{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:05Z","timestamp":1753893845395,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We analyze the structure of the algebra $\\mathbb{K}\\langle\\mathbf{x}\\rangle^{\\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\\mathbb{K}[\\mathbf{x}]^{\\mathfrak{S}_n}$, 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 $\\mathbb{K}\\langle\\mathbf{x}\\rangle^{\\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.  R\u00e9sum\u00e9. Nous analysons la structure de l'alg\u00e8bre $\\mathbb{K}\\langle\\mathbf{x}\\rangle^{\\mathfrak{S}_n}$ des polyn\u00f4mes sym\u00e9triques en des variables non-commutatives pour obtenir des analogues des r\u00e9sultats classiques concernant  la structure de l'anneau $\\mathbb{K}[\\mathbf{x}]^{\\mathfrak{S}_n}$ des polyn\u00f4mes sym\u00e9triques en des variables commutatives. Plus pr\u00e9cis\u00e9ment, au moyen de \"l'action par positions\", on r\u00e9alise  $\\mathbb{K}[\\mathbf{x}]^{\\mathfrak{S}_n}$ comme sous-module de $\\mathbb{K}\\langle\\mathbf{x}\\rangle^{\\mathfrak{S}_n}$. On d\u00e9couvre alors une nouvelle d\u00e9composition de $\\mathbb{K}\\langle\\mathbf{x}\\rangle^{\\mathfrak{S}_n}$ comme produit tensorial, obtenant ainsi un analogues des th\u00e9or\u00e8mes classiques de Chevalley et Shephard-Todd. <\/jats:p>","DOI":"10.37236\/438","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:58:55Z","timestamp":1578715135000},"source":"Crossref","is-referenced-by-count":0,"title":["Invariant and Coinvariant Spaces for the Algebra of Symmetric Polynomials in Non-Commuting Variables"],"prefix":"10.37236","volume":"17","author":[{"given":"Fran\u00e7ois","family":"Bergeron","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aaron","family":"Lauve","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2010,12,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r166\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r166\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:21:27Z","timestamp":1579303287000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r166"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/438","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,12,10]]},"article-number":"R166"}}