Sylvester versus Gundelfinger

Let Vn be the SL₂-module of binary forms of degree n and let V=V₁⊕V₃⊕V₄. We show that the minimum number of generators of the algebra R=C[V]SL₂ of polynomial functions on V invariant under the action of SL₂ equals 63. This settles a 143-year old question.

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2012
Hauptverfasser:	Brouwer, A.E., Popoviciu, M.
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spelling	Brouwer, A.E. Popoviciu, M. 2019-02-18T18:08:45Z 2019-02-18T18:08:45Z 2012 Sylvester versus Gundelfinger / A.E. Brouwer, M. Popoviciu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13A15; 68W30 DOI: http://dx.doi.org/10.3842/SIGMA.2012.075 https://nasplib.isofts.kiev.ua/handle/123456789/148715 Let Vn be the SL₂-module of binary forms of degree n and let V=V₁⊕V₃⊕V₄. We show that the minimum number of generators of the algebra R=C[V]SL₂ of polynomial functions on V invariant under the action of SL₂ equals 63. This settles a 143-year old question. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html. The second author is partially supported by the Swiss National Science Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Sylvester versus Gundelfinger Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Sylvester versus Gundelfinger
spellingShingle	Sylvester versus Gundelfinger Brouwer, A.E. Popoviciu, M.
title_short	Sylvester versus Gundelfinger
title_full	Sylvester versus Gundelfinger
title_fullStr	Sylvester versus Gundelfinger
title_full_unstemmed	Sylvester versus Gundelfinger
title_sort	sylvester versus gundelfinger
author	Brouwer, A.E. Popoviciu, M.
author_facet	Brouwer, A.E. Popoviciu, M.
publishDate	2012
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
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description	Let Vn be the SL₂-module of binary forms of degree n and let V=V₁⊕V₃⊕V₄. We show that the minimum number of generators of the algebra R=C[V]SL₂ of polynomial functions on V invariant under the action of SL₂ equals 63. This settles a 143-year old question.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148715
citation_txt	Sylvester versus Gundelfinger / A.E. Brouwer, M. Popoviciu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 20 назв. — англ.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 075, 7 pages Sylvester versus Gundelfinger? Andries E. BROUWER † and Mihaela POPOVICIU ‡ † Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: aeb@cwi.nl URL: http://www.win.tue.nl/~aeb/ ‡ Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland E-mail: mihaela.popoviciu@unibas.ch Received July 18, 2012, in final form October 12, 2012; Published online October 19, 2012 http://dx.doi.org/10.3842/SIGMA.2012.075 Abstract. Let Vn be the SL2-module of binary forms of degree n and let V = V1⊕V3⊕V4. We show that the minimum number of generators of the algebra R = C[V ]SL2 of polynomial functions on V invariant under the action of SL2 equals 63. This settles a 143-year old question. Key words: invariants; covariants; binary forms 2010 Mathematics Subject Classification: 13A15; 68W30 1 Introduction In 1868 Gordan [4] proved that the algebra of invariants of binary forms of given degree is finitely generated. This came as a surprise to Cayley and Sylvester, who had believed that the quintic and septimic had covariant resp. invariant rings that were not finitely generated. The number of invariants is first infinite in the case of a