Geometric combinatorial algebras: cyclohedron and simplex

Abstract

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.

Pseudograph associahedra

“…Forcey and Springfield [9] show a fine factorization of the Tonks cellular projection through a series of connected graph associahedra, and then an extension of the projection to disconnected graphs. Several of these cellular projections through polytopes are also shown to be algebra and coalgebra homomorphisms.…”

“…For any collection E of edges of G, denote Θ E as the composition of projections {Θ e | e ∈ E}. Let Γ n be the complete graph on n numbered nodes, and let E be the set of all edges of Γ n except for the path in consecutive order from nodes 1 to n. Then Θ E is equivalent to the Tonks projection [9].…”

Shuffles and concatenations in the construction of graphs

“…However, the connection is not clear. Similar connections have been made in Forcey and Springfield (2010). The polyhedra of Section 5 have been previously investigated in a manner comparable to ours in Stasheff (1997, …”

“…The maps T and L for a given graph Γ with n vertices induce an equivalence relation on the set of vertices of the (n−1)-dimensional permutohedron (see Section 5). While the collapsing studied in Tonks (1997) and Forcey and Springfield (2010) involves all the faces, our collapsing is a particular case of collapsing involving only the vertices. The concluding results of Section 5 (Propositions 5.6 and 5.7, and their consequence mentioned in the last paragraph) may be inferred from issues considered in Forcey and Springfield (2010, Section 3.3) † .…”

“…If G, X is the connected graph q q q q x y z u obtained from the graph in Example 5.1.4 by omitting the edge {x, y}, we find the sixteen S-trees in T G, X that label the vertices of the following polyhedron: Appendix B), Tonks (1997), Carr and Devadoss (2006), Devadoss (2009), Devadoss and Forcey (2008), Armstrong et al (2009), Forcey andSpringfield (2010) and Bloom (2011). Further references on antecedents of this approach may be found in Postnikov (2009), Postnikov et al (2008 and Došen and Petrić (2011).…”