Related papers: RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2…

In this paper we prove that an RGD-system over $\mathbb{F}_2$ with prescribed commutation relations exists if and only if the commutation relations are Weyl-invariant and can be realized in the group $U_+$. This result gives us a machinery…

An RGD system $\mathcal{D}$ is called \emph{linear w.r.t. a root basis $\mathcal{B}$} if the commutation relations between the root groups of $\mathcal{D}$ are `linear' in a certain sense. Moreover, $\mathcal{D}$ is called…

Given a connected isotropic reductive not necessarily split $k$-group $\mathcal{G}$ with irreducible relative root system, we construct root group data (RGD) system of affine type for significant subgroups of $\mathcal{G}(k[t,t^{-1}])$,…

Motivated by the observation that $2+2=4$, we consider four-dimensional $\mathcal{N}=2$ superconformal field theories on $S^2\times\Sigma$, turning on a suitable rigid supergravity background. On the one hand, reduction of a…

We define the triple Riordan group, whose elements consist of $4$-tuples of power series $(g, f_1, f_2, f_3)$ with $g\in \mathbf{R}[[x^3]]$, and $f_1, f_2, f_3 \in x\mathbf{R}[[x^3]]$, for an appropriate ring $\mathbf{R}$. The construction…

Let $ a,b \in {\bf Z}^+$, $r=a + b$, and let $T$ be a tree with parts $U = \{u_1,u_2,\dots,u_s\}$ and $V = \{v_1,v_2,\dots,v_t\}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all…

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups…

Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an…

Tree-graded spaces are a generalization of $\mathbb{R}$-trees and play an important role in describing the large-scale geometry of relatively hyperbolic groups. We consider a subclass of tree-graded spaces that we call "disjointly…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying…

This article is devoted to the investigation of semidirect products of groups of loops and groups of diffeomorphisms of finite and infinte dimensional real, complex and quaternion manifolds. Necessary statements about quaternion manifolds…

A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group-subgroup chains is presented. The canonical U(4)->U(3)->U(2)->U(1) coupling and recoupling coefficients are applicable to…

We prove some conditions for the existence of higher dimensional algebraic fibering of group extensions. This leads to various corollaries on incoherence of groups and some geometric examples of algebraic fibers of type $F_n$ but not…

Let $G$ be a discrete Coxeter group, $G^+$ its alternating subgroup and $\tilde{G}^+$ the spinor cover of $G^+$. A presentation of the groups $G^+$ and $\tilde{G}^+$ is proved for an arbitrary Coxeter system $(G,S)$; the generators are…

We introduce and study the class of groups graded by root systems. We prove that if {\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\Phi}, then under certain natural conditions on the grading,…

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which…

We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

We analyse the subgroup structure of direct products of groups. Earlier work on this topic has revealed that higher finiteness properties play a crucial role in determining which groups appear as subgroups of direct products of free groups…

We find an explicit presentation of relative odd unitary Steinberg groups constructed by odd form rings and of relative doubly laced Steinberg groups over commutative rings, i.e. the Steinberg groups associated with the Chevalley group…