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Báo cáo toán học: "Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables. | Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables Francois Bergeron LaCIM Universite du Quebec a Montreal Montreal Quebec H3C 3P8 CANADA bergeron.francois@uqam.ca Aaron Lauve Department of Mathematics Texas A M University College Station TX 77843 USA lauve@math.tamu.edu Submitted Oct 2 2009 Accepted Nov 26 2010 Published Dec 10 2010 Mathematics Subject Classification 05E05 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. Resume. Nous analysons la structure de l algebre K x Sn des polynomes syme-triques en des variables non-commutatives pour obtenir des analogues des resultats classiques concernant la structure de l anneau K x Sn des polynômes symetriques en des variables commutatives. Plus precisement au moyen de Faction par positions on realise K x Sn comme sous-module de K x Sn. On decouvre alors une nouvelle decomposition de K x Sn comme produit tensorial obtenant ainsi un analogues des theoremes classiques de Chevalley et Shephard-Todd. 1 Introduction One of the more striking results of invariant theory is certainly the following if W is a finite group of n X n matrices over some field K containing Q then there is a W-module decomposition of the polynomial ring S K x in variables x xi X2 . xn as a tensor product S Sw Sw 1 F. Bergeron is supported by NSERC-Canada and FQRNT-Quebec. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R166 1 if and only if W is a group generated by pseudo reflections. As usual S is afforded a natural W-module structure by considering it as the symmetric space on the defining vector space X for W .