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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.
Keywords: associahedron, multiplihedron, composihedron, binary tree, cofree coalgebra, Hopf algebra, operad, species
Berry, Lisa 1 ; Forcey, Stefan 2 ; Ronco, Maria 3 ; Showers, Patrick 2
@article{10_2140_agt_2019_19_1019,
author = {Berry, Lisa and Forcey, Stefan and Ronco, Maria and Showers, Patrick},
title = {Species substitution, graph suspension, and graded {Hopf} algebras of painted tree polytopes},
journal = {Algebraic and Geometric Topology},
pages = {1019--1078},
publisher = {mathdoc},
volume = {19},
number = {2},
year = {2019},
doi = {10.2140/agt.2019.19.1019},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1019/}
}
TY - JOUR AU - Berry, Lisa AU - Forcey, Stefan AU - Ronco, Maria AU - Showers, Patrick TI - Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes JO - Algebraic and Geometric Topology PY - 2019 SP - 1019 EP - 1078 VL - 19 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1019/ DO - 10.2140/agt.2019.19.1019 ID - 10_2140_agt_2019_19_1019 ER -
%0 Journal Article %A Berry, Lisa %A Forcey, Stefan %A Ronco, Maria %A Showers, Patrick %T Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes %J Algebraic and Geometric Topology %D 2019 %P 1019-1078 %V 19 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1019/ %R 10.2140/agt.2019.19.1019 %F 10_2140_agt_2019_19_1019
Berry, Lisa; Forcey, Stefan; Ronco, Maria; Showers, Patrick. Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes. Algebraic and Geometric Topology, Tome 19 (2019) no. 2, pp. 1019-1078. doi: 10.2140/agt.2019.19.1019
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