quantic of the seventh order, or septic; the number of covariants is first infinite in the case of a quantic of the fifth order, or quintic. (Cayley [2]) However, finding a minimal set of generators for these algebras is even today an open problem in all but a few small cases. In the case of V4 ⊕ V4, Gordan found a generating set of size 30, and Sylvester [17] showed that two of these generators are superfluous. He added J’ajouterai seulement que cette preuve éclatante de l’insuffisance de la méthode de M. Gordan et de son école, pour séparer les formes véritablement élémentaires des formes superflues qui s’y rattachent (insuffisance reconnue par M. Gordan lui-même de la manière la plus loyale dans son discours inaugural prononcé à Erlangen), n’ôte rien à la valeur immense du service qu’il a rendu à l’Algèbre, en ayant le premier démontré l’existence d’une limite au nombre de ces formes. This note focuses on the covariants of V3⊕V4, a case which illustrates the controversy between the German and English schools in the 19th century. The German school, following Clebsch and Gordan, was able to construct a system of generators for the algebra of invariants of binary forms, with no guarantee that the system was minimal. The English school, following Cayley and Sylvester, aimed to determine the number of independent generators. Sylvester used in his computations his ‘fundamental postulate’ (not defined here), which turned out to hold only in small cases. Counterexamples were given by Hammond [7] and Morley [10]. ?This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html mailto:aeb@cwi.nl http://www.win.tue.nl/~aeb/ mailto:mihaela.popoviciu@unibas.ch http://dx.doi.org/10.3842/SIGMA.2012.075 http://www.emis.de/journals/SIGMA/SDE2012.html 2 A.E. Brouwer and M. Popoviciu In 1869 Gundelfinger, a student of Clebsch, wrote a thesis [6] where he constructed generators for the covariants of V3 ⊕ V4 ‘in ordinary symbolic notation’, after Clebsch had given him this system as computed by Gordan in his ‘obscure’ notation (cf. [5, pp. 270–272]). He found 20 generators for the invariants and 64 for the covariants. Sylvester used the Poincaré series together with his fundamental postulate to show that there could be only 61 independent generators for the covariants of V3 ⊕ V4, and wrote a series of papers [15, 18, 19, 20] showing the superiority of the English methods over the German. In the first paper Sylvester uses his method (which he calls ‘tamisage’) to derive the numbers of generators of given degrees in the coefficients of V4 and those of V3, and given order in the variables x, y. The following table is taken from [20]: In the second paper he observes that it follows from the Poincaré series that there are 8 li- nearly independent covariants of order 2 and multidegree (4,3). Next, he constructs 8 reducible such covariants (products of covariants of lower degree) and argues that these are linearly inde- pendent. However, the forms are dependent and only seven are independent. He finishes with the announcement In the third paper he observes that it follows from the Poincaré series that there are 12 linearly independent covariants of order 1 and multidegree (5,4). Next, he constructs 12 reducible such covariants and argues that these are linearly independent. However, the forms are dependent and only eleven are independent. He concludes Sylvester versus Gundelfinger 3 (false theorem omitted) Here we show that the actual minimal number of generators for the covariants of V3 ⊕ V4 is 63. Our results coincide with those of Sylvester and Gundelfinger, with two exceptions: we show that one needs one generating covariant of order 1 and multidegree (5,4), and (only) one generating covariant of order 2 and multidegree (4,3). For completeness we give the corrected version of Sylvester’s table. The two corrected entries are underlined. Deg. in Deg. in coeff’s Order coeff’s of of quartic. cubic. 