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Decomposable forms and automorphisms
Journal of Number Theory 99 (2003) 232–254
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Decomposable forms and automorphisms Leonhard Summerer Departement Mathematik, ETH Zurich, Zurich, Switzerland Received 26 April 2000; revised 19 August 2002
Abstract In the late 19th century Jordan initiated the study of forms of higher degree and derived (see Memoire sur l’equivalence des formes, Oeuvres III, Gauthier Villars, Paris, 1962) the finiteness of the automorphism group Autð f Þ of complex forms of degree X3 and non-zero discriminant. This result has been extended to forms over arbitrary fields by Schneider (J. Algebra 27 (1973) 112), see also Curtis and Reiner (Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962) for related topics. Orlik and Solomon gave some bounds for the cardinality of Autð f Þ using cohomological arguments in Orlik and Solomon (Math. Ann. 231 (1978) 229); besides this, little seems to be known about this group in general. In connection with his study (Monatsh. Math., submitted for publication) of representations of forms by linear forms, the author was led to an investigation of the group of automorphisms of decomposable forms f through the permutations of the linear factors these automorphisms induce. The main result (Theorem 4.2 in Chapter 4) states that almost all forms in kX2 variables of degree dXmaxf5; k þ 2g have only the trivial automorphisms that consist in multiplying each variable by the same dth root of unity. The case k ¼ 2; d ¼ 4 has already been studied (see Survey in Algebraic Geometry, Part 2, Invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer, Berlin, 1994); however, it is treated in full detail to illustrate the elaborated techniques. The first chapters are devoted to the proof of some general results concerning the structure of the permutation group associated to a form f which also help to understand the case of forms with non-trivial automorphisms. In a few special cases, this allows to determine this group explicitly; in general we give a bound for the cardinality of Autð f Þ depending only on the degree of f which is relevant for some diophantine problems (see e.g. Ann. Math. 155 (2002) 553). The author is indebted to G. Wuestholz for his substantial help and encouragement during the redaction of the paper, he also wishes to thank V. Popov for several helpful remarks. r 2002 Elsevier Science (USA). All rights reserved.
E-mail address: [email protected]. 0022-314X/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 7 3 - 2
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1. Group actions and automorphisms For an algebraically closed field K of characteristic 0 we consider a vector space V of dimension k over K and define Fd to be the set of decomposable forms of degree d > 0 with variables vAV and coefficients in K: This means that any f AFd admits a decomposition f ¼: l1 ?ld
with
l1 ; y; ld AV n :
We consider the map ðV n Þd -Fd ; ðl1 ; y; ld Þ /l1 ?ld ; it is surjective since K is algebraically closed but obviously not injective. However, the decomposition of f AFd in l1 ?ld is unique up to a scalar factor of the li and their order, which leads us to consider the following groups acting on f and its given decomposition. Let Sd be the symmetric group of order d and T :¼ fm :¼ ðm1 ; y; md ÞAðK n Þd : m1 ?md ¼ 1g be a torus subgroup. Then Sd acts on T by ðm; sÞ/ms :¼ ðms1 ðiÞ Þi¼1;y;d :
&
If ½l; t and ½m; s denote two elements of the semi-direct product TsSd with respect to the above action of Sd on T; the composition in this group becomes ½l; t %½m; s ¼ ½lmt ; ts : Note that this group contains ½1; Sd DSd and ½T; id DT and as subgroups, the latter as a normal factor by definition of semi-direct products. If we put 1 ½m; s ðl1 ; y; ld Þ :¼ ðm1 1 ls1 ð1Þ ; y; md ls1 ðdÞ Þ;
the definition of the composition % yields 1 1 1 ½l; t ð½m; s ðl1 ; y; ld ÞÞ ¼ ðl1 i mt1 ðiÞ ; s ðt ðiÞÞÞi¼1;y;d
¼ ððlmt Þ1 ðiÞ; ðtsÞ1 ðiÞÞi¼1;y;d ¼ ð½l; t %½m; s Þ ðl1 ; y; ld Þ; so that we obtain in this way an action of TsSd on the linear forms in the decomposition of f AFd and more generally on vn AðV n Þd :
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Besides the action of TsSd we also consider the dual of the natural action ðg; vÞ/gv of GLðV Þ on V and extend it componentwise to ðV n Þd to obtain ðGLðV Þ; ðV n Þd Þ -ðV n Þd ; ðg; vn Þ /g vn :¼ ðvn1 3 g1 ; y; vnd 3 g1 Þ: Although the two actions on ðV n Þd introduced so far seem quite different, we will explain that they are in fact two special cases of one more general action which shall be introduced now. The interpretation of mAT as diagonal matrix Dm ASLd ðKÞ; respectively, of sASd as permutation matrix Ps AOd ðKÞ defined by Ps ðei Þ ¼ esðiÞ leads to two subgroups of GLd ðKÞ and composition with GLd ðKÞ -GLd ðGLðV ÞÞ; M /M#idV yielding an embedding I of TsSd in GLd ðGLðV ÞÞ for which ið½m; s Þ ¼ Dm Ps and we denote this image by ðm; sÞ: In this way, the multiplication % in TsSd translates to composition in the algebraic group GLd ðGLðV ÞÞ restricted to iðTsSd Þ ¼: S since we have ið½l; t %½m; s Þ ¼ ðlmt ; tsÞ ¼ Dlmt Pts ¼ Dl Dmt Pt Ps ¼ Dl Pt Dm Ps ¼ ðl; tÞðm; sÞ ¼ ið½l; t Þið½m; s Þ: On the other hand, the diagonal embedding D of GLðV Þ in GLd ðGLðV ÞÞ carries over the usual composition in GLðV Þ to its image DðGLðV ÞÞ: This motivates the definition of an action of GLd ðGLðV ÞÞ on ðV n Þd as follows: ðGLd ðGLðV Þ; ðV n Þd Þ -ðV n Þd ; ðA; vn Þ /A vn :¼ vn 3 A1 :
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Note that vn 3 A1 is defined as the matrix multiplication of vn viewed as row vector with the inverse matrix of A where the multiplication of the respective components and entries is composition. One checks immediately that this really yields an action since ðABÞ vn ¼ vn 3 ðABÞ1 ¼ vn 3 B1 A1 ¼ ðvn 3 B1 Þ 3 A1 ¼ A ðB vn Þ: Moreover, the restriction of this action to S; respectively, DðGLðV ÞÞ extends the previously defined actions as claimed. From this point of view, the following statement becomes quite obvious: Lemma 1.1. The actions of ½m; s ATsSd and gAGLðV Þ +D GLd ðGLðV ÞÞ on ðV n Þd commute, that is ½m; s ðg vn Þ ¼ g ð½m; s vn Þ:
Proof. Passing to the images of ½m; s and g in GLd ðGLðV ÞÞ; the statement follows directly from DðGLðV ÞÞ ¼ ZGLd ðGLðV ÞÞ ðSÞ; the centralizer of S in GLd ðGLðV ÞÞ: In fact, this stems from the combination of the two facts that any element of S has only entries of the form mi idV ; which commute with GLðV Þ and that DðyÞ lies in the center of GLd ðyÞ: At this point, it is time to apply the established facts about the action of GLd ðGLðV ÞÞ on ðV n Þd to Fd : We have an identification of decomposable forms with their linear factors modulo the action of TsSd and thus an isomorphism B
Fd -ðV n Þd =S;
f /Svn : Moreover, the action of GLðV Þ on ðV n Þd factorizes to an action Svn /g ðSvn Þ : ¼ Sðg vn Þ on ðV n Þd =SDFd since the image is well defined by Lemma 1.1. In this way, we obtain the well-known action ðGLðV Þ; Fd Þ -Fd ; ðg; f Þ /g f :¼ f 3 g1 ; that is related to the concept of automorphisms by:
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Definition 1.2. The automorphism group Autð f Þ of f AFd is defined as Autð f Þ ¼ fgAGLðV Þ : g f ¼ f g; the stabilizer of f in GLðV Þ: The outcome of our more general approach is the fact that the action of GLd ðGLðV ÞÞ on Fd relates the actions of DðGLðV ÞÞ (resp. S) on ðV n Þd in the following way: for f ¼ l1 ?ld ¼ Svn and gAAutð f Þ the invariance of f under the action of g; namely g ðSvn Þ ¼ Sðg vn Þ ¼ Svn implies the existence of some ðl; tÞg AS with g ðl1 ; y; ld Þ ¼ ðl; tÞg ðl1 ; y; ld Þ:
ð%Þ
Note that ðl; tÞg depends on ðl1 ; y; ld Þ; this is subject of the next lemma. Lemma 1.3. For every decomposition f ¼ l1 ?ld of f AFd ; Eq. ð%Þ defines a homomorphism rl : Autð f Þ -S; g /rl ðgÞ :¼ ðl; tÞ1 g : If ðl1 ; y; ld Þ ðm1 ; y; md Þ mod S (i.e. f ¼ l1 ?ld ¼ m1 ?md ), the respective maps rl and rm are conjugate over S: Proof. For the first statement, we check that for g; g0 AAutð f Þ; rl ðgg0 Þ ¼ rl ðgÞrl ðg0 Þ: 0
With rl ðgÞ ¼ ðl; tÞ1 and rl ðg0 Þ ¼ ðl; tÞ 1 Eq. ð%Þ yields ðgg0 Þ ðl1 ; y; ld Þ ¼ g ðg0 ðl1 ; y; ld ÞÞ ¼ g ððl; tÞ0 ðl1 ; y; ld ÞÞ ¼ ðl; tÞ0 ðg ðl1 ; y; ld ÞÞ by Lemma 1:1 ¼ ðl; tÞ0 ððl; tÞ ðl1 ; y; ld ÞÞ ¼ ððl; tÞ0 ðl; tÞÞ ðl1 ; y; ld Þ
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and inversion reverses the order of ðl; tÞ0 and ðl; tÞ: Applying this to ðgg1 Þ ðl1 ; y; ld Þ ¼ ðl1 ; y; ld Þ implies 1 rl ðg1 Þ ¼ ðl; tÞ1 g1 ¼ ðl; tÞg ¼ rl ðgÞ
and rl is indeed a homomorphism. Now let ðm; sÞAS with ðm; sÞ ðl1 ; y; ld Þ ¼ ðm1 ; y; md Þ: For gAAutð f Þ we use the commutativity argument once again to check that g ðm1 ; y; md Þ ¼ g ððm; sÞ ðl1 ; y; ld ÞÞ ¼ ðm; sÞðg ðl1 ; y; ld ÞÞ ¼ ðm; sÞ ððl; tÞ ðl1 ; y; ld ÞÞ ¼ ððm; sÞðl; tÞÞððm; sÞ1 ðm1 ; y; md ÞÞ ¼ ððm; sÞðl; tÞðm; sÞ1 Þðm1 ; y; md Þ; from which we conclude rm ðgÞ ¼ ðm; sÞrl ðgÞðm; sÞ1 and the homomorphisms are conjugate over S:
&
In order to examine some properties of the above homomorphism, we need to introduce the concept of rank for forms f AFd : Definition 1.4. For vn ¼ ðvn1 ; y; vnd ÞAðV n Þd and any I ¼ fi1 ; y; ir gDf1; y; dg we consider the homomorphism vnI :¼ pI 3 vn : vn
pI
V - Kd - KI ; where pI denotes the projection on the r-dimensional coordinate subspace K I of K d defined by I: The weight wðvn Þ shall be defined by 8rpd
ðwðvn ÞXr38IDf1; y; dg; jIj ¼ r : rankðvnI Þ ¼ rÞ:
Lemma 1.5. For vn ¼ ðvn1 ; y; vnd ÞAðV n Þd the following conditions are equivalent: (1) wðvn ÞXr; (2) any r components vni1 ; y; vnir are linearly independent. In particular, we have wn ASvn ) wðwn Þ ¼ wðvn Þ
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so that for a form f ¼ Sðl1 ; y; ld Þ the quantity rankð f Þ :¼ wðl1 ; y; ld Þ is well defined and bounded by k ¼ dimðV Þ from above. Proof. For the equivalence of (1) and (2) we observe that the linear independence of vni1 ; y; vnir yields the existence of vi1 ; y; vir AV such that vnij ðvik Þ ¼ djk and thus for I ¼ fi1 ; y; ir g the homomorphism pI 3 vn has rank Xr: Conversely, if for I ¼ fi1 ; y; ir g the rank of pI 3 vn is Xr; this homomorphism is surjective so that vnij ðvik Þ ¼ djk has a solution and the vnij are linearly independent. As I was arbitrary, this holds for any r components of vn : For the last statement, note that wn ASvn implies that the components of wn are, up to scalar multiples, just a permutation of the ones of vn ; by (2) this does not affect the weight function. Moreover, rankðvnI Þprankðvn ÞpdimðV Þ: & The relation between the parameters k and d has a significant influence on rl and thus on Autð f Þ; especially for forms of rank k; which we will call non-degenerate. Lemma 1.6. Let f AFd be a form of rank r such that 2prpdimðV Þod and rl the homomorphism corresponding to the decomposition f ¼ l1 ?ld : If for some gAAutð f Þ we have rl ðgÞ ¼ ðl; idÞ; then li ¼ x ði ¼ 1; y; dÞ for some xAmd ; the group of dth roots of unity, and there exists an r-dimensional subspace V0 of V with gjV0 ¼ x idV0 : In particular, if f is non-degenerate, g ¼ x idV and kerðrl Þ is finite. Proof. For t ¼ id; ð%Þ becomes g ðl1 ; y; ld Þ ¼ ðl1 l1 ; y; ld ld Þ; which we may split up into g l i ¼ li l i
for i ¼ 1; y; d:
Suppose further that rankðl1 ; y; ld Þ ¼ r and choose fl1 ; y; lr g as a basis for the subspace of V n spanned by fðl1 ; y; ld Þg: Consequently, lrþ1 ¼
r X
aj lj
and
g lrþ1 ¼ lrþ1 lrþ1
j¼1
together imply lrþ1
r X j¼1
aj lj ¼
r X
aj lj l j :
j¼1
At this point, the linear independence of any set of r linear forms out of fl1 ; y; lrþ1 g guarantees aj a0; 1pjpr; and thus lj ¼ lrþ1 for j ¼ 1; y; r: Repeated use of this argument yields l1 ¼ ? ¼ ld ¼: x and l1 yld ¼ xd ¼ 1 shows xAmd as desired.
