Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give seve...

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Bibliographic Details
Published in:Advances in applied mathematics 2013-07, Vol.51 (2), p.177-227
Main Authors: Menous, Frédéric, Novelli, Jean-Christophe, Thibon, Jean-Yves
Format: Article
Language:English
Subjects:
Algebra
Calculus
Combinatorial analysis
Combinatorial Hopf algebras
Combinatorics
Cones
Construction
Functions (mathematics)
Mathematical analysis
Mathematics
Molds
Mould calculus
Noncommutative symmetric functions
Polyhedral cones
Quasi-symmetric functions
Resurgent functions
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Summary:We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2013.02.003