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

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled 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 Γ ≀ S n and...

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Bibliographic Details
Published in:Discrete mathematics 2010-12, Vol.310 (24), p.3584-3606
Main Authors: Novelli, Jean-Christophe, Thibon, Jean-Yves
Format: Article
Language:English
Subjects:
Algebra
Algebraic combinatorics
Associative rings and algebras
Combinatorial Hopf algebras
Combinatorics
Combinatorics. Ordered structures
Descent algebras
Exact sciences and technology
Functional analysis
Manifolds and cell complexes
Mathematical analysis
Mathematics
Noncommutative symmetric functions
Quasi-symmetric functions
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled 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 Γ ≀ S 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 .
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2010.09.008