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We may thus rewrite ð%Þ as g ðl1 ; y; ld Þ ¼ x idV ðl1 ; y; ld Þ 3 ðl1 ; y; ld Þ 3 g1 ¼ l1 ; y; ld 3 x1 idV and call vn the composition ðl1 ; y; ld Þ 3 ðg1 x1 idV Þ: Then vn 0 and thus we conclude imðg1 x1 idV ÞDkerðl1 ; y; ld Þ: By assumption kerðl1 ; y; ld Þ is a ðk rÞdimensional subspace of V ; so that V0 :¼ kerðg1 x1 idV Þ is an r-dimensional subspace on which g ¼ x idV : If, moreover, f is non-degenerate V0 ¼ V and g is equivalent to some multiple of idV : Since this is contained in the center of GlðV0 Þ we conclude g ¼ x idV : This leaves only finitely many choices for g corresponding to the d possible xAmd : Obviously, any of these lie in kerðrl Þ and everything is proved. & From now on, we will focus on non-degenerate forms. In this case, we consider the composition of rl with the projection p : S-S=iðTÞDiðSd Þ; which makes sense since iðTÞoS is normal and we obtain a homomorphism r* l : Autð f Þ -Sd ; g /tg :¼ i1 ðpðrl ð f ÞÞÞ associated to the decomposition f ¼ l1 ?ld : Note that this reduction does not involve any loss of information since gAkerðr* l Þ3 i1 ðpðrl ð f ÞÞÞ ¼ id 3 pðrl ð f ÞÞ ¼ Pid 3 (lAT : rl ð f Þ ¼ ðl; idÞ and this is equivalent to gAkerðrl Þ by Lemma 1.6. As an immediate consequence we have Corollary 1.7. For any non-degenerate f AFd the group Autð f ÞDGLðV Þ is finite and we have the estimate jAutð f Þjpdðd!Þ:
The finiteness of Autð f Þ for forms of degree dX3; not necessarily decomposable, was first claimed by Jordan [Jo]; however, in this more general case the proof involves other techniques. See also [CR,OS] for generalizations.
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The next step consists in the study of the image of Autð f Þ under r* l : Let f AFd be a form of rank r such that 2prpdimðV Þod and f ¼ l1 ?ld be a decomposition of f : Definition 1.8. The permutation group of f relative to this decomposition is the subgroup of Sd defined by Gl ð f Þ :¼ r* l ðAutð f ÞÞ:
With this notation, we may reformulate Lemma 1.6 as follows: For every decomposition f ¼ l1 ?ld we obtain a short exact sequence 1-md -Autð f Þ-Gl ð f Þ-1 with the obvious maps x/x idV and g/r* l ðgÞ ¼ tg : So far the above sequence has the drawback that the factor Gl ð f Þ is not uniquely determined for f AFd : However, the conjugacy of any two homomorphisms rl and rm over S translates to conjugacy of r* l and r* m over Sd so that Autð f ÞDmd sGl ð f Þ for any set of linear factors l1 ?ld : Consequently, we can say more about the elements in Autð f Þ by using the structure of the permutations tAGð f Þ that is invariant under conjugation: Definition 1.9. We say that a permutation tASd contains a cycle of length r > 1; if there exists an iAf1; y; dg such that tr ðiÞ ¼ i and tj ðiÞai for 1ojor: We write ðn Þ ðn Þ t ¼ ðr1 1 ; y; rs s Þ if r1 ; y; rs are the cycle P lengths of the cycles in t and n1 ; y; ns their respective multiplicities. For instance j nj rj ¼ d: Lemma 1.10. Let f AFd be non-degenerate and gAAutð f Þ with r* l ðgÞ ¼: tg AGl ð f Þ for ðn Þ
ðn Þ
some decomposition of f : If tg ¼ ðr1 1 ; y; rs s Þ; then g is a root of unity of order pd lcmðr1 ; y; rs Þ ¼: dR: Proof. First we note that the cycle lengths r1 ; y; rs are indeed independent of the special decomposition of f : For any i; 1pips; we use r* l ðgri Þ ¼ ðr* l ðgÞÞri ¼ trgi to conclude that trgi has ri fixed points. By Lemma 1.6 this yields gR ¼ xð1; idÞ ¼ x idV and the exact order of g depends on the order of xAmd ; which is bounded by d: &
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In order to say more about the possible structure of permutations tAGð f Þ we need the following refinement of Lemma 1.6 which will open the way to some interesting applications of the previous lemma. Proposition 1.11. With the assumptions of Lemma 1.10 we have: tg has more than k fixed points 3 g ¼ x idV with xd ¼ 1: Proof. As fixed points are precisely the cycles of length 1, their number is again independent of the chosen decomposition of f and the assumption is well defined. W.l.o.g. we may assume that f1; y; k þ 1g is among the fixed points of tg ; hence with rl ðgÞ ¼ ðl; tÞ we have g l i ¼ li l i
for i ¼ 1; y; k þ 1:
As dimðV Þ ¼ kok þ 1 and rankðl1 ; y; ld Þ ¼ k we conclude with the same arguments as in Lemma 1.6 that l1 ; y; lkþ1 are eigenvectors corresponding to one single eigenvalue x :¼ l1 ¼ ? ¼ lkþ1 Amd and the linear independence of any k of these vectors yields g li ¼ xli for i ¼ k þ 2; y; d as well. If gð f Þ denotes any permutation group in Sd arising for some f AFd and some decomposition of f ; this result enables us to obtain considerable restrictions on the possible cycle lengths of permutations tAGð f Þ: & ðn Þ
ðn Þ
Proposition 1.12. Let t ¼ tg ¼ ðr1 1 ; y; rs s Þ be in the permutation group of some f AFd and some gAAutð f Þ: Then for every subset frs1 ; y; rst g of fr1 ; y; rs g such that X nsj rsj > k; j
ð%Þ implies tlcmðrs1 ;y;rst Þ ¼: tRs ¼ id: In particular, all other cycles of t have an order dividing lcmðrs1 ; y; rst Þ ¼ Rs : Proof. Suppose frs1 ; y; rst g is such a collection of cycle lengths. P Then the total number of vectors li involved in the corresponding cycles is j nsj rsj and thus > k by assumption. This yields at least k þ 1 fixed points of tRs ; by Proposition 1.11 we thus have gRs ¼ x idV and the order of g is ordðxÞ Rs : Comparison with the estimate of ordðgÞ in Lemma 1.10 yields Rs ¼ lcmðrs1 ; y; rst Þ ¼ lcmðr1 ; y; rs Þ ¼ R and the last claim is immediate.
&
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Corollary 1.13. With the assumptions as in Proposition 1.12 suppose that there is an integer Tpd such that X rsi nðrsi Þ > k; rsi jT
then all cycles in t have an order r dividing T: Proof. Note that rs1 ; y; rst jT implies lcmðrs1 ; y; rst ÞjT: The result is now a direct consequence of the previous proposition. & In particular, this covers the case of permutations that contain one cycle of length t > k: However, if t ¼ k; we may even improve this result: Proposition 1.14. If t contains a cycle of length k; then ð%Þ implies that t consists only of cycles of order r; where rjk: Proof. Without loss of generality, we may suppose that g l1
¼ l2 l2 ; ^
g lk1 g lk
¼ lk l k ; ¼ l1 l1
is the cycle of length k: As before we conclude that there are k linearly independent eigenvectors for the action of gk and since they all belong to the same cycle, they all correspond to the same eigenvalue l1 ylk : Thus, the action of g is equivalent to the one of x idV where x :¼ l1 ylk Amd and once again tk ¼ id; the conclusion of Corollary 1.12 thus holds in this case as well. & The restrictions obtained so far allow to determine completely the possible structure of Gð f Þ in several interesting cases; we will devote the next section to some examples of how this may be achieved using the results of this section.
2. Some examples We first examine the case of forms whose degree is large compared to the number of variables and start by investigating when the condition of Corollary 1.12 is fulfilled. Proposition 2.1. Suppose f is a form in k variables of degree d > k2 k: Then for every tAGl ð f Þ for some decomposition of f there exists a positive integer tpd such that all cycles in t have an order dividing t:
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Proof. Suppose that for any tpd we have X lnðrÞpk: rjt
Then obviously t contains not more than ½k=t t-cycles for any tok and no t-cycle for kptpd: Since for the degree of f we have X d¼ tnðtÞ; tpd
the above assumption implies dp
X
t½k=t pkðk 1Þ:
tok
Consequently, for d > k2 k this assumption cannot be fulfilled for all tpd and Corollary 1.12 is applicable, which concludes the proof. & The situation is particularly simple if k is small, even though the number of possibilities for tAGð f Þ increases rapidly with k: We will examine the cases k ¼ 2 and 3 in detail. Corollary 2.2. Let f ¼ l1 ?ld be a form in 2 variables of degree dX4: Then any taid in Gl ð f Þ consists of: (1)
d t
t-cycles ðtjdÞ; or (2) d1 t t-cycles ðtjd 1Þ and 1 fixed point, or (3) d2 t t-cycles ðtjd 2Þ and 2 fixed points.
