Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

On the algebraic dimension of Riesz spaces

We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.

Removahedral Congruences versus Permutree Congruences

The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cones of the permutree fans.

An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

A High Order Accurate and Effective Scheme for Solving Markovian Switching Stochastic Models

In this paper, we propose a new weak order 2.0 numerical scheme for solving stochastic differential equations with Markovian switching (SDEwMS). Using the Malliavin stochastic analysis, we theoretically prove that the new scheme has local weak order 3.0 convergence rate. Combining the special property of Markov chain, we study the effects from the changes of state space on the convergence rate of the new scheme. Two numerical experiments are given to verify the theoretical results.