Document Type

Article

Publication Date

12-2010

Publication Title

Electronic Journal of Combinatorics

Volume

R166

Abstract

We analyze the structure of the algebra K⟨x⟩Sn of symmetric polynomials in non-commuting variables in so far as it relates to K[x]Sn, its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of K⟨x⟩Sn analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.

Résumé. Nous analysons la structure de l'algèbre K⟨x⟩Sn des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau K[x]Sn des polynômes symétriques en des variables commutatives. Plus précisément, au moyen de "l'action par positions", on réalise K[x]Sn comme sous-module de K⟨x⟩Sn. On découvre alors une nouvelle décomposition de K⟨x⟩Sn comme produit tensorial, obtenant ainsi un analogues des théorèmes classiques de Chevalley et Shephard-Todd.

Comments

Author Posting. © Electronic Journal of Combinatorics, 2010. This article is posted here by permission of the Electronic Journal of Combinatorics for personal use, not for redistribution. The article was published in Electronic Journal of Combinatorics, Volume R166, December 2010. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r166

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