no. 5
The higher transvectants are redundant
[Les transvectants d’ordre supérieur sont redondants]
Abdesselam, Abdelmalek1 ; Chipalkatti, Jaydeep2
1 University of Virginia Department of Mathematics Kerchof Hall P. O. Box 400137 Charlottesville, VA 22904-4137 (USA)
2 University of Manitoba Department of Mathematics 433 Machray Hall Winnipeg MB R3T 2N2 (Canada)
Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713.

Pour deux formes binaires génériques A,B, notons 𝔲 r =(A,B) r leur transvectant d’ordre r, tel que défini en théorie classique des invariants. Dans cet article, nous obtenons une classification complète des syzygies quadratiques entre les {𝔲 r }. Il en résulte que les transvectants d’ordre supérieur {𝔲 r :r≥2} sont redondants, en ce sens qu’ils peuvent être exprimés à partir de 𝔲 0 et 𝔲 1 . Ce résultat peut s’interpréter géométriquement en termes du plongement incomplet de Segre. Les calculs utilisés reposent sur la suite exacte de Cauchy en théorie des représentations de SL 2 , ainsi que sur la notion de symbole 9-j de la théorie quantique du moment angulaire.

Nous donnons des exemples de calculs explicites concernant SL 3 ,𝔤 2 et 𝔖 5 afin d’indiquer l’existence possible de résultats analogues pour d’autres catégories de représentations.

Let A,B denote generic binary forms, and let 𝔲 r =(A,B) r denote their r-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the {𝔲 r }. As a consequence, we show that each of the higher transvectants {𝔲 r :r≥2} is redundant in the sense that it can be completely recovered from 𝔲 0 and 𝔲 1 . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL 2 -representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.

We give explicit computational examples for SL 3 ,𝔤 2 and 𝔖 5 to show that this result has possible analogues for other categories of representations.

DOI : 10.5802/aif.2474
Classification : 13A50, 22E70
Keywords: Angular momentum in quantum mechanics, binary forms, Cauchy exact sequence, 9-j symbols, representations of $SL_2$, transvectants
Mot clés : théorie quantique du moment angulaire, formes binaires, suite exacte de Cauchy, représentation de $SL_2$, transvectants

Abdesselam, Abdelmalek 1 ; Chipalkatti, Jaydeep 2

1 University of Virginia Department of Mathematics Kerchof Hall P. O. Box 400137 Charlottesville, VA 22904-4137 (USA)
2 University of Manitoba Department of Mathematics 433 Machray Hall Winnipeg MB R3T 2N2 (Canada) 
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Abdesselam, Abdelmalek; Chipalkatti, Jaydeep. The higher transvectants are redundant. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713. doi : 10.5802/aif.2474. https://aif.centre-mersenne.org/articles/10.5802/aif.2474/

[1] Abdesselam, A. The combinatorics of classical invariant theory revisited by modern physics, Slides of Feb 2007 talk at the Montreal CRM workshop “Combinatorial Problems Raised by Statistical Mechanics”. Available at http://people.virginia.edu/~aa4cr/MontrealFeb07slides.pdf

[2] Abdesselam, A.; Chipalkatti, J. The bipartite Brill-Gordan locus and angular momentum, Transform. Groups, Volume 11 (2006) no. 3, pp. 341-370 | DOI | MR | Zbl

[3] Abdesselam, A.; Chipalkatti, J. Brill-Gordan Loci, transvectants and an analogue of the Foulkes conjecture, Adv. Math, Volume 208 (2007) no. 2, pp. 491-520 | DOI | MR | Zbl

[4] Akin, K.; Buchsbaum, D.; Weyman, J. Schur functors and Schur complexes, Adv. Math., Volume 44 (1982) no. 3, pp. 207-278 | DOI | MR | Zbl

[5] Ališauskas, S. J.; Jucys, A. P. Weight lowering operators and the multiplicity-free isoscalar factors for the group R 5 , J. Math. Phys., Volume 12 (1971) no. 4, pp. 594-605 | DOI | MR | Zbl

[6] Biedenharn, L. C.; Louck, J. D. Angular Momentum in Quantum Physics. Theory and Application, Encyclopedia of Mathematics and its Applications, 8, Addison-Wesley, 1981 | MR | Zbl

[7] Bourbaki, N. Topological Vector Spaces, (translated by H. G. Eggleston and S. Madan), Elements of Mathematics, Springer-Verlag, 1987 | MR | Zbl

[8] Brunnemann, J.; Thiemann, T. Simplification of the spectral analysis of the volume operator in loop quantum gravity, Class. Quant. Grav., Volume 23 (2006), pp. 1289-1346 | DOI | MR | Zbl

[9] Brussaard, P. J. Clebsch-Gordan- of Wigner coefficienten, Ned. Tijdschr. Natuurk., Volume 33 (1967), pp. 202-222

[10] Carter, J.; Flath, D.; Saito, M. The Classical and Quantum 6j-Symbols, Mathematical Notes, Princeton University Press, 1995 no. 43 | MR | Zbl

