Commutative combinatorial Hopf algebras

Abstract

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 non-commutative 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.

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…57 Some results of this section, e.g., Theorems 5.63 and 5.65, has been proved independently and in greater generality in [102]. What one can say about the polynomial Q n (β) :…”

On composition polynomials

“…The number of permutitions of length n is equal to the number of stalactic classes of parking functions of the same length, [4], on which a combinatorial Hopf algebra structure can also be defined. There is also a bijection between permutitions and noncrossing set compositions [3].…”

Strong forms of linearization for Hopf monoids in species

“…There is a natural Hopf algebra structure on this graded algebra (which we review in Sect. 5.2), and following [7,14,21,30], we call NCSym the Hopf algebra of symmetric functions in noncommuting variables. NCSym is one of the several noncommutative analogs of the more familiar and much-studied Hopf algebra of symmetric functions, which we denote Sym.…”