Hyperdéterminant d’un $S{L}_{2}$-homomorphisme
Annales Mathématiques Blaise Pascal, Tome 15 (2008) no. 1, pp. 81-86.

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.

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.

DOI : https://doi.org/10.5802/ambp.240
Classification : 14L30
Mots clés : Hyperdéterminant, fibrés de Steiner, $S{L}_{2}$ modules
@article{AMBP_2008__15_1_81_0,
author = {Vall\es, Jean},
title = {Hyperd\'eterminant d'un $SL\_{2}$-homomorphisme},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {81--86},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {15},
number = {1},
year = {2008},
doi = {10.5802/ambp.240},
mrnumber = {2418014},
zbl = {1141.14030},
language = {fr},
url = {www.numdam.org/item/AMBP_2008__15_1_81_0/}
}
Vallès, Jean. Hyperdéterminant d’un $SL_{2}$-homomorphisme. Annales Mathématiques Blaise Pascal, Tome 15 (2008) no. 1, pp. 81-86. doi : 10.5802/ambp.240. http://www.numdam.org/item/AMBP_2008__15_1_81_0/`

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