Plethysm and fast matrix multiplication

Tim Seynnaeve

doi:10.1016/j.crma.2017.11.012

Lie algebras/Computer science

Plethysm and fast matrix multiplication
[Pléthysme et multiplication rapide des matrices]

Présenté par : Michèle Vergne

Tim Seynnaeve ¹

¹ Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 52-55.

Résumé
Abstract

Motivés par la version symétrique de la multiplication des matrices, nous étudions le pléthysme $S^{k} ({sl}_{n})$ de la représentation adjointe ${sl}_{n}$ du groupe de Lie $S L_{n}$ . En particulier, pour $k = 3$ , nous décrivons la décomposition de cette représentation en composantes irréductibles, et nous trouvons les vecteurs de plus grand poids pour toutes ces dernières. Nous présentons les liens avec la multipliction rapide des matrices, notamment le tenseur de Coppersmith–Winograd.

Motivated by the symmetric version of matrix multiplication we study the plethysm $S^{k} ({sl}_{n})$ of the adjoint representation ${sl}_{n}$ of the Lie group $S L_{n}$ . In particular, we describe the decomposition of this representation into irreducible components for $k = 3$ , and find highest-weight vectors for all irreducible components. Relations to fast matrix multiplication, in particular the Coppersmith–Winograd tensor, are presented.

Reçu le : 2017-10-10
Accepté le : 2017-11-20
Publié le : 2017-12-06

DOI : 10.1016/j.crma.2017.11.012

Affiliations des auteurs :

Tim Seynnaeve ¹

¹ Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

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Tim Seynnaeve. Plethysm and fast matrix multiplication. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 52-55. doi : 10.1016/j.crma.2017.11.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.012/

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