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Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Lisa Berry, Stefan Forcey, Maria Ronco and Patrick Showers

Algebraic & Geometric Topology 19 (2019) 1019–1078

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

associahedron, multiplihedron, composihedron, binary tree, cofree coalgebra, Hopf algebra, operad, species
Mathematical Subject Classification 2010
Primary: 18D50, 52B11, 57T05
Received: 24 April 2018
Revised: 8 July 2018
Accepted: 22 August 2018
Published: 12 March 2019
Lisa Berry
Bio-Med Science Academy
Rootstown, OH
United States
Stefan Forcey
Department of Mathematics
The University of Akron
Akron, OH
United States
Maria Ronco
Department of Physics and Mathematics
The University of Talca
Patrick Showers
Department of Mathematics
The University of Akron
Akron, OH
United States