Related papers: Some open problems on Coxeter groups and unimodali…

An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it…

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…

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal $h^*$-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider…

This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a…

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the…

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild…

In this paper, we analyze the faithful representations of the dihedral groups, and prove that the Coxeter groups can be determined by the proper joint spectrum of their faithful representations.

The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups…

We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.

We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.

This paper presents an overview of the current state of knowledge in the field of equivariant map algebras and discusses some open problems in this area.

Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group.

Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications.

This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sification efforts we single out some relator types that render the…

We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros…

Some question about representations of $p$-adic groups are discussed.

The article investigates high-level general invertible-sequential processing in the digital and quantum domains. In particular it is shown that (i) invertible digital-sequential processes, constructed using a standard general-inversion…

A survey of results on the residual properties of Baumslag -- Solitar groups which have been obtained to date.

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

The paper investigates exterior and symmetric (co)homologies of groups. We introduce symmetric homology of groups and compute exterior and symmetric (co)homologies of some finite groups. We also compare the classical, exterior and symmetric…