2017/12/11 by Samuele Giraudo, Giraudo, Samuele · 1 citation
Computer Science · Mathematics · #05A15 #05C05 #05E15 #16T05 #18D50 #68Q42 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bialgebra #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Generalization #Hopf algebra #Mathematics #Morphism #Noncommutative geometry #Product (mathematics) #Pure mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #cs.DM #math.CO #math.QA #math.RA #msc:05A15 #msc:05C05 #msc:05E15 #msc:16T05 #msc:18D50 #msc:68Q42
paper · pdf · doi:10.48550/arxiv.1712.03782
published in arXiv (Cornell University) (Cornell University) · Habilitation thesis ("Habilitation à diriger des recherches"); 388 pages; Mainly in english
arxiv created 2017/12/11 · openalex publication_date 2017/12/11 · arxiv updated 2017/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, representations), to collect combinatorial information about the underlying objects. The algebraic structures the most encountered here are magmas, posets, associative algebras, dendriform algebras, Hopf bialgebras, operads, and pros. This work explores the aforementioned research direction and provides many constructions having the particularity to build algebraic structures on combinatorial objects. We develop for instance a functor from nonsymmetric colored operads to nonsymmetric operads, from monoids to operads, from unitary magmas to nonsymmetric operads, from finite posets to nonsymmetric operads, from stiff pros to Hopf bialgebras, and from precompositions to nonsymmetric operads. These constructions bring alternative ways to describe already known structures and provide new ones, as for instance, some of the deformations of the noncommutative Faà di Bruno Hopf bialgebra of Foissy and a generalization of the dendriform operad of Loday. We also use algebraic structures to obtain enumerative results. In particular, nonsymmetric colored operads are promising devices to define formal series generalizing the usual ones. These series come with several products (for instance a pre-Lie product, an associative product, and their Kleene stars) enriching the usual ones on classical power series. This provides a framework and a toolbox to strike combinatorial questions in an original way. The first two chapters pose the elementary notions of combinatorics and algebraic combinatorics used here. The last ten chapters contain our original research.