The higher transvectants are redundant
[Les transvectants d’ordre supérieur sont redondants]
Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713.

Pour deux formes binaires génériques A,B, notons 𝔲r=(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 {𝔲r}. Il en résulte que les transvectants d’ordre supérieur {𝔲r:r2} sont redondants, en ce sens qu’ils peuvent être exprimés à partir de 𝔲0 et 𝔲1. 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 SL2, ainsi que sur la notion de symbole 9-j de la théorie quantique du moment angulaire.

Nous donnons des exemples de calculs explicites concernant SL3,𝔤2 et 𝔖5 afin d’indiquer l’existence possible de résultats analogues pour d’autres catégories de représentations.

Let A,B denote generic binary forms, and let 𝔲r=(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 {𝔲r}. As a consequence, we show that each of the higher transvectants {𝔲r:r2} is redundant in the sense that it can be completely recovered from 𝔲0 and 𝔲1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL2-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.

We give explicit computational examples for SL3,𝔤2 and 𝔖5 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 SL2, transvectants
Mots-clés : théorie quantique du moment angulaire, formes binaires, suite exacte de Cauchy, représentation de SL2, 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/

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