Commutative combinatorial Hopf algebras

By Florent Hivert, Jean-Christophe Novelli, Jean-Yves Thibon. 2006

78 citations recorded by the source corpus.

Abstract We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative dual is realized in three different ways, in particular, as the Grossman–Larson algebra of heap-ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees, and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c023eb4f3f1ba520c404613ebc80c9be773459f6

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras (2010, background): “It is shown in [23] that these polynomials span a (commutative, cofree) Hopf algebra, denoted by SQSym.”
Combinatorial Hopf algebras (2008, background): “Among other examples we find: the Faà di Bruno algebra, variations of the symmetric functions: Sym, NSym, FSym, QSym, FQSym, PQSym, (Aguiar-Sottile [2, 3], Bergeron-Hohlweg [4], Chapoton-Livernet [12], Livernet [33], Malvenuto-Reutenauer [44], Hivert-Novelli-Thibon [25], Palacios-Ronco [48]), exampl”