2023/01/16 by Fløystad, Gunnar
#06A11 #Combinatorics (math.CO) #FOS: Mathematics #Primary: 16T30 #Rings and Algebras (math.RA) #Secondary: 05E99
paper · doi:10.48550/arxiv.2301.06479
We introduce new concepts and viewpoints on combinatorial Hopf species and algebras. We give a category \rm \bf setℕ whose objects are sets, and (dualizable) morphisms represented by matrices of non-negative integers. For a bimonoid species (B,Δ, μ) in \rm \bf setℕ we may then dualize the product μ to get two intertwined coproducts Δ, Δ^′. We consider restriction species S over \rm \bf setℕ accompanied by pairs of natural transformations π1, π2 : S → \rm Pre to the species of preorders. A simple construction associates two comonoid species Δ1 and Δ2, and we investigate when they are intertwined. We get new Hopf algebras: i. choosing an arbitrary set of permutations without global descents, we get associated a quotient Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations avoiding this chosen set, ii. a Hopf algebra of pairs of parking filtrations, and iii. three Hopf algebras of pairs of preorders.