The higher transvectants are redundant

Abdesselam, Abdelmalek; Chipalkatti, Jaydeep

doi:10.5802/aif.2474

The higher transvectants are redundant
[Les transvectants d’ordre supérieur sont redondants]

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)

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713.

Résumé
Abstract

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 ${𝔲_{r} : r \geq 2}$ 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 $S L_{2}$ , ainsi que sur la notion de symbole 9-j de la théorie quantique du moment angulaire.

Nous donnons des exemples de calculs explicites concernant $S L_{3}, 𝔤_{2}$ 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 ${𝔲_{r} : r \geq 2}$ 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 $S L_{2}$ -representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.

We give explicit computational examples for $S L_{3}, 𝔤_{2}$ and $𝔖_{5}$ to show that this result has possible analogues for other categories of representations.

MR Zbl

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

Affiliations des auteurs :

Abdesselam, Abdelmalek ¹ ; Chipalkatti, Jaydeep ²

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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. http://www.numdam.org/articles/10.5802/aif.2474/

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