The higher transvectants are redundant

[Les transvectants d’ordre supérieur sont redondants]

Abdesselam, Abdelmalek ; Chipalkatti, Jaydeep

Annales de l'Institut Fourier, Tome 59 (2009), p. 1671-1713 / Harvested from Numdam

Résumé

Pour deux formes binaires génériques $A,B$, notons $\mathfrak{u}_r = (A,B)_r$ 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 $\lbrace \mathfrak{u}_r\rbrace $. Il en résulte que les transvectants d’ordre supérieur $\lbrace \mathfrak{u}_r: r \ge 2\rbrace $ sont redondants, en ce sens qu’ils peuvent être exprimés à partir de $\mathfrak{u}_0$ et $\mathfrak{u}_1$. 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 $SL_3, \mathfrak{g}_2$ et $\mathfrak{S}_5$ afin d’indiquer l’existence possible de résultats analogues pour d’autres catégories de représentations.

Abstract

Let $A,B$ denote generic binary forms, and let $\mathfrak{u}_r = (A,B)_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\lbrace \mathfrak{u}_r\rbrace $. As a consequence, we show that each of the higher transvectants $\lbrace \mathfrak{u}_r: r \ge 2\rbrace $ is redundant in the sense that it can be completely recovered from $\mathfrak{u}_0$ and $\mathfrak{u}_1$. 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 $SL_3, \mathfrak{g}_2$ and $\mathfrak{S}_5$ to show that this result has possible analogues for other categories of representations.

Détail

Publié le : 2009-01-01

MR 2573188 | Zbl 1189.13004

DOI : https://doi.org/10.5802/aif.2474
Classification:  13A50,  22E70
Mots clés: théorie quantique du moment angulaire, formes binaires, suite exacte de Cauchy, représentation de $SL_2$, transvectants

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