Hyperdeterminant of an S L 2 -homomorphism

Jean Vallès[1]

  • [1] Laboratoire de Mathématiques Pures et Appliquées Université de Pau et des Pays de l’Adour 64000 PAU FRANCE

Annales mathématiques Blaise Pascal (2008)

  • Volume: 15, Issue: 1, page 81-86
  • ISSN: 1259-1734

Abstract

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Let A 1 , ⋯ , A s ( s ≥ 3 ) be non-trivial S L 2 ( ℂ ) -modules with dimensions n 1 + 1 ≥ ⋯ ≥ n s + 1 (such that n 1 = n 2 + ⋯ + n s ) and φ ∈ ℒ ( A 2 ⊗ ⋯ ⊗ A s , A 1 * ) an S L 2 ( ℂ ) -homomorphism. We show that the hyperdeterminant of φ is null except if the modules A i are irreducibles and the homomorphism is the multiplication of homogeneous polynomials with two variables.

How to cite

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Vallès, Jean. "Hyperdéterminant d’un $SL_{2}$-homomorphisme." Annales mathématiques Blaise Pascal 15.1 (2008): 81-86. <http://eudml.org/doc/10554>.

@article{Vallès2008,
abstract = {Etant donnés $A_\{1\},\cdots ,A_\{s\}$ ($ s\ge 3 $) des $SL_\{2\}(\{\mathbb\{C\}\})$-modules non triviaux de dimensions respectives $n_\{1\}+1\ge \cdots \ge n_\{s\}+1$ (avec $n_\{1\}= n_\{2\}+\cdots +n_\{s\}$) et $\phi \in \mathcal\{L\}(A_\{2\}\otimes \cdots \otimes A_\{s\}, A_\{1\}^\ast )$ un $SL_\{2\}(\{\mathbb\{C\}\})$-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.},
affiliation = {Laboratoire de Mathématiques Pures et Appliquées Université de Pau et des Pays de l’Adour 64000 PAU FRANCE},
author = {Vallès, Jean},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Hyperdeterminant; Steinerbundles; $SL_\{2\}$ modules},
language = {fre},
month = {1},
number = {1},
pages = {81-86},
publisher = {Annales mathématiques Blaise Pascal},
title = {Hyperdéterminant d’un $SL_\{2\}$-homomorphisme},
url = {http://eudml.org/doc/10554},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Vallès, Jean
TI - Hyperdéterminant d’un $SL_{2}$-homomorphisme
JO - Annales mathématiques Blaise Pascal
DA - 2008/1//
PB - Annales mathématiques Blaise Pascal
VL - 15
IS - 1
SP - 81
EP - 86
AB - Etant donnés $A_{1},\cdots ,A_{s}$ ($ s\ge 3 $) des $SL_{2}({\mathbb{C}})$-modules non triviaux de dimensions respectives $n_{1}+1\ge \cdots \ge n_{s}+1$ (avec $n_{1}= n_{2}+\cdots +n_{s}$) et $\phi \in \mathcal{L}(A_{2}\otimes \cdots \otimes A_{s}, A_{1}^\ast )$ un $SL_{2}({\mathbb{C}})$-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.
LA - fre
KW - Hyperdeterminant; Steinerbundles; $SL_{2}$ modules
UR - http://eudml.org/doc/10554
ER -

References

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  1. V Ancona, G Ottaviani, Unstable hyperplanes for Steiner bundles and multidimensional matrices, Advances in Geometry 1 (2001), 165-192 Zbl0983.14034MR1840220
  2. A Cayley, On the theory of linear transformations, Cambridge Math. J. (1845), 193-209 
  3. C Dionisi, Stabilizers for nondegenerate matrices of boundary format and Steiner bundles, Rev.Mat.Complut. 17 (2004), 459-469 Zbl1068.14019MR2083965
  4. I Dolgachev, M M Kapranov, Arrangement of hyperplanes and vector bundles on ℙ n , Duke Math. J. (1993), 633-664 Zbl0804.14007MR1240599
  5. I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser Zbl0827.14036MR1264417
  6. R L E. Schwarzenberger, Vector bundles on the projective plane, Proc. London Math. Soc. 11 (1961), 623-640 Zbl0212.26004MR137712

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