Proof. For k ¼ 2 the existence of any cycle of length X2 already implies that all cycles have the same length (indeed all divisors > 1 would be at least 2 and thus themselves equal to the order of all cycles). Since there are at most 2 fixed points, the three cases above are the only possibilities to split up d into cycles of equal length and fixed points. & Corollary 2.3. Let f ¼ l1 ?ld be a form in 3 variables of degree dX5: Then any taid in Gl ð f Þ consists of: (1)
d t
t-cycles ðtjdÞ; or (2) d1 t t-cycles ðtjd 1Þ and 1 fixed point, or
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(3)
d2 t
t-cycles ðtjd 2Þ and 2 fixed points,
or (4)
d3 t
t-cycles ðtjd 3Þ and 3 fixed points,
or (5)
d2 t
(6)
d3 t
t-cycles ðtjd 2Þ and one 2-cycle,
or t-cycles ðtjd 3Þ; one 2-cycle and 1 fixed point, where the last two cases are possible only for even t:
Proof. For k ¼ 3 the existence of any cycle of length tX3 already implies that all cycles of length X3 have the same length t by the same argument as in the previous corollary. If t contains two 2-cycles, all other cycles have order 2 as well or are fixed points by Corollary 1.12. Since t contains at most 3 fixed points, the above restrictions leave only the mentioned six cases, where we notice that (5) and (6) may only happen when t is even (Corollary 1.12 again), since tt ¼ id must imply t2 ¼ id: We also give an example of how to use Proposition 1.11 if Corollary 1.12 fails. Let d ¼ 37 and t ¼ ð10; 12; 15Þ: If kp15; it is obvious that teGð f Þ for a form of degree d in k variables. (Take T ¼ 15:) For 15okp24 we argue as follows: let l1 ¼ 10; l2 ¼ 15; so l1 þ l2 ¼ 25 > 24 ¼ k and tlcmð10;15Þ ¼ t30 ¼ id by the proposition. But 12 [30 and thus the permutation ð10; 12; 15Þ does not appear in any Gð f Þ if kp24: &
3. Generic forms—a special case Having examined the structure of Autð f Þ in general in the previous two sections, we now turn to the question: Do forms in general have non-trivial automorphisms? This is not only of interest by itself, but also has considerable influence on representations of forms by linear forms as explained in [Su]. Definition 3.1. A form f of degree d is called generic if Autð f ÞDmd ; that is gAAutð f Þ ) (xAmd : g ¼ xidV : In the case of non-degenerate forms, this condition is equivalent to the statement that the permutation group of f for some (and thus any) decomposition f ¼ l1 ?ld reduces to fidg by Lemma 1.6. In order to determine whether there exist generic forms in k variables of degree d; and if this is the case, to find out if this is rather exceptional or maybe true for almost all (non-degenerate) forms, we start by a rather heuristical analysis of system ð%Þ:
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If f ¼ l1 ?ld and rl ¼ ðl; tÞ; the fact that f is generic translates to the solutions of ð%Þ in the following way: 8taid;
g li ¼ ltðiÞ ltðiÞ
for i ¼ 1; y; d;
has only the trivial solution l1 ¼ ? ¼ ld ¼ 0 and g ¼ 0 and it seems natural to distinguish the following cases: (a) kX2 and dpk þ 1: For this choice of parameters, ð%Þ has less equations than variables since dpk þ 1 ¼
k2 1 3ðk 1Þdpk2 13kd þ 1pk2 þ d: k1
Consequently, there is no hope to expect the majority of forms to be generic in this case since there will be infinitely many solutions to ð%Þ in general (only the presence of contradictory equations would lead to generic forms). (b) kX2 and dXk þ 2: Comparing again the number of equations and variables in ð%Þ; we find that for k ¼ 2; d ¼ 4; the number of equations equals the number of variables ðkd ¼ 8 ¼ k2 þ dÞ; k > 2 or da4; there are strictly fewer variables than equations. This suggests that generic forms are more or less the rule, although for k ¼ 2 and d ¼ 4 an exceptional behavior can be expected. The latter is confirmed in [Po, Section 7.2, Example 1]; however, for the convenience of the reader, we use the case of binary forms f of degree 4 to illustrate the concepts introduced so far. We write such forms as f ðX ; Y Þ ¼ a0 X 4 þ a1 X 3 Y þ ? þ a4 Y 4 ¼ a0
4 Y
ðX þ ri Y Þ;
i¼1
with 0aa0 AK and by assumption the ri lie in K as well and are pairwise disjoint. In this setup, we may express the definition of f being generic explicitly by replacing li by X þ ri Y in ð%Þ to the coefficient matrix of the system arising from ð%Þ compute for any matrix A ¼
a b g d
a li 3 A ¼ ð1; ri Þ g
in Autð f Þ to obtain
b d
! ¼ ða þ gri ; b þ dri Þ ¼ ðltðiÞ ; ltðiÞ rtðiÞ Þ ¼ ltðiÞ ltðiÞ
for i ¼ 1; 2; 3; 4: The resulting four equations in the first component determine the li in dependence of a; b; g; d and we may focus on the remaining homogeneous system in these four variables after plugging in a þ gri for ltðiÞ : The resulting coefficient matrix Cf
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is given by 0
rtð1Þ
B B rtð2Þ Cf :¼ B Br @ tð3Þ rtð4Þ
1 r1 rtð1Þ
r1
1
1 r3 rtð3Þ
C r2 C C: r3 C A
1 r4 rtð4Þ
r4
1 r2 rtð2Þ
This shows that the statement f is generic and is equivalent to 8taid : det Cf a0 since the homogeneous system ð%Þ has a non-trivial solution precisely when the rank of the above coefficient matrix is o4 for some t: In fact, det Cf ¼ 0 implies the existence of a solution a; b; g; d of the reduced system for which l1 yl4 ¼ 1: This is clear if ltðiÞ ¼ a þ gri a0 for i ¼ 1; 2; 3; 4; otherwise, if a þ gr1 ¼ 0; say, we immediately conclude b þ dr1 ¼ 0 as well, a contradiction to AAGLk ðKÞ: With this characterization of generic forms we obtain: Proposition 3.2. Let f ðX ; Y Þ ¼ a0 X 4 þ a1 X 3 Y þ ? þ a4 Y 4 be a non-degenerate form of degree 4 with a0 a0 and decomposition
f ðX ; Y Þ ¼ a0
4 Y ðX þ ri Y Þ: i¼1
Then Gð f Þ is a group of order 4, 8 or 12. Consequently, Gð f Þ and thus Autð f Þ are never trivial and all the binary forms of degree 4 are non-generic. Proof. We simply compute Cf for all the different permutations t in S4 : To start with, let t consist of two 2-cycles, say t ¼ ð1; 2Þð3; 4Þ: In this case, r2 r 1 Cf ¼ r4 r 3
1
r1 r2
1
r1 r2
1 1
r3 r4 r3 r4
r1 r2 ¼ 0; r3 r4
independent of ri ; since the difference of rows 1 and 2 is proportional to the difference of rows 3 and 4. Thus, Gð f Þ contains all the permutations consisting of two 2-cycles and we conclude jGð f ÞjX4; so that it remains to check for which other tAS4 the determinant Cf may vanish.
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247
Next, let t have a fixed point together with a 3-cycle, say t ¼ ð2; 3; 4Þ: In this case, r1 1 r21 r1 r 1 r r r 3 2 3 2 Cf ¼ ¼ a22 3a1 a3 þ 12a0 a4 ; r4 1 r3 r4 r3 r 1 r r r 2
4 2
4
as a direct computation shows. This expression is independent of the chosen 3-cycle, which means that either all or no such t is contained in Gð f Þ; depending on whether a22 3a1 a3 þ 12a0 a4 ¼ 0 or not. Now let t consist of a 4-cycle, say t ¼ ð1; 2; 3; 4Þ: We then compute r2 1 r1 r2 r1 r 1 r r r 3 2 3 2 Cf ¼ ¼ ðr1 r2 Þðr3 r4 Þ½3ðr1 r3 þ r2 r4 Þ a2 : r4 1 r3 r4 r3 r 1 r r r 1
4 1
4
In this product, the first two factors are non-zero by assumption and Gð f Þ contains t ¼ ð1; 2; 3; 4Þ and its inverse precisely when 3ðr1 r3 þ r2 r4 Þ a2 ¼ 0: Gð f Þ then contains no other 4-cycle; in fact, if t ¼ ð1; 3; 2; 4Þ were in Gð f Þ; we would have 3ðr1 r2 þ r3 r4 Þ a2 ¼ 0 as well, and this would already imply the equality of two roots of f ; a contradiction to f non-degenerate. As a consequence, we always have Gð f ÞaS4 : & Altogether, we may summarize our observations in the following listing (let Di;j denote the expression 3ðri rj þ rk rl Þ a2 for fi; j; k; lg ¼ f1; 2; 3; 4gÞ: a22 3a1 a3 þ 12a0 a4 a0 and Di;j a0 8ði; jÞ ) jGð f Þj ¼ 4; a22 3a1 a3 þ 12a0 a4 a0 and Di;j ¼ 0 for some ði; jÞ ) jGð f Þj ¼ 8; a22 3a1 a3 þ 12a0 a4 ¼ 0 and Di;j a0 8ði; jÞ ) jGð f Þj ¼ 12: Note that these cases are the only possible ones, since a22 3a1 a3 þ 12a0 a4 ¼ 0 and Di;j ¼ 0 would already imply the equality of two roots of f :
4. Generic forms—the general case Now that the special case k ¼ 2; d ¼ 4 is settled, we are left with proving the conjecture that for k ¼ 2 and dX5; respectively, kX3 and dXk þ 2 almost all forms are generic, as suggested by the comparison of the number of equations and variables in ð%Þ:
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For this purpose, we consider the Zariski Topology on ðV n Þd in order to state the theorem that formulates the above conjecture in a precise terminology and mention a result about constructible sets which we will use several times. For reference see [Hs, p. 94, Example 3.18]. Proposition 4.1. A constructible subset of an irreducible Zariski space is dense if and only if it contains the generic point. Furthermore, in that case it contains a non-empty open subset. Theorem 4.2. The set X of d-tuples vn ¼ ðl1 ; y; ld ÞAðV n Þd for which f ¼ l1 ?ld is generic is constructible. For dXmaxf5; k þ 2g X is Zariski dense and thus contains some non-empty Zariski open set O: Before we proceed to the proof of this statement, we formulate two lemmas that will be needed in the argument. Lemma 4.3. The set of degenerate forms is a closed subvariety of ðV n Þd : Proof. Recall that f ¼ Svn was called degenerate if wðvn Þok and by Lemma 1.5 this is equivalent to (ICf1; y; dg
with jIj ¼ k : rankðvnI Þok;
where vnI : V -K jIj is the composition of vn with pI : But this may be expressed as Y
det vnI ¼ 0;
I:jIj¼k
which defines a Zariski closed set in ðV n Þd : & In what follows, we need to consider the variety ZðtÞCGLðV Þ ðV n Þd T defined by ð%Þ; that is ðg; vn ; lÞAZðtÞ3 g vn ¼ ðl; tÞ vn 3 rvn ðgÞ ¼ ðl; tÞ with ðl; tÞAS: Similarly, for any ICf1; y; dg we define ZI ðtÞ by ðg; vn ; lÞAZI ðtÞ3rvn ðgÞ ¼ ðm; sÞ with ððm; sÞ vn ÞI ¼ ððl; tÞ vn ÞI : Note that this in particular implies sjI ¼ ljI and that ZI ðtÞ-ZJ ðtÞ ¼ ZI,J ðtÞ: The projection from ZðtÞ to the second component ðV n Þd is denoted by p and this notation is kept for its restriction to ZI ðtÞ:
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Lemma 4.4. Let vn ¼ l1 ; y; ld ApðZðtÞÞ with wðvn Þ ¼ k and ICf1; y; dg with jIj ¼ k þ 1: Then p1 ðvn Þ-ZI ðtÞ is independent of I; thus p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ since
S
jIj¼kþ1
I ¼ f1; y; dg and both sets have cardinality d:
Proof. First we notice that ZI ðtÞ+ZðtÞ implies p1 ðvn Þ-ZI ðtÞ+p1 ðvn Þ-ZðtÞ; so that it suffices to show that p1 ðvn Þ-ZI ðtÞDp1 ðvn Þ-ZðtÞ: To see this we consider two distinct elements in p1 ðvn Þ-ZI ðtÞ for which we have rvn ðgÞ ¼ ðm; sÞ
and
rvn ðg0 Þ ¼ ðm0 ; s0 Þ
with the required restrictions on I: By definition of ZI ðtÞ; we have sjI ¼ tjI ¼ s0 jI ; i.e. s1 s0 jI ¼ idjI and since 1 0 0 n n rg1 g0 ðvn Þ ¼ r1 g ðv Þrg0 ðv Þ ¼ ðm; sÞ ðm ; s Þ;
restriction to I yields r* g1 g0 ðvn ÞjI ¼ idjI ; so that r* g1 g0 ðvn Þ has at least k þ 1 fixed points. Lemma 1.11 yields g0 ¼ xg and m0 ¼ xm for some x in md from which we conclude that jp1 ðvn Þ-ZðtÞjpjp1 ðvn Þ-ZI ðtÞjpd: Further p1 ðvn Þ-ZðtÞ is non-empty by assumption and has cardinality d; indeed the implication ðg; vn ; lÞAZðtÞ ) ðxg; vn ; x1 lÞAZðtÞ for xAmd shows that md acts faithfully on ZðtÞ by ðg; vn ; lÞ-ðxg; vn ; x1 lÞ so that we get an injection md +ZðtÞ: This implies dpjp1 ðvn Þ-ZðtÞj and we obtain p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ as desired.
