28 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
Celebrating Loday’s associahedron
We survey Jean-Louis Loday’s vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry, and algebra. We present in particular four topics where it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal
Brick polytopes, lattices and Hopf algebras
Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytop
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
Lattice theory of torsion classes
The aim of this paper is to establish a lattice theoretical framework to
study the partially ordered set of torsion
classes over a finite-dimensional algebra . We show that
is a complete lattice which enjoys very strong
properties, as bialgebraicity and complete semidistributivity. Thus its Hasse
quiver carries the important part of its structure, and we introduce the brick
labelling of its Hasse quiver and use it to study lattice congruences of
. In particular, we give a
representation-theoretical interpretation of the so-called forcing order, and
we prove that is completely congruence
uniform. When is a two-sided ideal of , is a lattice quotient of which is
called an algebraic quotient, and the corresponding lattice congruence is
called an algebraic congruence. The second part of this paper consists in
studying algebraic congruences. We characterize the arrows of the Hasse quiver
of that are contracted by an algebraic
congruence in terms of the brick labelling. In the third part, we study in
detail the case of preprojective algebras , for which
is the Weyl group endowed with the weak
order. In particular, we give a new, more representation theoretical proof of
the isomorphism between and the Cambrian
lattice when is a Dynkin quiver. We also prove that, in type , the
algebraic quotients of are exactly its
Hasse-regular lattice quotients.Comment: 65 pages. Many improvements compared to the first version (in
particular, more discussion about complete congruence uniform lattices
The facial weak order and its lattice quotients
We investigate a poset structure that extends the weak order on a finite
Coxeter group to the set of all faces of the permutahedron of . We call
this order the facial weak order. We first provide two alternative
characterizations of this poset: a first one, geometric, that generalizes the
notion of inversion sets of roots, and a second one, combinatorial, that uses
comparisons of the minimal and maximal length representatives of the cosets.
These characterizations are then used to show that the facial weak order is in
fact a lattice, generalizing a well-known result of A. Bj\"orner for the
classical weak order. Finally, we show that any lattice congruence of the
classical weak order induces a lattice congruence of the facial weak order, and
we give a geometric interpretation of their classes. As application, we
describe the facial boolean lattice on the faces of the cube and the facial
Cambrian lattice on the faces of the corresponding generalized associahedron.Comment: 40 pages, 13 figure
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
Combinatorial generation via permutation languages. II. Lattice congruences
This paper deals with lattice congruences of the weak order on the symmetric
group, and initiates the investigation of the cover graphs of the corresponding
lattice quotients. These graphs also arise as the skeleta of the so-called
quotientopes, a family of polytopes recently introduced by Pilaud and Santos
[Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra,
associahedra, hypercubes and several other polytopes. We prove that all of
these graphs have a Hamilton path, which can be computed by a simple greedy
algorithm. This is an application of our framework for exhaustively generating
various classes of combinatorial objects by encoding them as permutations. We
also characterize which of these graphs are vertex-transitive or regular via
their arc diagrams, give corresponding precise and asymptotic counting results,
and we determine their minimum and maximum degrees. Moreover, we investigate
the relation between lattice congruences of the weak order and pattern-avoiding
permutations