Related papers: EL-Shellability and Noncrossing Partitions Associa…

We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same…

For a finite group $G$, a $G$-crossed braided fusion category is $G$-graded fusion category with additional structures, namely a $G$-action and a $G$-braiding. We develop the notion of $G$-crossed braided zesting: an explicit method for…

For $\textrm{SL}(n,\mathbb{R})$ ($n\geq3$), $\textrm{SO}(n+1,n)$ ($n\geq2$), $\textrm{Sp}(2n,\mathbb{R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{G}_2$, we exhibit non-uniform lattices in which we…

A weak order on the set of maximal chains of the non-crossing partition lattice is introduced and studied. A $0$-Hecke algebra action is used to compute the radius of the graph on these chains in which two chains are adjacent if they differ…

We follow the dual approach to Coxeter systems and show for Weyl groups a criterium which decides whether a set of reflections is generating the group depending on the root and the coroot lattice. Further we study special generating sets…

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of…

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-JM partitions" (see \cite{F}) that are also $\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas…

Let $\pi$ and $\lambda$ be two set partitions with the same number of blocks. Assume $\pi$ is a partition of $[n]$. For any integer $l, m \geq 0$, let $\mathcal{T}(\pi, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last…

Let $G$ be a group and $Sol(G)=\{x \in G : \langle x,y \rangle \text{ is solvable for all } y \in G\}$. We associate a graph $\mathcal{NS}_G$ (called the non-solvable graph of $G$) with $G$ whose vertex set is $G \setminus Sol(G)$ and two…

We investigate the poset structure on the alternating group that arises when the latter is generated by 3-cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the…

Studies of the Deep Inelastic Scattering (DIS) have provided fundamental information of the nucleon structure for decades. The electron-ion collider (EIC) will be the first collider capable of DIS study with both polarized lepton and…

There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split of a coarser ECS. However, an ECS admiting a maximal modulus which is divisible by at most two distinct primes, primely splits a coarser ECS.…

We show that for some classes of groups $G$, the homotopy fiber $E_{\mathrm{com}} G$ of the inclusion of the classifying space for commutativity $E_{\mathrm{com}} G$ into the classifying space $BG$, is contractible if and only if $G$ is…

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v…

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs,…

We construct finitely generated Engel branch groups, answering a question of Fern\'andez-Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of…

This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite…

We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A*_\Theta G$ by twisted partial actions $\Theta$ of $G$ on…

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural…

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of…