0 1 2 3 4 5 0 0 1 1 2 1 4 1 1 2 3 2 1 6 1 3 2 1 1 1 1 1 3 2 3 2 1 5 1 2 2 1 Deg. in Deg. in coeff’s Order coeff’s of of quartic. cubic. 0 1 2 3 2 2 1 2 2 1 4 2 2 1 3 1 1 1 1 1 3 1 1 1 1 4 0 1 1 2 1 1 1 5 1 1 1 6 0 1 1.1 Invariants and covariants Let V be a finite-dimensional vector space over a field k, with basis e1, . . . , em. Let xi be the coordinate function defined by xi( ∑ ξheh) = ξi. The algebra k[V ] of polynomial functions on V is by definition the algebra generated by the xi. (It does not depend on the choice of basis e1, . . . , em.) Let G be a group of linear transformations of V . It acts on k[V ] via the action (g · f)(v) = f(g−1v). Invariant theory studies k[V ]G, the algebra of G-invariant polynomial functions on V , i.e., the f ∈ k[V ] such that g · f = f for all g ∈ G. A covariant of order m and degree d of V is a G-equivariant homogeneous polynomial map φ : V → Vm of degree d. In other words, φ(g · v) = g · φ(v), for all g ∈ G, and φ(tv) = tdφ(v), for all t ∈ k. In particular, the covariants of V of order 0 are the invariants of V . Below we shall take k = C, G = SL2(k), and V = Vn1 ⊕ · · · ⊕ Vnp , where Vn is the vector space (of dimension n + 1) consisting of 0 and the binary forms of degree n, that is, of the homogeneous polynomials of degree n v(x, y) = a0x n + ( n 1 ) a1x n−1y + · · ·+ ( n n− 1 ) an−1xy n−1 + any n, in two variables. This Vn is the n-th graded part of k[W ], where W is a 2-dimensional vector space over C with natural action of SL2, hence has a natural action of SL2. The main way to construct covariants is via transvectants (Überschiebungen). These are derived from the Clebsch–Gordan decomposition of the SL2-module Vm ⊗ Vn, with m ≥ n: Vm ⊗ Vn ' Vm+n ⊕ Vm+n−2 ⊕ · · · ⊕ Vm−n. This decomposition defines for each p, 0 ≤ p ≤ n, an SL2-equivariant linear map Vm ⊗ Vn → Vm+n−2p, denoted (g, h) 7→ (g, h)p, and called the p-th transvectant. It is given explicitly by the 4 A.E. Brouwer and M. Popoviciu following formula: (g, h) 7→ (g, h)p := (m− p)!(n− p)! m!n! p∑ i=0 (−1)i ( p i ) ∂pg ∂xp−i∂yi ∂ph ∂xi∂yp−i (see [11, Chapter 5]). The covariants of V can be identified with the invariants of V1⊕V : we have V1⊕V ' V ∗1 ⊕V as SL2-representations and the set of covariants of V is isomorphic to k[V ∗1 ⊕ V ]SL2 (see [12, Chapter 15]). Each covariant φ of V of order m corresponds to the invariant of V1 ⊕ V defined by the transvectant (φ(v), lm)m, where l ∈ V1. 2 The generators of the invariants of V1 ⊕ V3 ⊕ V4 We identify the covariants of V3⊕V4 with the invariants of V1⊕V3⊕V4 and show that a minimal set of generators for the algebra of invariants of this module has size 63. Doing this type of work requires finding dependencies. Gundelfinger did not try to do this exhaustively, but following Gordan he only noted the obvious ones. Sylvester tried, and made some mistakes, no doubt because he already knew what answer he wanted. For us this is relatively easy – a modern computer has no problems