&
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As we have seen in the proof ðg; vn ; lÞ/ðxg; vn ; x1 lÞ leaves r* vn ðgÞ invariant and thus defines an action of md on ZðtÞ: This action leaves vn invariant and may be extended to a Z :¼ ZðtÞ: tASd
Note that ZðtÞ consists of the ðg; vn ; lÞ in Z for which r* vn ðgÞ ¼ t and that the extension of p to Z is still the projection to ðV n Þd : In order to relate this setup to the concept of generic forms, we consider the subvariety C :¼ ðGLðV Þ\fmd idV gÞ ðV n Þd T of GLðV Þ ðV n Þd T: Proof of Theorem 4.2. The first step consists in showing that X is constructible. By definition f ¼ Svn is non-generic if there exists some gaxidV in GLðV Þ and some ðl; tÞ in S such that ðg; vn ; lÞAZðtÞ: This amounts to say p1 ðvn Þ-Z-Ca|; so that vn is generic precisely if ðg; vn ; lÞ ¼ p1 ðvn ÞAZ\ðZ-CÞ and this is a constructible set in GLðV Þ ðV n Þd T since all open and closed sets as well as finite unions, intersections and complements are in this class. By a theorem of Chevalley (see [Hs, p. 94, Example 3.19]) constructibility is conserved under the projection p and thus X ¼ pðZ\ðZ-CÞÞ is constructible as well. For the second part, we proceed by contraposition. Suppose X was not dense for the indicated range of parameters. Its complement ðV n Þd \X is constructible as well and thus by Proposition 4.1 would contain some dense open set O: By Lemma 4.3 (intersect O with fvn : wðvn Þ ¼ kg) we may assume that f ¼ Svn is non-degenerate for vn AO: This allows to apply Lemma 1.6 which yields p1 ðvn Þ-Z-CDp1 ðvn Þ-ðZ\ZðidÞÞ and since the left-hand side is non-empty (vn eX) this implies the existence of some taid with p1 ðvn Þ-ZðtÞa| for vn AO: Consequently, we get a covering [ O¼ pðp1 ðvn Þ-ZðtÞÞ taid
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251
of O by constructible sets. By Proposition 4.1 at least one of the covering sets must contain some non-empty open subset Ot DO; so for the corresponding t we have p1 ðvn Þ-ZðtÞa| for all vn AOt : If we let T be the set consisting of the identity and of all t subject to this restriction, i.e. T :¼ ftASd : (|aOt DO open; 8vn AOt : p1 ðvn Þ-ZðtÞa|g; we have thus shown TPfidg: T We claim: T is a normal subgroup of Sd :1 Indeed O :¼ tAT Ot is open, nonempty and for vn AO; t; sAT there exist ðg; vn ; lÞAp1 ðvn Þ-ZðtÞ and
ðg0 ; vn ; mÞAp1 ðvn Þ-ZðsÞ
by definition of O: We thus have r* vn ðg1 g0 Þ ¼ t1 s; so that p1 ðvn Þ-Zðt1 sÞa|; i.e. t1 sAT and T is indeed a subgroup of Sd : Moreover, with t0 ¼ sts1 we have r* vn ðð1; sÞgÞ ¼ t0 (see Lemma 1.3), which shows that T is normal. For dX5; the only non-trivial normal subgroups of Sd are Ad and Sd itself, so there are only these two T groups to consider. Now the assumption dXk þ 2 implies the existence of some tAT that induces a non-trivial permutation on I :¼ f1; y; k þ 1g (note that either ð1; y; k þ 1Þ or ð1; y; kÞðk þ 1Þ lies in TjI ) and we intend to show that any such t is determined uniquely by its restriction to I: To do so, note that Lemma 4.4 yields p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ for vn AO: But this set is disjoint to p1 ðvn Þ-Zðt0 Þ for any other t0 AT: indeed ðg; vn ; lÞAZðtÞ-Zðt0 Þ yields g vn ¼ ðl; tÞ vn
and
g vn ¼ ðl; t0 Þ vn
and by Lemma 1.3 ðg1 gÞ vn ¼ vn ¼ ðl; tÞ1 ðl; t0 Þ vn : This shows the existence of some nAT such that ðidV ; vn ; nÞAZðt1 t0 Þ; which implies that any two coordinates li and lt1 t0 ðiÞ of vn are proportional for i ¼ 1; y; d: Of course, as vn is non-degenerate, this is only possible if t ¼ t0 : 1
The author is indebted to Y. Neretin for this observation.
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In other words, we have jI : StabðIÞ-T -Skþ1 -TjI ; s /sjI is injective. If T ¼ Sd this yields ðk þ 1Þ!ðd k 1Þ! ¼ jStabðIÞjpjSkþ1 j ¼ ðk þ 1Þ! and thus d k 1p1 which implies d ¼ k þ 2: On the other hand, for T ¼ Ad it implies ðk þ 1Þ! ðd k 1Þ! ðk þ 1Þ! ¼ jStabðIÞ-Ad jI jpjSkþ1 -Ad jI j ¼ ; 2 2 2 showing that d k 1p2 which is only possible for d ¼ k þ 2 or k þ 3 and we are reduced to handle only these three special cases. Let us consider d ¼ k þ 2 first and let t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ: Then if tAT we have p1 ðvn Þ-ZðtÞa| for any vn ¼ ðl1 ; y; lkþ2 ÞAO: We denote by prI the projection from ðV n Þd to the first k þ 1 coordinates and observe that prI ðvn Þ K
lkþ2 defines a closed subset of ðV n Þd : Consequently, the complement of this set in ðV n Þd is Zariski open and thus has non-empty intersection with O: So O contains 0 Þ for which some wn ¼ ðl10 ; y; lkþ2 li0 ¼ li for iAI
and
0 lkþ2 ac lkþ2 ; cAK:
By definition of wn ðg; vn ; lÞAp1 ðvn Þ-ZI ðtÞ3ðg; wn ; lÞAp1 ðwn Þ-ZI ðtÞ and Lemma 4.4 yields ðg; vn ; lÞAp1 ðvn Þ-ZðtÞ3ðg; wn ; lÞAp1 ðwn Þ-ZðtÞ; 0 0 ¼ lkþ2 lkþ2 : As l1 ; y; lk form a basis of ðV n Þd ; we so that glkþ2 ¼ lkþ2 lkþ2 and glkþ2 may write
lkþ2 ¼
k X
ai l i ;
i¼1
0 lkþ2 ¼
k X i¼1
bi li
L. Summerer / Journal of Number Theory 99 (2003) 232–254
253
0 and in combination with the above equations for lkþ2 and lkþ2 we obtain k X
ai ltðiÞ ltðiÞ ¼ lkþ2
i¼1
k X
k X
ai l i ;
i¼1
btðiÞ ltðiÞ ¼ lkþ2
i¼1
k X
b i li :
i¼1
A comparison of the coefficients (note that tAStabðf1; y; kgÞ and thus both sums are linear combinations of the chosen basis vectors) yields at1 ðiÞ li ¼ lkþ2 ai ; bt1 ðiÞ li ¼ lkþ2 bi : This leads to asðiÞ =bsðiÞ ¼ ai =bi for i ¼ 1; y; k and all s in the cyclic group /tS generated by t: Thus, the quotient is 0 independent of i; i.e. lkþ2 and lkþ2 are proportional, a contradiction to the choice n of w : This argument works only if tAT; which is certainly true if T is the whole permutation group. However, it is also applicable to the case T ¼ Ad ; as kX3 implies that either t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ or
t0 ¼ ð1; 2; y; k 1ÞðkÞðk þ 1Þðk þ 2Þ
is in T: In the latter case, the aforementioned argument shows that ai =bi is independent of i for i ¼ 1; y; k 1: If this ratio was different from ak =bk ; the 0 decomposition of lkþ2 (resp. lkþ2 ) in the basis fl1 ; y; lk g would yield a linear 0 dependence of lk ; lkþ2 and lkþ2 : With respect to the basis fl1 ; y; lk1 ; lkþ1 g the same 0 and finally to computation would lead to a linear dependence of lkþ1 ; lkþ2 and lkþ2 the vanishing of a non-trivial linear combination of lk ; lkþ1 and lkþ2 : As kX3 this contradicted the non-degeneracy of vn : Finally, if d ¼ k þ 3; either t ¼ ð1; y; kÞðk þ 1Þðk þ 2Þðk þ 3Þ
or
t0 ¼ ð1; y; kÞðk þ 1; k þ 2Þðk þ 3Þ
0 lies in T: With I ¼ f1; y; k þ 2g and lkþ3 ac lkþ3 the same strategy as explained for d ¼ k þ 2; t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ yields the result. (Note that Lemma 4.4 holds for any I with jIj > k þ 1 as well.) &
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References [CR] [Hs] [Jo] [OS] [Po] [Su]
C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962. R. Hartshorne, Algebraic Geometry, Springer Graduate Texts, Springer, Berlin. C. Jordan, Memoire sur l’equivalence des formes, Oeuvres III, Gauthier Villars, Paris, 1962. P. Orlik, L. Solomon, Automorphisms of forms, Math. Ann. 231 (1978) 229–240. V. Popov, Survey in Algebraic Geometry, Part 2, Invariant Theory, in: Encyclopaedia of Mathematical Sciences, Vol. 55, Springer, Berlin, 1994. L. Summerer, Forms represented by linear forms, Monatsh. Math., submitted for publication.
