objekttyp: Nebe, Gabriele Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 43 (1997) Heft 1-: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: 911017 Persistenter Link: http://doiorg/105169/seals-637 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften Die Rechte liegen in der Regel bei den Herausgebern Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 809 Zürich, Schweiz, wwwlibraryethzch http://wwwe-periodicach
THE NORMALISER ACTION AND STRONGLY MODULAR LATTICES by Gabriele Nebe ABSTRACT We dérive group theoretical methods to test if a lattice is strongly modular We then apply thèse methods to the lattices of rational irreducible maximal finite groups 1 Introduction Let L Ç R^ be an even intégral lattice in the Euclidean space of dimension d and let L # Ç R^ be its dual lattice Let tt(l) be the set of ail intermediate lattices L<L' <L# that are inverse images of sums of Sylow subgroups of the finite abelian group L#/LL# /L Then, L is said to be strongly modular if L is similar to L' for ail U G tt(l) (cf [Que 96]) Recall that L and l! are called similar if there exists sg GL(RL) and a G R>o such that Ls =L1 and (vsws) = a(vw) for ail vw G RL, where () product The automorphism group dénotes the Euclidean scalar is conjugate to a finite subgroup of GL^(Z) Since G acts as group automorphisms on L?/L it préserves the lattices L' G tt(l) In Section 3 it is shown that the similarities L'? > L normalise G So one may use the normaliser *) Supported by the DFG
of G in GLd (Q) to test strong modularity of L In the next section we dérive some methods for explicitly constructing éléments of NglaQ)^) Every flnite subgroup of GL d (Q) is a subgroup of the automorphism group of an intégral lattice In particular the maximal finite subgroups of GLd (Q) are automorphism groups of distinguished lattices A subgroup of GL d (Q) is called rational irreducible if it does not préserve a proper subspace {0} of Q d The rational irreducible maximal finite, abbreviated to rimf, subgroups of GLd (Q) are classified for d < 3 (cf [PIN 95], [NeP 95], [Neb 95], [Neb 96], [Neb 96a]) Their invariant lattices provide many examples of strongly modular lattices The following theorem is proved by applying the methods derived in Section 4 THEOREM In dimension d < 3, ail even lattices L Ç K d that are preserved by a rimf group and satisfy L# /L = (Z/lZ) d / for some IGN the the lattices ]i6 are strongly modular, except for rimf group [± Alt6 of in GL l6 (Q) (cf [NeP 95]; -]i6 PRELIMINAIRES AND NOTATION The main strategy in this paper is the application of the following normaliser principle Let G be a group acting on a set S, H a subgroup of the group of trans formationsof S Then the normaliser of G in H acts on the set of G-orbits In our situation G = Aut(L) is the automorphism group of an intégral lattice Lin the Euclidean space RL = Rd By writing the action of Gon RL with respect to a Z-basis ( >i,, bd ) of L, G becomes a finite subgroup of GL d (Z) Then G= Aut(F) = {g G GLd (Z) gfg tr =F} where F is the \ = {ib i^bj)) i ==1 of L For the rest of this article let H = GL^(Q), G < //, be a finite subgroup of H, and let N := Nh(G) be its normaliser We also assume that G contains the négative unit matrix,?id G G We apply the normaliser principle to the following three situations Gram matrix F (i) S={LÇQd L=?=i Zfe / for a basis (b u,b d ) of Q^}, the of Z -lattices of rank din Q d, and the action of HonSis right Lh := {lh \le L} Then the set of multiplication :SxH->S, (L,h) set G-fixed points is the set of G -invariant lattices
