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The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free. (English) Zbl 1188.16030

It is a classical result that the algebra of symmetric polynomials in commuting variables is freely generated by the elementary symmetric polynomials. Much work has been done on getting analogous results for the \(Q\)-algebra NCSym of symmetric functions in non-commutative variables, in particular to show that it is freely generated by various monomial bases. Recently, the authors, together with M. Rosas and C. Reutenauer, gave a Hopf algebra structure on NCSym [Can. J. Math. 60, No. 2, 266-296 (2008; Zbl 1180.16025)], and this was used by F. Hivert, J.-C. Novelli and J.-Y. Thibon to give a proof that NCSym is both free and co-free (i.e., the graded dual of NCSym is free) [J. Algebr. Comb. 28, No. 1, 65-95 (2008; Zbl 1181.16031)].
The paper under review shows that NCSym is free and co-free by exhibiting bases which generate the algebra and coalgebra. This involves set partitions and Lyndon words. The generating bases have simple and natural rules for the product of two elements. The authors also study the algebra NCQSym of quasi-symmetric functions in non-commutative variables. This is ordered by set compositions (ordered set partitions). It is also a Hopf algebra, and analogous results are obtained in showing that it is both free and co-free. Again the product of two basis elements has a simple and elegant expression.

MSC:

16T30 Connections of Hopf algebras with combinatorics
05E05 Symmetric functions and generalizations
16T05 Hopf algebras and their applications
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References:

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