http://www.elsevier.com/locate/jnt
Decomposable forms and automorphisms Leonhard Summerer Departement Mathematik, ETH Zurich, Zurich, Switzerland Received 26 April 2000; revised 19 August 2002
Abstract In the late 19th century Jordan initiated the study of forms of higher degree and derived (see Memoire sur l’equivalence des formes, Oeuvres III, Gauthier Villars, Paris, 1962) the finiteness of the automorphism group Autð f Þ of complex forms of degree X3 and non-zero discriminant. This result has been extended to forms over arbitrary fields by Schneider (J. Algebra 27 (1973) 112), see also Curtis and Reiner (Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962) for related topics. Orlik and Solomon gave some bounds for the cardinality of Autð f Þ using cohomological arguments in Orlik and Solomon (Math. Ann. 231 (1978) 229); besides this, little seems to be known about this group in general. In connection with his study (Monatsh. Math., submitted for publication) of representations of forms by linear forms, the author was led to an investigation of the group of automorphisms of decomposable forms f through the permutations of the linear factors these automorphisms induce. The main result (Theorem 4.2 in Chapter 4) states that almost all forms in kX2 variables of degree dXmaxf5; k þ 2g have only the trivial automorphisms that consist in multiplying each variable by the same dth root of unity. The case k ¼ 2; d ¼ 4 has already been studied (see Survey in Algebraic Geometry, Part 2, Invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer, Berlin, 1994); however, it is treated in full detail to illustrate the elaborated techniques. The first chapters are devoted to the proof of some general results concerning the structure of the permutation group associated to a form f which also help to understand the case of forms with non-trivial automorphisms. In a few special cases, this allows to determine this group explicitly; in general we give a bound for the cardinality of Autð f Þ depending only on the degree of f which is relevant for some diophantine problems (see e.g. Ann. Math. 155 (2002) 553). The author is indebted to G. Wuestholz for his substantial help and encouragement during the redaction of the paper, he also wishes to thank V. Popov for several helpful remarks. r 2002 Elsevier Science (USA). All rights reserved.
E-mail address: [email protected]. 0022-314X/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 7 3 - 2
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1. Group actions and automorphisms For an algebraically closed field K of characteristic 0 we consider a vector space V of dimension k over K and define Fd to be the set of decomposable forms of degree d > 0 with variables vAV and coefficients in K: This means that any f AFd admits a decomposition f ¼: l1 ?ld
with
l1 ; y; ld AV n :
We consider the map ðV n Þd -Fd ; ðl1 ; y; ld Þ /l1 ?ld ; it is surjective since K is algebraically closed but obviously not injective. However, the decomposition of f AFd in l1 ?ld is unique up to a scalar factor of the li and their order, which leads us to consider the following groups acting on f and its given decomposition. Let Sd be the symmetric group of order d and T :¼ fm :¼ ðm1 ; y; md ÞAðK n Þd : m1 ?md ¼ 1g be a torus subgroup. Then Sd acts on T by ðm; sÞ/ms :¼ ðms1 ðiÞ Þi¼1;y;d :
&
If ½l; t and ½m; s denote two elements of the semi-direct product TsSd with respect to the above action of Sd on T; the composition in this group becomes ½l; t %½m; s ¼ ½lmt ; ts : Note that this group contains ½1; Sd DSd and ½T; id DT and as subgroups, the latter as a normal factor by definition of semi-direct products. If we put 1 ½m; s ðl1 ; y; ld Þ :¼ ðm1 1 ls1 ð1Þ ; y; md ls1 ðdÞ Þ;
the definition of the composition % yields 1 1 1 ½l; t ð½m; s ðl1 ; y; ld ÞÞ ¼ ðl1 i mt1 ðiÞ ; s ðt ðiÞÞÞi¼1;y;d
¼ ððlmt Þ1 ðiÞ; ðtsÞ1 ðiÞÞi¼1;y;d ¼ ð½l; t %½m; s Þ ðl1 ; y; ld Þ; so that we obtain in this way an action of TsSd on the linear forms in the decomposition of f AFd and more generally on vn AðV n Þd :
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Besides the action of TsSd we also consider the dual of the natural action ðg; vÞ/gv of GLðV Þ on V and extend it componentwise to ðV n Þd to obtain ðGLðV Þ; ðV n Þd Þ -ðV n Þd ; ðg; vn Þ /g vn :¼ ðvn1 3 g1 ; y; vnd 3 g1 Þ: Although the two actions on ðV n Þd introduced so far seem quite different, we will explain that they are in fact two special cases of one more general action which shall be introduced now. The interpretation of mAT as diagonal matrix Dm ASLd ðKÞ; respectively, of sASd as permutation matrix Ps AOd ðKÞ defined by Ps ðei Þ ¼ esðiÞ leads to two subgroups of GLd ðKÞ and composition with GLd ðKÞ -GLd ðGLðV ÞÞ; M /M#idV yielding an embedding I of TsSd in GLd ðGLðV ÞÞ for which ið½m; s Þ ¼ Dm Ps and we denote this image by ðm; sÞ: In this way, the multiplication % in TsSd translates to composition in the algebraic group GLd ðGLðV ÞÞ restricted to iðTsSd Þ ¼: S since we have ið½l; t %½m; s Þ ¼ ðlmt ; tsÞ ¼ Dlmt Pts ¼ Dl Dmt Pt Ps ¼ Dl Pt Dm Ps ¼ ðl; tÞðm; sÞ ¼ ið½l; t Þið½m; s Þ: On the other hand, the diagonal embedding D of GLðV Þ in GLd ðGLðV ÞÞ carries over the usual composition in GLðV Þ to its image DðGLðV ÞÞ: This motivates the definition of an action of GLd ðGLðV ÞÞ on ðV n Þd as follows: ðGLd ðGLðV Þ; ðV n Þd Þ -ðV n Þd ; ðA; vn Þ /A vn :¼ vn 3 A1 :
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Note that vn 3 A1 is defined as the matrix multiplication of vn viewed as row vector with the inverse matrix of A where the multiplication of the respective components and entries is composition. One checks immediately that this really yields an action since ðABÞ vn ¼ vn 3 ðABÞ1 ¼ vn 3 B1 A1 ¼ ðvn 3 B1 Þ 3 A1 ¼ A ðB vn Þ: Moreover, the restriction of this action to S; respectively, DðGLðV ÞÞ extends the previously defined actions as claimed. From this point of view, the following statement becomes quite obvious: Lemma 1.1. The actions of ½m; s ATsSd and gAGLðV Þ +D GLd ðGLðV ÞÞ on ðV n Þd commute, that is ½m; s ðg vn Þ ¼ g ð½m; s vn Þ:
Proof. Passing to the images of ½m; s and g in GLd ðGLðV ÞÞ; the statement follows directly from DðGLðV ÞÞ ¼ ZGLd ðGLðV ÞÞ ðSÞ; the centralizer of S in GLd ðGLðV ÞÞ: In fact, this stems from the combination of the two facts that any element of S has only entries of the form mi idV ; which commute with GLðV Þ and that DðyÞ lies in the center of GLd ðyÞ: At this point, it is time to apply the established facts about the action of GLd ðGLðV ÞÞ on ðV n Þd to Fd : We have an identification of decomposable forms with their linear factors modulo the action of TsSd and thus an isomorphism B
Fd -ðV n Þd =S;
f /Svn : Moreover, the action of GLðV Þ on ðV n Þd factorizes to an action Svn /g ðSvn Þ : ¼ Sðg vn Þ on ðV n Þd =SDFd since the image is well defined by Lemma 1.1. In this way, we obtain the well-known action ðGLðV Þ; Fd Þ -Fd ; ðg; f Þ /g f :¼ f 3 g1 ; that is related to the concept of automorphisms by:
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Definition 1.2. The automorphism group Autð f Þ of f AFd is defined as Autð f Þ ¼ fgAGLðV Þ : g f ¼ f g; the stabilizer of f in GLðV Þ: The outcome of our more general approach is the fact that the action of GLd ðGLðV ÞÞ on Fd relates the actions of DðGLðV ÞÞ (resp. S) on ðV n Þd in the following way: for f ¼ l1 ?ld ¼ Svn and gAAutð f Þ the invariance of f under the action of g; namely g ðSvn Þ ¼ Sðg vn Þ ¼ Svn implies the existence of some ðl; tÞg AS with g ðl1 ; y; ld Þ ¼ ðl; tÞg ðl1 ; y; ld Þ:
ð%Þ
Note that ðl; tÞg depends on ðl1 ; y; ld Þ; this is subject of the next lemma. Lemma 1.3. For every decomposition f ¼ l1 ?ld of f AFd ; Eq. ð%Þ defines a homomorphism rl : Autð f Þ -S; g /rl ðgÞ :¼ ðl; tÞ1 g : If ðl1 ; y; ld Þ ðm1 ; y; md Þ mod S (i.e. f ¼ l1 ?ld ¼ m1 ?md ), the respective maps rl and rm are conjugate over S: Proof. For the first statement, we check that for g; g0 AAutð f Þ; rl ðgg0 Þ ¼ rl ðgÞrl ðg0 Þ: 0
With rl ðgÞ ¼ ðl; tÞ1 and rl ðg0 Þ ¼ ðl; tÞ 1 Eq. ð%Þ yields ðgg0 Þ ðl1 ; y; ld Þ ¼ g ðg0 ðl1 ; y; ld ÞÞ ¼ g ððl; tÞ0 ðl1 ; y; ld ÞÞ ¼ ðl; tÞ0 ðg ðl1 ; y; ld ÞÞ by Lemma 1:1 ¼ ðl; tÞ0 ððl; tÞ ðl1 ; y; ld ÞÞ ¼ ððl; tÞ0 ðl; tÞÞ ðl1 ; y; ld Þ
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and inversion reverses the order of ðl; tÞ0 and ðl; tÞ: Applying this to ðgg1 Þ ðl1 ; y; ld Þ ¼ ðl1 ; y; ld Þ implies 1 rl ðg1 Þ ¼ ðl; tÞ1 g1 ¼ ðl; tÞg ¼ rl ðgÞ
and rl is indeed a homomorphism. Now let ðm; sÞAS with ðm; sÞ ðl1 ; y; ld Þ ¼ ðm1 ; y; md Þ: For gAAutð f Þ we use the commutativity argument once again to check that g ðm1 ; y; md Þ ¼ g ððm; sÞ ðl1 ; y; ld ÞÞ ¼ ðm; sÞðg ðl1 ; y; ld ÞÞ ¼ ðm; sÞ ððl; tÞ ðl1 ; y; ld ÞÞ ¼ ððm; sÞðl; tÞÞððm; sÞ1 ðm1 ; y; md ÞÞ ¼ ððm; sÞðl; tÞðm; sÞ1 Þðm1 ; y; md Þ; from which we conclude rm ðgÞ ¼ ðm; sÞrl ðgÞðm; sÞ1 and the homomorphisms are conjugate over S:
&
In order to examine some properties of the above homomorphism, we need to introduce the concept of rank for forms f AFd : Definition 1.4. For vn ¼ ðvn1 ; y; vnd ÞAðV n Þd and any I ¼ fi1 ; y; ir gDf1; y; dg we consider the homomorphism vnI :¼ pI 3 vn : vn
pI
V - Kd - KI ; where pI denotes the projection on the r-dimensional coordinate subspace K I of K d defined by I: The weight wðvn Þ shall be defined by 8rpd
ðwðvn ÞXr38IDf1; y; dg; jIj ¼ r : rankðvnI Þ ¼ rÞ:
Lemma 1.5. For vn ¼ ðvn1 ; y; vnd ÞAðV n Þd the following conditions are equivalent: (1) wðvn ÞXr; (2) any r components vni1 ; y; vnir are linearly independent. In particular, we have wn ASvn ) wðwn Þ ¼ wðvn Þ
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so that for a form f ¼ Sðl1 ; y; ld Þ the quantity rankð f Þ :¼ wðl1 ; y; ld Þ is well defined and bounded by k ¼ dimðV Þ from above. Proof. For the equivalence of (1) and (2) we observe that the linear independence of vni1 ; y; vnir yields the existence of vi1 ; y; vir AV such that vnij ðvik Þ ¼ djk and thus for I ¼ fi1 ; y; ir g the homomorphism pI 3 vn has rank Xr: Conversely, if for I ¼ fi1 ; y; ir g the rank of pI 3 vn is Xr; this homomorphism is surjective so that vnij ðvik Þ ¼ djk has a solution and the vnij are linearly independent. As I was arbitrary, this holds for any r components of vn : For the last statement, note that wn ASvn implies that the components of wn are, up to scalar multiples, just a permutation of the ones of vn ; by (2) this does not affect the weight function. Moreover, rankðvnI Þprankðvn ÞpdimðV Þ: & The relation between the parameters k and d has a significant influence on rl and thus on Autð f Þ; especially for forms of rank k; which we will call non-degenerate. Lemma 1.6. Let f AFd be a form of rank r such that 2prpdimðV Þod and rl the homomorphism corresponding to the decomposition f ¼ l1 ?ld : If for some gAAutð f Þ we have rl ðgÞ ¼ ðl; idÞ; then li ¼ x ði ¼ 1; y; dÞ for some xAmd ; the group of dth roots of unity, and there exists an r-dimensional subspace V0 of V with gjV0 ¼ x idV0 : In particular, if f is non-degenerate, g ¼ x idV and kerðrl Þ is finite. Proof. For t ¼ id; ð%Þ becomes g ðl1 ; y; ld Þ ¼ ðl1 l1 ; y; ld ld Þ; which we may split up into g l i ¼ li l i
for i ¼ 1; y; d:
Suppose further that rankðl1 ; y; ld Þ ¼ r and choose fl1 ; y; lr g as a basis for the subspace of V n spanned by fðl1 ; y; ld Þg: Consequently, lrþ1 ¼
r X
aj lj
and
g lrþ1 ¼ lrþ1 lrþ1
j¼1
together imply lrþ1
r X j¼1
aj lj ¼
r X
aj lj l j :
j¼1
At this point, the linear independence of any set of r linear forms out of fl1 ; y; lrþ1 g guarantees aj a0; 1pjpr; and thus lj ¼ lrþ1 for j ¼ 1; y; r: Repeated use of this argument yields l1 ¼ ? ¼ ld ¼: x and l1 yld ¼ xd ¼ 1 shows xAmd as desired.