F positive definite}, the set of positive definite symmetric matrices, where x tr dénotes the transposed matrix of (ii) S={FG Mrf(Q) F= F tr ', x G M d(q) and the action of H on S is 5 x H -> 5, (F, A) h-> /zf/z'r Then the set of G-fixed points is Note that (R >o )J?r >o(g) is the set of G-invariant Euclidean scalar products on R d Gis called uniform, if there is essentially oné G-invariant Euclidean structure on R J, that is if j^>0 (G) = {af o<agq} for some F G Af^(Q) (iii) S = Mj(Q), and the action of H is conjugation: S x H -^ S, (c,h) \-^ h~ l ch Then the set of G-flxed points is the commuting algebra of G The following two remarks follow immediately from the normaliser principle REMARK 1 Assume that G is uniform and let F G J> 0(G) Then for each n G N, the matrix nfn tr is also G-invariant and therefore nfn tr = (det(«)) F Hence n induces a similarity of F REMARK G-invariant For nen and LG Z(G), the lattice Ln G Z(G) is also 3 Similarities Normalise In this section we show that if G is the automorphism group of a (strongly modular) lattice L then the similarities between L and L' G tt(l) are éléments of N PROPOSITION 3 Let G= Aut(F) < GL d(z) be the full automorphism group of a lattice L Assume that L is an intégral lattice Let L' G n(l) and n G GL d(q) which induces a similarity from L' to L, Le Un = L and nfn tr = af, (a G N) Then a~ l n eg and nen Proof The matrix a l n is clearly orthogonal with respect to F Therefore to prove that a~ l n GGwe only hâve to show that La~ l n =L Now L' = Ln~ l, hence its dual lattice is
For /G L,v GQd we hâve vf{ln- l ) tr = va~ l nfl tr and hence (Z/) # = L # an~ l Since L7L7 G tt(l) one has L 1 =L#Da ILl L Using this one obtains since (L') # /L is the orthogonal complément of /L in Ll/LL1 if/l with respect to the induced quadratic form with values in Q/Z So a~ l n eg Finally we check that ngn Let ggg, then «! pn is in G= Aut(F) since Ln~ l gn = L f gn = Z/«=L and 4 Obtaining Elements of N Now we give examples as to how one may construct éléments n of the normaliser N To obtain similarities we are interested in n G TV of déterminant ±p d / for some (squarefree) natural number p such that p~ x n GG The first method is an application of the normaliser principle to the situation (iii) described in Section : PROPOSITION 4 Let U<Gbea normal subgroup of G and assume that the commuting algebra K := CM d (Q)(U) is isomorphic to a number field If c G K satisfies =p G Q*/^, then c lies in N cc Proof Since G normalises U, it acts by conjugation (and hence as Galois automorphisms) on the abelian number field K Now let c G K, with cc =: p g Q*/</ and g G G Then g stabilises the subfield Q[c] and hence g~ l cg = ±c, which is équivalent to c-i^c =±ggg Therefore cg A/", since we assumed that?/^ GG? The following construction described in [PIN 95] Proposition (II4) also allows us to find éléments of N For i= 1, let G t < GL di (Q) be finite rational irreducible matrix groups with commuting algebras Ai C M^(Q) Also let Q be a maximal common subalgebra of dimension zofai and A Let d:= and view the G; as subgroups of G\ Gi < GL <S> d(q) If there exist éléments at G NN GG Ld (Q){Gi)
centralising Gj and q(l</ <) and a squarefree natural number p 0 such that p~ l af G G;, the group generated by the éléments of G x OG and lai1 p aia is, a finite subgroup of Q GL d(q) containing G\ (g) GG as a subgroup of index Q is an élément of For d<3l and p>l we only need the case where aa 0?) (p) the enveloping algebra of G Then Gis denoted by G\ <8> GG (or Gi IX) GG ) according to whether a\ is (or is not) a rational linear combination of éléments of G x Using this notation one immediately has the following proposition PROPOSITION 5 For i= 1, /ze matrix at is an élément of déterminant jn t fo e normauser NofG -\-pd/ A common feature of the situations in Propositions 4 and 5 is that we extend the natural représentation of G to a projective représentation which is réalisable as a linear représentation over a quadratic extension of Q PROPOSITION 6 Let G<E be a supergroup containing G of index Assume that C Md (Q)(G) = Q and that the natural character of G extends to E with character field Q[y/p], where p G Z is not a square Then there exists n Nof déterminant ±p d / with p~ l n GG Proof By Clifford theory one may extend the natural représentation A of G to a représentation ô\ 06 E -> (Q[^/p] Af d(q))*, where : 6\ and 66 are projective représentations 6\(G)? {1} and (<5 ) G =A Let ee E\G Then since e G Therefore ee Bx{e8 x {e) G Q Replacing B\{é) by a suitable 6(e)6 rational multiple (and multiplying (e) by the inverse) one may assume that 6 { (e) = p~ ] 5(^)5 Then n:= (^) is an élément of the normaliser N with the desired properties?
