Related papers: A Poset Structure on the Alternating Group Generat…

The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…

We explore the application of generating symmetries, i.e. symmetries that depend on a parameter, to integrable hyperbolic third order equations, and in particular to consistent pairs of such equations as introduced by Adler and Shabat (AS).…

Interval-closed sets of a poset are a natural superset of order ideals. We initiate the study of interval-closed sets of finite posets from enumerative and dynamical perspectives. In particular, we use the generalized toggle group to define…

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a…

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq…

For any finite poset we define a generating polynomial counting upsets, downsets, and their intersection. We investigate the behaviour of this polynomial with respect to poset operations, show that it distinguishes series-parallel posets,…

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By…

Cyclic poset are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of arXiv:1209.1879 are…

In this thesis, we construct and classify planar noncommutative phase spaces by the coadjoint orbit method on the anisotropic and absolute time kinematical groups. We show that noncommutative symplectic structures can be generated in the…

In 2004, Cs\"{o}rg\H{o} constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product…

We examine higher-form symmetries of quantum lattice gauge theories through the lens of homotopy theory and operator algebras. We show that in the operator-algebraic approach both higher-form symmetries and 't Hooft anomalies arise from…

We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1 s_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what…

The present thesis studies structural properties of non-crossing partitions associated to finite Coxeter groups from both algebraic and geometric perspectives. On the one hand, non-crossing partitions are lattices, and on the other hand, we…

Polariton condensates have proved to be model systems to investigate topological defects, as they allow for direct and non-destructive imaging of the condensate complex order parameter. The fundamental topological excitations of such…

We study the equivalence relation on the set of acyclic orientations of an undirected graph G generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver…

Building on the work of Cassels we prove the existence of infinite families of compact orbits of the diagonal group in the space of lattices which accumulate only on the divergent orbit of the standard lattice. As a consequence, we prove…

We demonstrate, using examples from $2$ and $3$-dimensions, a systematic method of finding all possible periodic arrangements of a given molecule or molecules such that the arrangements have the symmetry of a given space group. The…

This article examines noncrossing partitions of the unit circle in the complex plane; we call these continuous noncrossing partitions. More precisely, we focus on the degree-$d$ continuous noncrossing partitions where unit complex numbers…

A poset is Macaulay if its partial order and an additional total order interact well. Analogously, a ring is Macaulay if the partial order defined on its monomials by division interacts nicely with any total monomial order. We investigate…

We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by…