118 research outputs found
Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
Given any polytope and any generic linear functional , one
obtains a directed graph by taking the 1-skeleton of and
orienting each edge from to for .
This paper raises the question of finding sufficient conditions on a polytope
and generic cost vector so that the graph will
not have any directed paths which revisit any face of after departing from
that face. This is in a sense equivalent to the question of finding conditions
on and under which the simplex method for linear programming
will be efficient under all choices of pivot rules. Conditions on and are given which provably yield a corollary of the desired face
nonrevisiting property and which are conjectured to give the desired property
itself. This conjecture is proven for 3-polytopes and for spindles having the
two distinguished vertices as source and sink; this shows that known
counterexamples to the Hirsch Conjecture will not provide counterexamples to
this conjecture.
A part of the proposed set of conditions is that be the
Hasse diagram of a partially ordered set, which is equivalent to requiring non
revisiting of 1-dimensional faces. This opens the door to the usage of
poset-theoretic techniques. This work also leads to a result for simple
polytopes in which is the Hasse diagram of a lattice L that the
order complex of each open interval in L is homotopy equivalent to a ball or a
sphere of some dimension. Applications are given to the weak Bruhat order, the
Tamari lattice, and more generally to the Cambrian lattices, using realizations
of the Hasse diagrams of these posets as 1-skeleta of permutahedra,
associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition
substantially improved throughou
Shelling Coxeter-like Complexes and Sorting on Trees
In their work on `Coxeter-like complexes', Babson and Reiner introduced a
simplicial complex associated to each tree on nodes,
generalizing chessboard complexes and type A Coxeter complexes. They
conjectured that is -connected when the tree has
leaves. We provide a shelling for the -skeleton of , thereby
proving this conjecture.
In the process, we introduce notions of weak order and inversion functions on
the labellings of a tree which imply shellability of , and we
construct such inversion functions for a large enough class of trees to deduce
the aforementioned conjecture and also recover the shellability of chessboard
complexes with . We also prove that the existence or
nonexistence of an inversion function for a fixed tree governs which networks
with a tree structure admit greedy sorting algorithms by inversion elimination
and provide an inversion function for trees where each vertex has capacity at
least its degree minus one.Comment: 23 page
Regular cell complexes in total positivity
This paper proves a conjecture of Fomin and Shapiro that their combinatorial
model for any Bruhat interval is a regular CW complex which is homeomorphic to
a ball. The model consists of a stratified space which may be regarded as the
link of an open cell intersected with a larger closed cell, all within the
totally nonnegative part of the unipotent radical of an algebraic group. A
parametrization due to Lusztig turns out to have all the requisite features to
provide the attaching maps. A key ingredient is a new, readily verifiable
criterion for which finite CW complexes are regular involving an interplay of
topology with combinatorics.Comment: accepted to Inventiones Mathematicae; 60 pages; substantially revised
from earlier version
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