3,813 research outputs found
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
Controllability and observability of grid graphs via reduction and symmetries
In this paper we investigate the controllability and observability properties
of a family of linear dynamical systems, whose structure is induced by the
Laplacian of a grid graph. This analysis is motivated by several applications
in network control and estimation, quantum computation and discretization of
partial differential equations. Specifically, we characterize the structure of
the grid eigenvectors by means of suitable decompositions of the graph. For
each eigenvalue, based on its multiplicity and on suitable symmetries of the
corresponding eigenvectors, we provide necessary and sufficient conditions to
characterize all and only the nodes from which the induced dynamical system is
controllable (observable). We discuss the proposed criteria and show, through
suitable examples, how such criteria reduce the complexity of the
controllability (respectively observability) analysis of the grid
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Computing a pyramid partition generating function with dimer shuffling
We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by
computing the generating function for pyramid partitions. Pyramid partitions
are closely related to Aztec Diamonds; their generating function turns out to
be the partition function for the Donaldson--Thomas theory of a non-commutative
resolution of the conifold singularity {x1x2 -x3x4 = 0}. The proof does not
require algebraic geometry; it uses a modified version of the domino shuffling
algorithm of Elkies, Kuperberg, Larsen and Propp.Comment: 19 pages, 13 figures. v2: fixed minor typos, updated references and
future work; added some definitions to Section
- …