5 research outputs found
Minkowski Decomposition of Associahedra and Related Combinatorics
Realisations of associahedra with linearly non-isomorphic normal fans can be
obtained by alteration of the right-hand sides of the facet-defining
inequalities from a classical permutahedron. These polytopes can be expressed
as Minkowski sums and differences of dilated faces of a standard simplex as
described by Ardila, Benedetti & Doker (2010). The coefficients of such a
Minkowski decomposition can be computed by M\"obius inversion if tight
right-hand sides are known not just for the facet-defining inequalities
of the associahedron but also for all inequalities of the permutahedron that
are redundant for the associahedron.
We show for certain families of these associahedra: (a) how to compute tight
values for the redundant inequalities from the values for the
facet-defining inequalities; (b) the computation of the values of Ardila,
Benedetti & Doker can be significantly simplified and at most four values
, , and are needed to compute ;
(c) the four indices , , and are determined by the
geometry of the normal fan of the associahedron and are described
combinatorially; (d) a combinatorial interpretation of the values using a
labeled -gon. This last result is inspired from similar interpretations for
vertex coordinates originally described originally by J.-L. Loday and
well-known interpretations for the -values of facet-defining inequalities.Comment: 30 pages; 21 figures; changed title; minor stylistic change
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Hochschild polytopes
The -multiplihedron is a polytope whose faces correspond to
-painted -trees, and whose oriented skeleton is the Hasse diagram of the
rotation lattice on binary -painted -trees. Deleting certain inequalities
from the facet description of the -multiplihedron, we construct the
-Hochschild polytope whose faces correspond to -lighted -shades,
and whose oriented skeleton is the Hasse diagram of the rotation lattice on
unary -lighted -shades. Moreover, there is a natural shadow map from
-painted -trees to -lighted -shades, which turns out to define a
meet semilattice morphism of rotation lattices. In particular, when , our
Hochschild polytope is a deformed permutahedron whose oriented skeleton is the
Hasse diagram of the Hochschild lattice.Comment: 32 pages, 25 figures, 7 tables. Version 2: Minor correction
Shard polytopes
For any lattice congruence of the weak order on permutations, N. Reading
proved that gluing together the cones of the braid fan that belong to the same
congruence class defines a complete fan, called a quotient fan, and V. Pilaud
and F. Santos showed that it is the normal fan of a polytope, called a
quotientope. In this paper, we provide a simpler approach to realize quotient
fans based on Minkowski sums of elementary polytopes, called shard polytopes,
which have remarkable combinatorial and geometric properties. In contrast to
the original construction of quotientopes, this Minkowski sum approach extends
to type .Comment: 73 pages, 35 figures; Version 2: minor corrections for final versio