computing the rank of a 40000 by 600000 matrix (which is what is needed in the most straightforward approach). We had a different problem: up to which degree should we compute covariants or inva- riants? Gundelfinger ‘just’ followed Gordan’s algorithm, but as far as we know that has not been implemented yet. The secret knowledge known today but not in the 19th century, is that the ring R of invariants of V1 ⊕ V3 ⊕ V4 (or any such ring) is Cohen–Macaulay (see [9]). It has a homogeneous system of parameters (hsop) j1, . . . , jr, algebraically independent, and finitely many further generators i1, . . . , is, such that every invariant can be uniquely written as a linear combination of products imjm1 · · · jmh . It follows that the Poincaré series P (t) = ∑ dit i, where di is the dimension of the degree i part of R, is of the form P (t) = ta1 + · · ·+ tas (1− tb1) · · · (1− tbr) , where the ah and bh are the degrees of the ih and jh. For the module V = Vn1 ⊕ · · · ⊕ Vnp , with ∑ i ni ≥ 3, one has P (t−1) = (−1)d−3tdP (t) where d = ∑ i(ni + 1) (by [14, Corollary 2] for p = 1, and by [1, Theorem 2] in general), so that maxh ah − ∑ j bj = − ∑ i(ni + 1). Therefore, in order to find maxh{ah, bh} it suffices to find the bh. The power series P (t) = ∑ dit i is known from Cayley [2] and Sylvester [16] (cf. [13, 3.3.4]). In the present case, P (t) = 1 + t2 + 2t3 + 5t4 + 10t5 + 18t6 + 31t7 + 55t8 + 92t9 + 144t10 + 223t11 + 341t12 + 499t13 + 725t14 + 1031t15 + 1436t16 + 1978t17 + 2685t18 + 3592t19 + 4761t20 + 6235t21 + 8078t22 + 10379t23 + 13226t24 + 16698t25 + 20937t26 + 26069t27 + 32230t28 + 39614t29 + 48401t30 + · · · = a(t) (1− t3)(1− t4)2(1− t5)2(1− t6)2(1− t7) , Sylvester versus Gundelfinger 5 where a(t) = 1 + t2 + t3 + 3t4 + 7t5 + 12t6 + 21t7 + 32t8 + 47t9 + 58t10 + 72t11 + 83t12 + 89t13 + 94t14 + 94t15 + 89t16 + 83t17 + 72t18 + 58t19 + 47t20 + 32t21 + 21t22 + 12t23 + 7t24 + 3t25 + t26 + t27 + t29, and it follows that computing invariants up to degree 29 suffices if we know that there is a hsop with degrees 3, 4, 4, 5, 5, 6, 6, 7. 2.1 Finding a hsop Hilbert introduced in the 19th century the notion of nullcone. If V is an SL2-module, then the nullcone N (V ) of V is the set of elements of V on which all invariants of V of positive degree vanish. The elements of N (V ) are called nullforms. One can show that a binary form f ∈ Vn is a nullform if and only if f has a root of multiplicity > n 2 (this is a consequence of the Hilbert– Mumford criterion, see [3, § 2.4.1]). Similarly, if we have p binary forms f1, . . . , fp of degrees n1, . . . , np, then (f1, . . . , fp) ∈ N (Vn1⊕· · ·⊕Vnp) if and only if f1, . . . , fp have a common root that has multiplicity > ni 2 in fi, for all i = 1, . . . , p. In our particular case, if (l, c, q) ∈ V1 ⊕ V3 ⊕ V4, then (l, c, q) ∈ N (V1 ⊕ V3 ⊕ V4) if and only if l2 \| c and l3 \| q. Let V(J) stand for the vanishing locus of J . The following result, due to Hilbert, gives a characterisation of homogeneous systems of parameters of k[Vn1 ⊕ · · · ⊕ Vnp ]SL2 as sets that define the nullcone of N (Vn1 ⊕ · · · ⊕ Vnp): Proposition 1 (Hilbert [8]). Let V = Vn1 ⊕ · · · ⊕ Vnp, and R = k[V ]SL2, and m = n1 + · · · + np +p−3 > 0. A set {j1, . . . , jm} of homogeneous elements of R is a system of parameters of R if and only if V(j1, . . . , jm) = N (V ). Let our binary forms l ∈ V1, c ∈ V3, q ∈ V4 be l = c0x+ c1y, c = a0x 3 + 3a1x 2y + 3a2xy 2 + a3y 3, q = b0x 4 + 4b1x 3y + 6b2x 2y2 + 