Proof of the Theorem 5 In this section we prove the theorem stated in the Introduction The principle the of proof is given in the following remark G= Aut(F) < GLd (Z) be a uniform automorphism group L=Zd and let nen, where N is the normaliser of G Then REMARK 7 Let the lattice Ln Remark says that the lattice by Remark 1, n induces a similarity such that det(w) = [L : (LHLn)) [Ln : G U Let G also -invariant tt(l) Ln is l l Then [Z/ : (Z/ n Ln)} = [Ln : (L'HLn)} So (LnLn)]~ equals [L' L]~ there is no other G -invariant lattice M in the loyer conclude that, one of L^ : if may of L' (Le with [L 1 : (M n L')] = [M : (M n L')] ), then Ln = L' The last uniqueness condition is fulfilled if CMd (Q)(G) = Q, ail lattices in Z(G) are even, and G is lattice sparse according to the following définition In this case Z(G) = {al! L G tt(l), a G Q*} for any G -invariant lattice L # Note that this implies that the exponent exp(l /L) is square free 1 \ DEFINITION 8 prime then a finite group G< GLd (Q) is If called p -lattice sparse if any lattice L G Z(G) can be obtained from any other lattice in Z{G) that contains L of p-power index by a combination of the five opérations : taking sums, intersections, the dual lattice or the even sublattice with respect to some F G JSoCG), or multiplying by invertible éléments of CMd (Q)(G) The group G is called lattice sparse if G is p -lattice pisa sparse for ail primes p Since the proof of the Theorem is similar for ail rimf groups, we only = 16 and 4 deal with the most interesting cases d The rimf groups of degree 16 and 4 fixing strongly modular lattices, which are not proper tensor products, are displayed in the table The first column gives the number of the group under which it is referred to in [NeP 95] or [Neb 95] and [Neb 96] The second column contains a name for the matrix group as partially explained in the paragraph preceding Proposition 5 In the notation there we additional make the following abbreviations If z = d\ or d, we omit x and Q in the symbols Also (1) is omitted if The division algebra Q is abbreviated as a, if Q = Q[a], by the set of ramified primes, if Q is a quaternion algebra over Q, and omitted if Q = Q For the finite simple and quasisimple groups we use the notation of [CCNPW 85] p=l
except that the alternating group is denoted by Alt n to avoid confusion with A n, which also dénotes the automorphism group of the root lattice A n The next three columns give information about the invariant strongly modular lattice L First the déterminant det(l) is given as the product of the abelian invariants of the Sylow subgroups of # /L The next column contains min(l) the minimum of the square lengths of the non zéro vectors in L From thèse two columns one may see whether L is an extremal lattice as defined in [Que 96] The number of vectors of square length min(l) decomposed as a sum of the orbit lengths under Aut(L) is displayed in the fifth column The last column contains the primes p for which Aut(L) is p -lattice sparse A + indicates that Aut(L) is lattice sparse L#/LL Proof of the Theorem The commuting algebras of the groups in the table are ail isomorphic to Q except for the one of [6Alt7 : ] 4 which is Q[\/? 