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
Mots-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/

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