4b3xy 3 + b4y 4, and consider the following invariants: k2 = (q, q)4, k3 = ((q, q)2, q)4, k4,1 = ((c, c)2, (c, c)2)2, k4,2 = (lc, lc)4, k4,3 = (c, l3)3, k5,1 = ( (q, (q, q)2)1, c 2 ) 6 , k5,2 = ( (q, c2)2, c 2 ) 6 , k5,3 = ( q, l4 ) 4 , k6,1 = ( [(c, c)2] 2, (q, q)2 ) 4 , k6,2 = ((lc, lc)2, lc)4, k6,3 = ( (q, q)2, l 4 ) 4 , k7 = ( c4, q3 ) 12 . We prove the following Proposition 2. With the notations above, the invariants j1 = k3, j2 = k4,1 + k22, j3 = k4,2 + k4,3 − k22, j4 = k5,1 + k5,2, j5 = k5,3, j6 = k6,1 + k6,2, j7 = k6,3, j8 = k7, (of degrees 3, 4, 4, 5, 5, 6, 6, 7, respectively) form a system of parameters of k[V1⊕V3⊕V4]SL2. 6 A.E. Brouwer and M. Popoviciu Proof. We show that V(j1, . . . , j8) = N (V1 ⊕ V3 ⊕ V4). Consider three cases. Case 1: q = 0. In this case, the vanishing of j1, . . . , j8 reduces to k4,1 = k4,2 +k4,3 = k6,2 = 0, which implies that (l, c) ∈ N (V1 ⊕ V3). Indeed, if k4,1 = 0, then c is a nullform, and, without loss of generality, we may suppose that x2 \| c, i.e. a2 = a3 = 0. But then k6,2 ∼ a31c31 (we use ‘∼’ for equality up to a nonzero constant). If c1 = 0, then (l, c) ∈ N (V1⊕V3). If c1 6= 0, then a1 = 0 and k4,2 + k4,3 ∼ a0c31. Hence a0 = 0, so that c = 0 and (l, c) ∈ N (V1 ⊕ V3). Case 2: l = 0. In this case, the vanishing of j1, . . . , j8 reduces to k2 = k3 = k4,1 = k5,1+k5,2 = k6,1 = k7 = 0, which implies that (c, q) ∈ N (V3⊕V4). Indeed, the vanishing of k2, k3, k4,1 implies that c and q are nullforms. If c or q vanish identically, then the statement is clear. Otherwise, if the double zero of c and the triple zero of q do not coincide, we may suppose, without loss of generality, that x2 \| c and y3 \| q, i.e. a2 = a3 = b0 = b1 = b2 = 0. Then k6,1 ∼ a41b 2 3. If a1 = 0, then k5,1 + k5,2 ∼ a20b 3 3, and k7 ∼ a40b 3 4, which contradicts the assumption c, q 6= 0. If b3 = 0, then k5,1 + k5,2 ∼ a41b4 and k7 ∼ a40b34, which again contradicts the assumption c, q 6= 0. Case 3: q, l 6= 0. In this case, j5 = 0 implies that q and l have a common root (up to a constant, j5 is the resultant of q and l). Without loss of generality, we can suppose that the common factor of q and l is x, i.e., c1 = b4 = 0 and c0 6= 0. Then j7 ∼ b23c 4 0, which implies b3 = 0. Then j1 ∼ b32, which implies b2 = 0. Then a3 becomes a factor of j8. If a3 = 0, then j3 ∼ a22c 2 0, which implies a2 = 0, and then (l, c, q) ∈ N (V1 ⊕ V3 ⊕ V4). If a3 6= 0, we may take a3 = c0 = 1. Now j3 ∼ 3a22 − 3a1 − 2, and it follows that a1 = a22 − 2 3 . Then j6 ∼ 27a32 − 54a2 − 27a0 − 256b21, and it follows that a0 = a32 − 2a2 − 256 27 b 2 1. Then j4 ∼ 36b0 − 144a2b1 − 949b31, and it follows that b0 = 4a2b1 + 949 36 b 3 1. Then j2 ∼ 27− 2048b41, j8 ∼ b51 ( 33205248− 4273351745b41 ) . But j2 = j8 = 0 has no solution. This settles Case 3. By Proposition 1, it follows that these eight invariants form a hsop of the ring of invariants of V1 ⊕ V3 ⊕ V4. � 2.2 The degrees of the generators The Poincaré series of the ring of invariants of V1⊕V3⊕V4 tells us which is the maximal degree in which we have to look for generators, namely 29. For each i ≤ 29 we do the following: multiply invariants of smaller degrees to see what part of the vector space of invariants of degree i is known. The Poincaré series tells us how big the dimension of this vector space is, and if the known invariants do not yet span this vector space, one constructs in some way further invariants, until they do span. In the following table i denotes the degree