6] So ail thèse groups are uniform Since the arguments are similar 4 for ail groups Gwe only deal with G= (3) i [SL (5) K) SL (9)\ r 6 00,3 extensively Let L G Z(G) be a G-invariant lattice There is a unique F G FyoiL) such that {lifl r /i,/ GL}=Z The déterminant of L with respect to F is L# /L =38?58 and its minimum is 10 If this lattice is strongly modular, then it is an extremal strongly modular lattice ([Que 96]) Since G is of the form described in Proposition 5 with p= 3, there is an élément a\ G N = NGL]6 (Q)(G) with \a\ G G Since G is lattice sparse and r = 8 # one has La x =3L HLe Z(G) and aifa\r =3F (by Remarks # induces and 1) Hence a\ a similarity between L and 3L HL det(fli) Next consider the normal subgroup U := SL (5) muting algebra Cm (Q)(U)?'- K is isomorphic to 16 0 SL (9) 00,3 < G The c0m Q[\/s] From Proposition 4 one obtains an élément c G N with = 5 As above one concludes that # c yields a similarity between L and 5L DL The product U\c G Af is of 8 déterminant ±15 and gives a similarity between L and 15L# Therefore L cc is strongly modular Most of the other groups can be dealt with similarly One has to use Proposition 6 to construct an additional élément of TV for the rimf groups 4 and 14 of GL 16 (Q) For G = [ Alt lo ]i6 (number 14), one obtains ne N of déterminant ±58,5 8, since the character extends to S\ç> with character field Qh/±s] l s (cf [CCNPW 85]) Analogously, for F4 F4 = _+ O^() (num 5 the character extends to }+ 0 () with character field Q[^±] ber4) The strong modularity for the lattices of the rimf groups 9 and 1 of GLiô(Q) (in particular the similarity of L with the lattice corresponding to
the Sylow--subgroup of L#/L)L# /L) may be derived from the equality and using Proposition 5 Similarly one uses Proposition 5 to show the -modularity of the lattices of the rimf group 6 in GLa(Q) using the description For the groups 44 and 64, which are the only groups which are not p -lattice sparse for a relevant prime p (=), one has to note that the invariant sublattice 1 of index 1 in Lis unique The theorem now follows from the next lemma LEMMA 9 383 The lattices (of déterminant 8? 5 8 J of the rimf sub group G := [±Alt 6 ]i 6 < GLi 6 (Q) (number 0 of [NeP 95]) are not (strongly) modular Proof Let Lbe such a G -invariant lattice and L' G n(l) Assume that there is a similarity s : L 1 -» L By Proposition 3, this similarity s normalises G Let U= Alt6 be the characteristic subgroup = Alt 6 of G Since the full automorphism group of U is already induced by conjugation with éléments of G, there exists g G G, such that n := gs G GLi 6(Q) centralises U Hence «G Cmi 6 (Q)(U) = Q[\/s] Since this number field does not contain an élément 8 So the lattice Lis neither similar of norm 3, one concludes that [L 1 to L # :L]= 585 nor to the lattice L' G tt(l) corresponding to the 3-Sylow subgroup of L*/L Note that if [L' :L]= 58,5 induces a similarity by Proposition 4 8, an élément xg C M[dQ) (U) with xx n =5, REFERENCES [CCNPWBS] Conway, J H, R T Curtis, S P Norton, R A Parker and R A WILSON Atlas of Finite Groups Oxford University Press, 1985 [Neb9s] Nebe, G Endliche rationale Matrixgruppen vom Grad 4 Dissertation RWTH Aachen Aachener Beitràge zur Mathematik 1 (1995)
[Neb96] NEBE, G Finite subgroups of GL 4(Q) Exp Math 5 (1996), 163-195 [Neb96a] Finite subgroups of GL n (Q) for 5 <n< 31 Comm Alg 4 (7) (1996), 341-397 [NeP9s] NEBE, G and W PLESKEN Finite rational matrix groups of degree 16 [PIN 95] [Que 95] [Que 96] AMS Memoirs, vol 116, No 556 (1995) PLESKEN, W and G NEBE Finite rational matrix groups AMS Memoirs, vol 116, No 556 (1995) QUEBBEMANN, H-G Modular lattices in Euclidean spaces / Number Theory 54 (1995), 190-0 Atkin-Lehner eigenforms and strongly modular lattices L'Ens Math 43 (1997), 55-65 (Reçu le 5 juin 1996; version révisée reçue le novembre 1996) Gabriele Nebe Lehrstuhl B fur Mathematik RWTH Aachen Templergraben 64 506 Aachen Germany E-mail : gabi@willimathrwth-aachende