183 research outputs found
Geometric combinatorial algebras: cyclohedron and simplex
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
Malvenuto-Reutenauer 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 (one-sided) with
basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices,
with journal correction
Bergman Complexes, Coxeter Arrangements, and Graph Associahedra
Tropical varieties play an important role in algebraic geometry. The Bergman
complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M
generalize to matroids the notions of the tropical variety and positive
tropical variety associated to a linear ideal. Our main result is that if A is
a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi,
then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi)
equals the nested set complex of A. In addition, we prove that for any
orientable matroid M, one can find |mu(M)| different reorientations of M such
that the corresponding positive Bergman complexes cover B(M), where mu(M)
denotes the Mobius function of the lattice of flats of M.Comment: 24 pages, 4 figures, new result and new proofs adde
Graph properties of graph associahedra
A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction
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