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Invariant And Coinvariant Spaces For The Algebra Of Symmetric Polynomials In Non-Commuting Variables, Francois Bergeron, Aaron Lauve

Mathematics and Statistics: Faculty Publications and Other Works

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 …