Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

Details Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

2008-06-23

arxiv.org/abs/0806.3682v1

Mathematics

Combinatorics

34 pages; LaTEX

Scientific paper

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $\Gamma\wr\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.

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