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On descent algebras and twisted bialgebras. (English) Zbl 1103.16026

Summary: Bialgebras in the category of tensor species (twisted bialgebras) deserve a particular attention, in particular in view of applications to algebraic combinatorics. In order to study these bialgebras, a new class of descent algebras is introduced. The fine structure of Barratt’s permutation bi-ring (the direct sum of the symmetric group algebras) is investigated in detail from this point of view, leading to the definition of an enveloping algebra structure on it.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
17B01 Identities, free Lie (super)algebras
17B35 Universal enveloping (super)algebras
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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