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We may thus rewrite ð%Þ as g ðl1 ; y; ld Þ ¼ x idV ðl1 ; y; ld Þ 3 ðl1 ; y; ld Þ 3 g1 ¼ l1 ; y; ld 3 x1 idV and call vn the composition ðl1 ; y; ld Þ 3 ðg1 x1 idV Þ: Then vn 0 and thus we conclude imðg1 x1 idV ÞDkerðl1 ; y; ld Þ: By assumption kerðl1 ; y; ld Þ is a ðk rÞdimensional subspace of V ; so that V0 :¼ kerðg1 x1 idV Þ is an r-dimensional subspace on which g ¼ x idV : If, moreover, f is non-degenerate V0 ¼ V and g is equivalent to some multiple of idV : Since this is contained in the center of GlðV0 Þ we conclude g ¼ x idV : This leaves only finitely many choices for g corresponding to the d possible xAmd : Obviously, any of these lie in kerðrl Þ and everything is proved. & From now on, we will focus on non-degenerate forms. In this case, we consider the composition of rl with the projection p : S-S=iðTÞDiðSd Þ; which makes sense since iðTÞoS is normal and we obtain a homomorphism r* l : Autð f Þ -Sd ; g /tg :¼ i1 ðpðrl ð f ÞÞÞ associated to the decomposition f ¼ l1 ?ld : Note that this reduction does not involve any loss of information since gAkerðr* l Þ3 i1 ðpðrl ð f ÞÞÞ ¼ id 3 pðrl ð f ÞÞ ¼ Pid 3 (lAT : rl ð f Þ ¼ ðl; idÞ and this is equivalent to gAkerðrl Þ by Lemma 1.6. As an immediate consequence we have Corollary 1.7. For any non-degenerate f AFd the group Autð f ÞDGLðV Þ is finite and we have the estimate jAutð f Þjpdðd!Þ:
The finiteness of Autð f Þ for forms of degree dX3; not necessarily decomposable, was first claimed by Jordan [Jo]; however, in this more general case the proof involves other techniques. See also [CR,OS] for generalizations.
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The next step consists in the study of the image of Autð f Þ under r* l : Let f AFd be a form of rank r such that 2prpdimðV Þod and f ¼ l1 ?ld be a decomposition of f : Definition 1.8. The permutation group of f relative to this decomposition is the subgroup of Sd defined by Gl ð f Þ :¼ r* l ðAutð f ÞÞ:
With this notation, we may reformulate Lemma 1.6 as follows: For every decomposition f ¼ l1 ?ld we obtain a short exact sequence 1-md -Autð f Þ-Gl ð f Þ-1 with the obvious maps x/x idV and g/r* l ðgÞ ¼ tg : So far the above sequence has the drawback that the factor Gl ð f Þ is not uniquely determined for f AFd : However, the conjugacy of any two homomorphisms rl and rm over S translates to conjugacy of r* l and r* m over Sd so that Autð f ÞDmd sGl ð f Þ for any set of linear factors l1 ?ld : Consequently, we can say more about the elements in Autð f Þ by using the structure of the permutations tAGð f Þ that is invariant under conjugation: Definition 1.9. We say that a permutation tASd contains a cycle of length r > 1; if there exists an iAf1; y; dg such that tr ðiÞ ¼ i and tj ðiÞai for 1ojor: We write ðn Þ ðn Þ t ¼ ðr1 1 ; y; rs s Þ if r1 ; y; rs are the cycle P lengths of the cycles in t and n1 ; y; ns their respective multiplicities. For instance j nj rj ¼ d: Lemma 1.10. Let f AFd be non-degenerate and gAAutð f Þ with r* l ðgÞ ¼: tg AGl ð f Þ for ðn Þ
ðn Þ
some decomposition of f : If tg ¼ ðr1 1 ; y; rs s Þ; then g is a root of unity of order pd lcmðr1 ; y; rs Þ ¼: dR: Proof. First we note that the cycle lengths r1 ; y; rs are indeed independent of the special decomposition of f : For any i; 1pips; we use r* l ðgri Þ ¼ ðr* l ðgÞÞri ¼ trgi to conclude that trgi has ri fixed points. By Lemma 1.6 this yields gR ¼ xð1; idÞ ¼ x idV and the exact order of g depends on the order of xAmd ; which is bounded by d: &
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In order to say more about the possible structure of permutations tAGð f Þ we need the following refinement of Lemma 1.6 which will open the way to some interesting applications of the previous lemma. Proposition 1.11. With the assumptions of Lemma 1.10 we have: tg has more than k fixed points 3 g ¼ x idV with xd ¼ 1: Proof. As fixed points are precisely the cycles of length 1, their number is again independent of the chosen decomposition of f and the assumption is well defined. W.l.o.g. we may assume that f1; y; k þ 1g is among the fixed points of tg ; hence with rl ðgÞ ¼ ðl; tÞ we have g l i ¼ li l i
for i ¼ 1; y; k þ 1:
As dimðV Þ ¼ kok þ 1 and rankðl1 ; y; ld Þ ¼ k we conclude with the same arguments as in Lemma 1.6 that l1 ; y; lkþ1 are eigenvectors corresponding to one single eigenvalue x :¼ l1 ¼ ? ¼ lkþ1 Amd and the linear independence of any k of these vectors yields g li ¼ xli for i ¼ k þ 2; y; d as well. If gð f Þ denotes any permutation group in Sd arising for some f AFd and some decomposition of f ; this result enables us to obtain considerable restrictions on the possible cycle lengths of permutations tAGð f Þ: & ðn Þ
ðn Þ
Proposition 1.12. Let t ¼ tg ¼ ðr1 1 ; y; rs s Þ be in the permutation group of some f AFd and some gAAutð f Þ: Then for every subset frs1 ; y; rst g of fr1 ; y; rs g such that X nsj rsj > k; j
ð%Þ implies tlcmðrs1 ;y;rst Þ ¼: tRs ¼ id: In particular, all other cycles of t have an order dividing lcmðrs1 ; y; rst Þ ¼ Rs : Proof. Suppose frs1 ; y; rst g is such a collection of cycle lengths. P Then the total number of vectors li involved in the corresponding cycles is j nsj rsj and thus > k by assumption. This yields at least k þ 1 fixed points of tRs ; by Proposition 1.11 we thus have gRs ¼ x idV and the order of g is ordðxÞ Rs : Comparison with the estimate of ordðgÞ in Lemma 1.10 yields Rs ¼ lcmðrs1 ; y; rst Þ ¼ lcmðr1 ; y; rs Þ ¼ R and the last claim is immediate.
&
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Corollary 1.13. With the assumptions as in Proposition 1.12 suppose that there is an integer Tpd such that X rsi nðrsi Þ > k; rsi jT
then all cycles in t have an order r dividing T: Proof. Note that rs1 ; y; rst jT implies lcmðrs1 ; y; rst ÞjT: The result is now a direct consequence of the previous proposition. & In particular, this covers the case of permutations that contain one cycle of length t > k: However, if t ¼ k; we may even improve this result: Proposition 1.14. If t contains a cycle of length k; then ð%Þ implies that t consists only of cycles of order r; where rjk: Proof. Without loss of generality, we may suppose that g l1
¼ l2 l2 ; ^
g lk1 g lk
¼ lk l k ; ¼ l1 l1
is the cycle of length k: As before we conclude that there are k linearly independent eigenvectors for the action of gk and since they all belong to the same cycle, they all correspond to the same eigenvalue l1 ylk : Thus, the action of g is equivalent to the one of x idV where x :¼ l1 ylk Amd and once again tk ¼ id; the conclusion of Corollary 1.12 thus holds in this case as well. & The restrictions obtained so far allow to determine completely the possible structure of Gð f Þ in several interesting cases; we will devote the next section to some examples of how this may be achieved using the results of this section.
2. Some examples We first examine the case of forms whose degree is large compared to the number of variables and start by investigating when the condition of Corollary 1.12 is fulfilled. Proposition 2.1. Suppose f is a form in k variables of degree d > k2 k: Then for every tAGl ð f Þ for some decomposition of f there exists a positive integer tpd such that all cycles in t have an order dividing t:
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Proof. Suppose that for any tpd we have X lnðrÞpk: rjt
Then obviously t contains not more than ½k=t t-cycles for any tok and no t-cycle for kptpd: Since for the degree of f we have X d¼ tnðtÞ; tpd
the above assumption implies dp
X
t½k=t pkðk 1Þ:
tok
Consequently, for d > k2 k this assumption cannot be fulfilled for all tpd and Corollary 1.12 is applicable, which concludes the proof. & The situation is particularly simple if k is small, even though the number of possibilities for tAGð f Þ increases rapidly with k: We will examine the cases k ¼ 2 and 3 in detail. Corollary 2.2. Let f ¼ l1 ?ld be a form in 2 variables of degree dX4: Then any taid in Gl ð f Þ consists of: (1)
d t
t-cycles ðtjdÞ; or (2) d1 t t-cycles ðtjd 1Þ and 1 fixed point, or (3) d2 t t-cycles ðtjd 2Þ and 2 fixed points.
