Hyperdéterminant d’un $S{L}_{2}$-homomorphisme
[Hyperdeterminant of an $S{L}_{2}$-homomorphism]
Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, pp. 81-86.

Let ${A}_{1},\cdots ,{A}_{s}$ ($s\ge 3$) be non-trivial $S{L}_{2}\left(ℂ\right)$-modules with dimensions ${n}_{1}+1\ge \cdots \ge {n}_{s}+1$ (such that ${n}_{1}={n}_{2}+\cdots +{n}_{s}$) and $\phi \in ℒ\left({A}_{2}\otimes \cdots \otimes {A}_{s},{A}_{1}^{*}\right)$ an $S{L}_{2}\left(ℂ\right)$-homomorphism. We show that the hyperdeterminant of $\phi$ is null except if the modules ${A}_{i}$ are irreducibles and the homomorphism is the multiplication of homogeneous polynomials with two variables.

Etant donnés ${A}_{1},\cdots ,{A}_{s}$ ($s\ge 3$) des $S{L}_{2}\left(ℂ\right)$-modules non triviaux de dimensions respectives ${n}_{1}+1\ge \cdots \ge {n}_{s}+1$ (avec ${n}_{1}={n}_{2}+\cdots +{n}_{s}$) et $\phi \in ℒ\left({A}_{2}\otimes \cdots \otimes {A}_{s},{A}_{1}^{*}\right)$ un $S{L}_{2}\left(ℂ\right)$-homomorphisme, nous montrons que l’hyperdéterminant de $\phi$ est nul sauf si les modules ${A}_{i}$ sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

DOI: 10.5802/ambp.240
Classification: 14L30
Keywords: Hyperdeterminant, Steinerbundles, $S{L}_{2}$ modules
Vallès, Jean 1

1 Laboratoire de Mathématiques Pures et Appliquées Université de Pau et des Pays de l’Adour 64000 PAU FRANCE
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Vallès, Jean. Hyperdéterminant d’un $SL_{2}$-homomorphisme. Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 1, pp. 81-86. doi : 10.5802/ambp.240. http://www.numdam.org/articles/10.5802/ambp.240/`

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