of the generators, and di the number of generators of degree i needed: i 2 3 4 5 6 7 8 9 10 11 di 1 2 4 8 10 13 11 10 3 1 For 12 ≤ i ≤ 29 no further generators are needed, and it follows that the minimal number of generators is 63. Sylvester versus Gundelfinger 7 Acknowledgements The second author is partially supported by the Swiss National Science Foundation. References [1] Brion M., Invariants de plusieurs formes binaires, Bull. Soc. Math. France 110 (1982), 429–445. [2] Cayley A., A second memoir upon quantics, Phil. Trans. Royal Soc. London 146 (1856), 101–126. [3] Derksen H., Kemper G., Computational invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 130, Springer-Verlag, Berlin, 2002. [4] Gordan P., Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit nu- merischen Coeffizienten einer endlichen Anzahl solcher Formen ist, J. Reine Angew. Math. 69 (1868), 323–354. [5] Gordan P., Die simultanen Systeme binärer Formen, Math. Ann. 2 (1870), 227–280. [6] Gundelfinger S., Zur Theorie des simultanen Systems einer cubischen und einer biquadratischen binären Form, Habilitationsschrift, J.B. Metzler, Stuttgart, 1869. [7] Hammond J., Note on an exceptional case in which the fundamental postulate of professor Sylvester’s theory of tamisage fails, Proc. London Math. Soc. 14 (1882), 85–88. [8] Hilbert D., Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–373. [9] Hochster M., Roberts J.L., Rings of invariants of reductive groups acting on regular rings are Cohen– Macaulay, Adv. Math. 13 (1974), 115–175. [10] Morley R.K., On the fundamental postulate of tamisage, Amer. J. Math. 34 (1912), 47–68. [11] Olver P.J., Classical invariant theory, London Mathematical Society Student Texts, Vol. 44, Cambridge University Press, Cambridge, 1999. [12] Procesi C., Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007. [13] Springer T.A., Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin, 1977. [14] Springer T.A., On the invariant theory of SU2, Indag. Math. 42 (1980), 339–345. [15] Sylvester J.J., Détermination du nombre exact des covariants irréductibles du système cubo-biquadratique binaire, C. R. Acad. Sci. Paris 87 (1878), 477–481. [16] Sylvester J.J., Proof of the hitherto undemonstrated fundamental theorem of invariants, Phil. Mag. 5 (1879), 178–188. [17] Sylvester J.J., Sur le vrai nombre des covariants élémentaires d’un système de deux formes biquadratiques binaires, C. R. Acad. Sci. Paris 84 (1877), 1285–1289. [18] Sylvester J.J., Sur le vrai nombre des formes irréductibles du système cubo-biquadratique, C. R. Acad. Sci. Paris 87 (1878), 445–448. [19] Sylvester J.J., Sur les covariants fundamentaux d’un système cubo-quartique binaire, C. R. Acad. Sci. Paris 87 (1878), 287–289. [20] Sylvester J.J., Franklin F., Tables of the generating functions and groundforms for the binary quantics of the first ten orders, Amer. J. Math. 2 (1879), 223–251. http://dx.doi.org/10.1515/crll.1868.69.323 http://dx.doi.org/10.1007/BF01444021 http://dx.doi.org/10.1112/plms/s1-14.1.85 http://dx.doi.org/10.1007/BF01444162 http://dx.doi.org/10.1016/0001-8708(74)90067-X http://dx.doi.org/10.2307/2370111 http://dx.doi.org/10.1017/CBO9780511623660 http://dx.doi.org/10.2307/2369240 1 Introduction 1.1 Invariants and covariants 2 The generators of the invariants of V1V3 V4 2.1 Finding a hsop 2.2 The degrees of the generators References

Sylvester versus Gundelfinger

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