Proof. For k ¼ 2 the existence of any cycle of length X2 already implies that all cycles have the same length (indeed all divisors > 1 would be at least 2 and thus themselves equal to the order of all cycles). Since there are at most 2 fixed points, the three cases above are the only possibilities to split up d into cycles of equal length and fixed points. & Corollary 2.3. Let f ¼ l1 ?ld be a form in 3 variables of degree dX5: Then any taid in Gl ð f Þ consists of: (1)
d t
t-cycles ðtjdÞ; or (2) d1 t t-cycles ðtjd 1Þ and 1 fixed point, or
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244
(3)
d2 t
t-cycles ðtjd 2Þ and 2 fixed points,
or (4)
d3 t
t-cycles ðtjd 3Þ and 3 fixed points,
or (5)
d2 t
(6)
d3 t
t-cycles ðtjd 2Þ and one 2-cycle,
or t-cycles ðtjd 3Þ; one 2-cycle and 1 fixed point, where the last two cases are possible only for even t:
Proof. For k ¼ 3 the existence of any cycle of length tX3 already implies that all cycles of length X3 have the same length t by the same argument as in the previous corollary. If t contains two 2-cycles, all other cycles have order 2 as well or are fixed points by Corollary 1.12. Since t contains at most 3 fixed points, the above restrictions leave only the mentioned six cases, where we notice that (5) and (6) may only happen when t is even (Corollary 1.12 again), since tt ¼ id must imply t2 ¼ id: We also give an example of how to use Proposition 1.11 if Corollary 1.12 fails. Let d ¼ 37 and t ¼ ð10; 12; 15Þ: If kp15; it is obvious that teGð f Þ for a form of degree d in k variables. (Take T ¼ 15:) For 15okp24 we argue as follows: let l1 ¼ 10; l2 ¼ 15; so l1 þ l2 ¼ 25 > 24 ¼ k and tlcmð10;15Þ ¼ t30 ¼ id by the proposition. But 12 [30 and thus the permutation ð10; 12; 15Þ does not appear in any Gð f Þ if kp24: &
3. Generic forms—a special case Having examined the structure of Autð f Þ in general in the previous two sections, we now turn to the question: Do forms in general have non-trivial automorphisms? This is not only of interest by itself, but also has considerable influence on representations of forms by linear forms as explained in [Su]. Definition 3.1. A form f of degree d is called generic if Autð f ÞDmd ; that is gAAutð f Þ ) (xAmd : g ¼ xidV : In the case of non-degenerate forms, this condition is equivalent to the statement that the permutation group of f for some (and thus any) decomposition f ¼ l1 ?ld reduces to fidg by Lemma 1.6. In order to determine whether there exist generic forms in k variables of degree d; and if this is the case, to find out if this is rather exceptional or maybe true for almost all (non-degenerate) forms, we start by a rather heuristical analysis of system ð%Þ:
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If f ¼ l1 ?ld and rl ¼ ðl; tÞ; the fact that f is generic translates to the solutions of ð%Þ in the following way: 8taid;
g li ¼ ltðiÞ ltðiÞ
for i ¼ 1; y; d;
has only the trivial solution l1 ¼ ? ¼ ld ¼ 0 and g ¼ 0 and it seems natural to distinguish the following cases: (a) kX2 and dpk þ 1: For this choice of parameters, ð%Þ has less equations than variables since dpk þ 1 ¼
k2 1 3ðk 1Þdpk2 13kd þ 1pk2 þ d: k1
Consequently, there is no hope to expect the majority of forms to be generic in this case since there will be infinitely many solutions to ð%Þ in general (only the presence of contradictory equations would lead to generic forms). (b) kX2 and dXk þ 2: Comparing again the number of equations and variables in ð%Þ; we find that for k ¼ 2; d ¼ 4; the number of equations equals the number of variables ðkd ¼ 8 ¼ k2 þ dÞ; k > 2 or da4; there are strictly fewer variables than equations. This suggests that generic forms are more or less the rule, although for k ¼ 2 and d ¼ 4 an exceptional behavior can be expected. The latter is confirmed in [Po, Section 7.2, Example 1]; however, for the convenience of the reader, we use the case of binary forms f of degree 4 to illustrate the concepts introduced so far. We write such forms as f ðX ; Y Þ ¼ a0 X 4 þ a1 X 3 Y þ ? þ a4 Y 4 ¼ a0
4 Y
ðX þ ri Y Þ;
i¼1
with 0aa0 AK and by assumption the ri lie in K as well and are pairwise disjoint. In this setup, we may express the definition of f being generic explicitly by replacing li by X þ ri Y in ð%Þ to the coefficient matrix of the system arising from ð%Þ compute for any matrix A ¼
a b g d
a li 3 A ¼ ð1; ri Þ g
in Autð f Þ to obtain
b d
! ¼ ða þ gri ; b þ dri Þ ¼ ðltðiÞ ; ltðiÞ rtðiÞ Þ ¼ ltðiÞ ltðiÞ
for i ¼ 1; 2; 3; 4: The resulting four equations in the first component determine the li in dependence of a; b; g; d and we may focus on the remaining homogeneous system in these four variables after plugging in a þ gri for ltðiÞ : The resulting coefficient matrix Cf
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is given by 0
rtð1Þ
B B rtð2Þ Cf :¼ B Br @ tð3Þ rtð4Þ
1 r1 rtð1Þ
r1
1
1 r3 rtð3Þ
C r2 C C: r3 C A
1 r4 rtð4Þ
r4
1 r2 rtð2Þ
This shows that the statement f is generic and is equivalent to 8taid : det Cf a0 since the homogeneous system ð%Þ has a non-trivial solution precisely when the rank of the above coefficient matrix is o4 for some t: In fact, det Cf ¼ 0 implies the existence of a solution a; b; g; d of the reduced system for which l1 yl4 ¼ 1: This is clear if ltðiÞ ¼ a þ gri a0 for i ¼ 1; 2; 3; 4; otherwise, if a þ gr1 ¼ 0; say, we immediately conclude b þ dr1 ¼ 0 as well, a contradiction to AAGLk ðKÞ: With this characterization of generic forms we obtain: Proposition 3.2. Let f ðX ; Y Þ ¼ a0 X 4 þ a1 X 3 Y þ ? þ a4 Y 4 be a non-degenerate form of degree 4 with a0 a0 and decomposition
f ðX ; Y Þ ¼ a0
4 Y ðX þ ri Y Þ: i¼1
Then Gð f Þ is a group of order 4, 8 or 12. Consequently, Gð f Þ and thus Autð f Þ are never trivial and all the binary forms of degree 4 are non-generic. Proof. We simply compute Cf for all the different permutations t in S4 : To start with, let t consist of two 2-cycles, say t ¼ ð1; 2Þð3; 4Þ: In this case, r2 r 1 Cf ¼ r4 r 3
1
r1 r2
1
r1 r2
1 1
r3 r4 r3 r4
r1 r2 ¼ 0; r3 r4
independent of ri ; since the difference of rows 1 and 2 is proportional to the difference of rows 3 and 4. Thus, Gð f Þ contains all the permutations consisting of two 2-cycles and we conclude jGð f ÞjX4; so that it remains to check for which other tAS4 the determinant Cf may vanish.
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247
Next, let t have a fixed point together with a 3-cycle, say t ¼ ð2; 3; 4Þ: In this case, r1 1 r21 r1 r 1 r r r 3 2 3 2 Cf ¼ ¼ a22 3a1 a3 þ 12a0 a4 ; r4 1 r3 r4 r3 r 1 r r r 2
4 2
4
as a direct computation shows. This expression is independent of the chosen 3-cycle, which means that either all or no such t is contained in Gð f Þ; depending on whether a22 3a1 a3 þ 12a0 a4 ¼ 0 or not. Now let t consist of a 4-cycle, say t ¼ ð1; 2; 3; 4Þ: We then compute r2 1 r1 r2 r1 r 1 r r r 3 2 3 2 Cf ¼ ¼ ðr1 r2 Þðr3 r4 Þ½3ðr1 r3 þ r2 r4 Þ a2 : r4 1 r3 r4 r3 r 1 r r r 1
4 1
4
In this product, the first two factors are non-zero by assumption and Gð f Þ contains t ¼ ð1; 2; 3; 4Þ and its inverse precisely when 3ðr1 r3 þ r2 r4 Þ a2 ¼ 0: Gð f Þ then contains no other 4-cycle; in fact, if t ¼ ð1; 3; 2; 4Þ were in Gð f Þ; we would have 3ðr1 r2 þ r3 r4 Þ a2 ¼ 0 as well, and this would already imply the equality of two roots of f ; a contradiction to f non-degenerate. As a consequence, we always have Gð f ÞaS4 : & Altogether, we may summarize our observations in the following listing (let Di;j denote the expression 3ðri rj þ rk rl Þ a2 for fi; j; k; lg ¼ f1; 2; 3; 4gÞ: a22 3a1 a3 þ 12a0 a4 a0 and Di;j a0 8ði; jÞ ) jGð f Þj ¼ 4; a22 3a1 a3 þ 12a0 a4 a0 and Di;j ¼ 0 for some ði; jÞ ) jGð f Þj ¼ 8; a22 3a1 a3 þ 12a0 a4 ¼ 0 and Di;j a0 8ði; jÞ ) jGð f Þj ¼ 12: Note that these cases are the only possible ones, since a22 3a1 a3 þ 12a0 a4 ¼ 0 and Di;j ¼ 0 would already imply the equality of two roots of f :
4. Generic forms—the general case Now that the special case k ¼ 2; d ¼ 4 is settled, we are left with proving the conjecture that for k ¼ 2 and dX5; respectively, kX3 and dXk þ 2 almost all forms are generic, as suggested by the comparison of the number of equations and variables in ð%Þ:
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For this purpose, we consider the Zariski Topology on ðV n Þd in order to state the theorem that formulates the above conjecture in a precise terminology and mention a result about constructible sets which we will use several times. For reference see [Hs, p. 94, Example 3.18]. Proposition 4.1. A constructible subset of an irreducible Zariski space is dense if and only if it contains the generic point. Furthermore, in that case it contains a non-empty open subset. Theorem 4.2. The set X of d-tuples vn ¼ ðl1 ; y; ld ÞAðV n Þd for which f ¼ l1 ?ld is generic is constructible. For dXmaxf5; k þ 2g X is Zariski dense and thus contains some non-empty Zariski open set O: Before we proceed to the proof of this statement, we formulate two lemmas that will be needed in the argument. Lemma 4.3. The set of degenerate forms is a closed subvariety of ðV n Þd : Proof. Recall that f ¼ Svn was called degenerate if wðvn Þok and by Lemma 1.5 this is equivalent to (ICf1; y; dg
with jIj ¼ k : rankðvnI Þok;
where vnI : V -K jIj is the composition of vn with pI : But this may be expressed as Y
det vnI ¼ 0;
I:jIj¼k
which defines a Zariski closed set in ðV n Þd : & In what follows, we need to consider the variety ZðtÞCGLðV Þ ðV n Þd T defined by ð%Þ; that is ðg; vn ; lÞAZðtÞ3 g vn ¼ ðl; tÞ vn 3 rvn ðgÞ ¼ ðl; tÞ with ðl; tÞAS: Similarly, for any ICf1; y; dg we define ZI ðtÞ by ðg; vn ; lÞAZI ðtÞ3rvn ðgÞ ¼ ðm; sÞ with ððm; sÞ vn ÞI ¼ ððl; tÞ vn ÞI : Note that this in particular implies sjI ¼ ljI and that ZI ðtÞ-ZJ ðtÞ ¼ ZI,J ðtÞ: The projection from ZðtÞ to the second component ðV n Þd is denoted by p and this notation is kept for its restriction to ZI ðtÞ:
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Lemma 4.4. Let vn ¼ l1 ; y; ld ApðZðtÞÞ with wðvn Þ ¼ k and ICf1; y; dg with jIj ¼ k þ 1: Then p1 ðvn Þ-ZI ðtÞ is independent of I; thus p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ since
S
jIj¼kþ1
I ¼ f1; y; dg and both sets have cardinality d:
Proof. First we notice that ZI ðtÞ+ZðtÞ implies p1 ðvn Þ-ZI ðtÞ+p1 ðvn Þ-ZðtÞ; so that it suffices to show that p1 ðvn Þ-ZI ðtÞDp1 ðvn Þ-ZðtÞ: To see this we consider two distinct elements in p1 ðvn Þ-ZI ðtÞ for which we have rvn ðgÞ ¼ ðm; sÞ
and
rvn ðg0 Þ ¼ ðm0 ; s0 Þ
with the required restrictions on I: By definition of ZI ðtÞ; we have sjI ¼ tjI ¼ s0 jI ; i.e. s1 s0 jI ¼ idjI and since 1 0 0 n n rg1 g0 ðvn Þ ¼ r1 g ðv Þrg0 ðv Þ ¼ ðm; sÞ ðm ; s Þ;
restriction to I yields r* g1 g0 ðvn ÞjI ¼ idjI ; so that r* g1 g0 ðvn Þ has at least k þ 1 fixed points. Lemma 1.11 yields g0 ¼ xg and m0 ¼ xm for some x in md from which we conclude that jp1 ðvn Þ-ZðtÞjpjp1 ðvn Þ-ZI ðtÞjpd: Further p1 ðvn Þ-ZðtÞ is non-empty by assumption and has cardinality d; indeed the implication ðg; vn ; lÞAZðtÞ ) ðxg; vn ; x1 lÞAZðtÞ for xAmd shows that md acts faithfully on ZðtÞ by ðg; vn ; lÞ-ðxg; vn ; x1 lÞ so that we get an injection md +ZðtÞ: This implies dpjp1 ðvn Þ-ZðtÞj and we obtain p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ as desired.