[11] Cayley, A. On linear transformations, Collected Mathematical Works, vol. I, Cambridge University Press, 1889 no. 14

[12] Chipalkatti, J. On the invariant theory of the Bezoutiant, Beiträge Alg. Geom., Volume 47 (2006) no. 2, pp. 397-417 | MR | Zbl

[13] Clebsch, A. Theorie der Binaren Algebraischen Formen, Teubner, Leipzig, 1872 (MiH)

[14] Condon, E. U.; Shortley, G. H. The Theory of Atomic Spectra, Cambridge University Press, 1935 | Zbl

[15] Dolgachev, I. Lectures on Invariant Theory, London Mathematical Society Lecture Notes, Cambridge University Press, 2003 no. 296 | MR | Zbl

[16] Edmonds, A. R. Angular Momentum in Quantum Mechanics, Princeton University Press, 1957 | MR | Zbl

[17] Flath, D. The Clebsch-Gordan formulas, Enseign. Math. (2), Volume 29 (1983) no. 3–4, pp. 339-346 | MR | Zbl

[18] Fulton, W. Young Tableaux, London Mathematical Society Student Texts, Cambridge University Press, 1957 no. 35 | MR | Zbl

[19] Fulton, W.; Harris, J. Representation Theory, A First Course, Graduate Texts in Mathematics, Springer–Verlag, 1991 | MR | Zbl

[20] Glenn, O. The Theory of Invariants, Ginn and Co., Boston, 1915 (PG)

[21] Goldberg, L. Catalan numbers and branched coverings by the Riemann sphere, Adv. Math., Volume 85 (1991) no. 2, pp. 129-144 | DOI | MR | Zbl

[22] Gordan, P. Die Resultante Binärer Formen, Rend. Circ. Matem. Palermo, Volume XXII (1906), pp. 161-196 | DOI

[23] Grace, J. H.; Young, A. The Algebra of Invariants, 1903, Reprinted by Chelsea Publishing Co., New York, 1962 (MiH)

[24] Harris, J. Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer–Verlag, 1992 | MR | Zbl

[25] Hartshorne, R. Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, 1977 | MR | Zbl

[26] Huang, J.-S.; Zhu, C.-B. Weyl’s construction and tensor power decomposition for G 2 , Proc. Amer. Math. Soc., Volume 127 (1999) no. 3, pp. 925-934 | DOI | MR | Zbl

[27] Jahn, H. A.; Hope, J. Symmetry properties of the Wigner 9j symbol, Phys. Rev., Volume 93 (1954) no. 2, pp. 318-321 | DOI | Zbl

[28] Jeugt, J. Van der; Sangita, N. Pitre; Srinivasa Rao, K. Multiple hypergeometric functions and 9-j coefficients, J. Phys A: Math. Gen., Volume 27 (1994), pp. 5251-5264 | DOI | MR | Zbl

[29] Jucys, A. P.; Bandzaitis, A. A. Angular Momentum in Quantum Physics, 2nd Ed., Vilnius: Mokslas, 1977

[30] Kirillov, A. A. Elements of the Theory of Representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, 1976 | MR | Zbl

[31] Kung, J. P. S.; Rota, G.-C. The invariant theory of binary forms, Bulletin of the A.M.S., Volume 10 (1984) no. 1, pp. 27-85 | DOI | MR | Zbl

[32] Littlewood, D. E. Invariant theory, tensors and group characters, Philos. Trans. Roy. Soc. London. Ser. A, Volume 239 (1944) no. 807, pp. 305-365 | DOI | MR | Zbl

[33] MacDonald, I. G. Symmetric Functions and Hall polynomials, (2nd ed.), Oxford University Press, 1995 | MR | Zbl

[34] Mumford, D. Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, Princeton University Press, 1966 no. 59 | MR | Zbl

[35] Olver, P. Classical Invariant Theory, London Mathematical Society Student Texts, Cambridge University Press, 1999 | MR | Zbl

[36] Racah, G. Theory of complex spectra II, Phys. Rev., Volume 62 (1942), pp. 438-462 | DOI

[37] Rosengren, H. On the triple sum formula for Wigner 9j-symbols, J. Math. Phys., Volume 39 (1998) no. 12, pp. 6730-6744 | DOI | MR | Zbl

[38] Rosengren, H. Another proof of the triple sum formula for Wigner 9j-symbols, J. Math. Phys., Volume 40 (1999) no. 12, pp. 6689-6691 | DOI | MR | Zbl

[39] Salmon, G. Higher Algebra, Reprinted by Chelsea Publishing Co., New York, 1965

[40] Springer, T. A. Invariant Theory, Lecture Notes in Mathematics, Springer-Verlag, 1977 no. 585 | MR | Zbl

[41] Stroh, E. Entwicklung der Grundsyzyganten der binären Form fünfter Ordnung, Math. Ann., Volume 34 (1889), pp. 354-370 | DOI | MR

[42] Sturmfels, B. Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, 1993 | MR | Zbl

[43] Wigner, E. Group Theory and Its Application to the Quantum Theory of Atomic Spectra, Academic Press, 1959 | MR | Zbl

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