&
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As we have seen in the proof ðg; vn ; lÞ/ðxg; vn ; x1 lÞ leaves r* vn ðgÞ invariant and thus defines an action of md on ZðtÞ: This action leaves vn invariant and may be extended to a Z :¼ ZðtÞ: tASd
Note that ZðtÞ consists of the ðg; vn ; lÞ in Z for which r* vn ðgÞ ¼ t and that the extension of p to Z is still the projection to ðV n Þd : In order to relate this setup to the concept of generic forms, we consider the subvariety C :¼ ðGLðV Þ\fmd idV gÞ ðV n Þd T of GLðV Þ ðV n Þd T: Proof of Theorem 4.2. The first step consists in showing that X is constructible. By definition f ¼ Svn is non-generic if there exists some gaxidV in GLðV Þ and some ðl; tÞ in S such that ðg; vn ; lÞAZðtÞ: This amounts to say p1 ðvn Þ-Z-Ca|; so that vn is generic precisely if ðg; vn ; lÞ ¼ p1 ðvn ÞAZ\ðZ-CÞ and this is a constructible set in GLðV Þ ðV n Þd T since all open and closed sets as well as finite unions, intersections and complements are in this class. By a theorem of Chevalley (see [Hs, p. 94, Example 3.19]) constructibility is conserved under the projection p and thus X ¼ pðZ\ðZ-CÞÞ is constructible as well. For the second part, we proceed by contraposition. Suppose X was not dense for the indicated range of parameters. Its complement ðV n Þd \X is constructible as well and thus by Proposition 4.1 would contain some dense open set O: By Lemma 4.3 (intersect O with fvn : wðvn Þ ¼ kg) we may assume that f ¼ Svn is non-degenerate for vn AO: This allows to apply Lemma 1.6 which yields p1 ðvn Þ-Z-CDp1 ðvn Þ-ðZ\ZðidÞÞ and since the left-hand side is non-empty (vn eX) this implies the existence of some taid with p1 ðvn Þ-ZðtÞa| for vn AO: Consequently, we get a covering [ O¼ pðp1 ðvn Þ-ZðtÞÞ taid
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of O by constructible sets. By Proposition 4.1 at least one of the covering sets must contain some non-empty open subset Ot DO; so for the corresponding t we have p1 ðvn Þ-ZðtÞa| for all vn AOt : If we let T be the set consisting of the identity and of all t subject to this restriction, i.e. T :¼ ftASd : (|aOt DO open; 8vn AOt : p1 ðvn Þ-ZðtÞa|g; we have thus shown TPfidg: T We claim: T is a normal subgroup of Sd :1 Indeed O :¼ tAT Ot is open, nonempty and for vn AO; t; sAT there exist ðg; vn ; lÞAp1 ðvn Þ-ZðtÞ and
ðg0 ; vn ; mÞAp1 ðvn Þ-ZðsÞ
by definition of O: We thus have r* vn ðg1 g0 Þ ¼ t1 s; so that p1 ðvn Þ-Zðt1 sÞa|; i.e. t1 sAT and T is indeed a subgroup of Sd : Moreover, with t0 ¼ sts1 we have r* vn ðð1; sÞgÞ ¼ t0 (see Lemma 1.3), which shows that T is normal. For dX5; the only non-trivial normal subgroups of Sd are Ad and Sd itself, so there are only these two T groups to consider. Now the assumption dXk þ 2 implies the existence of some tAT that induces a non-trivial permutation on I :¼ f1; y; k þ 1g (note that either ð1; y; k þ 1Þ or ð1; y; kÞðk þ 1Þ lies in TjI ) and we intend to show that any such t is determined uniquely by its restriction to I: To do so, note that Lemma 4.4 yields p1 ðvn Þ-ZI ðtÞ ¼ p1 ðvn Þ-ZðtÞ for vn AO: But this set is disjoint to p1 ðvn Þ-Zðt0 Þ for any other t0 AT: indeed ðg; vn ; lÞAZðtÞ-Zðt0 Þ yields g vn ¼ ðl; tÞ vn
and
g vn ¼ ðl; t0 Þ vn
and by Lemma 1.3 ðg1 gÞ vn ¼ vn ¼ ðl; tÞ1 ðl; t0 Þ vn : This shows the existence of some nAT such that ðidV ; vn ; nÞAZðt1 t0 Þ; which implies that any two coordinates li and lt1 t0 ðiÞ of vn are proportional for i ¼ 1; y; d: Of course, as vn is non-degenerate, this is only possible if t ¼ t0 : 1
The author is indebted to Y. Neretin for this observation.
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In other words, we have jI : StabðIÞ-T -Skþ1 -TjI ; s /sjI is injective. If T ¼ Sd this yields ðk þ 1Þ!ðd k 1Þ! ¼ jStabðIÞjpjSkþ1 j ¼ ðk þ 1Þ! and thus d k 1p1 which implies d ¼ k þ 2: On the other hand, for T ¼ Ad it implies ðk þ 1Þ! ðd k 1Þ! ðk þ 1Þ! ¼ jStabðIÞ-Ad jI jpjSkþ1 -Ad jI j ¼ ; 2 2 2 showing that d k 1p2 which is only possible for d ¼ k þ 2 or k þ 3 and we are reduced to handle only these three special cases. Let us consider d ¼ k þ 2 first and let t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ: Then if tAT we have p1 ðvn Þ-ZðtÞa| for any vn ¼ ðl1 ; y; lkþ2 ÞAO: We denote by prI the projection from ðV n Þd to the first k þ 1 coordinates and observe that prI ðvn Þ K
lkþ2 defines a closed subset of ðV n Þd : Consequently, the complement of this set in ðV n Þd is Zariski open and thus has non-empty intersection with O: So O contains 0 Þ for which some wn ¼ ðl10 ; y; lkþ2 li0 ¼ li for iAI
and
0 lkþ2 ac lkþ2 ; cAK:
By definition of wn ðg; vn ; lÞAp1 ðvn Þ-ZI ðtÞ3ðg; wn ; lÞAp1 ðwn Þ-ZI ðtÞ and Lemma 4.4 yields ðg; vn ; lÞAp1 ðvn Þ-ZðtÞ3ðg; wn ; lÞAp1 ðwn Þ-ZðtÞ; 0 0 ¼ lkþ2 lkþ2 : As l1 ; y; lk form a basis of ðV n Þd ; we so that glkþ2 ¼ lkþ2 lkþ2 and glkþ2 may write
lkþ2 ¼
k X
ai l i ;
i¼1
0 lkþ2 ¼
k X i¼1
bi li
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0 and in combination with the above equations for lkþ2 and lkþ2 we obtain k X
ai ltðiÞ ltðiÞ ¼ lkþ2
i¼1
k X
k X
ai l i ;
i¼1
btðiÞ ltðiÞ ¼ lkþ2
i¼1
k X
b i li :
i¼1
A comparison of the coefficients (note that tAStabðf1; y; kgÞ and thus both sums are linear combinations of the chosen basis vectors) yields at1 ðiÞ li ¼ lkþ2 ai ; bt1 ðiÞ li ¼ lkþ2 bi : This leads to asðiÞ =bsðiÞ ¼ ai =bi for i ¼ 1; y; k and all s in the cyclic group /tS generated by t: Thus, the quotient is 0 independent of i; i.e. lkþ2 and lkþ2 are proportional, a contradiction to the choice n of w : This argument works only if tAT; which is certainly true if T is the whole permutation group. However, it is also applicable to the case T ¼ Ad ; as kX3 implies that either t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ or
t0 ¼ ð1; 2; y; k 1ÞðkÞðk þ 1Þðk þ 2Þ
is in T: In the latter case, the aforementioned argument shows that ai =bi is independent of i for i ¼ 1; y; k 1: If this ratio was different from ak =bk ; the 0 decomposition of lkþ2 (resp. lkþ2 ) in the basis fl1 ; y; lk g would yield a linear 0 dependence of lk ; lkþ2 and lkþ2 : With respect to the basis fl1 ; y; lk1 ; lkþ1 g the same 0 and finally to computation would lead to a linear dependence of lkþ1 ; lkþ2 and lkþ2 the vanishing of a non-trivial linear combination of lk ; lkþ1 and lkþ2 : As kX3 this contradicted the non-degeneracy of vn : Finally, if d ¼ k þ 3; either t ¼ ð1; y; kÞðk þ 1Þðk þ 2Þðk þ 3Þ
or
t0 ¼ ð1; y; kÞðk þ 1; k þ 2Þðk þ 3Þ
0 lies in T: With I ¼ f1; y; k þ 2g and lkþ3 ac lkþ3 the same strategy as explained for d ¼ k þ 2; t ¼ ð1; 2; y; kÞðk þ 1Þðk þ 2Þ yields the result. (Note that Lemma 4.4 holds for any I with jIj > k þ 1 as well.) &
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C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962. R. Hartshorne, Algebraic Geometry, Springer Graduate Texts, Springer, Berlin. C. Jordan, Memoire sur l’equivalence des formes, Oeuvres III, Gauthier Villars, Paris, 1962. P. Orlik, L. Solomon, Automorphisms of forms, Math. Ann. 231 (1978) 229–240. V. Popov, Survey in Algebraic Geometry, Part 2, Invariant Theory, in: Encyclopaedia of Mathematical Sciences, Vol. 55, Springer, Berlin, 1994. L. Summerer, Forms represented by linear forms, Monatsh. Math., submitted for publication.