Related papers: Connectivity Properties of McKay Quivers
Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of…
In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.
We establish a McKay correspondence for finite and linearly reductive subgroup schemes of $\mathrm{SL}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in…
We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection…
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…
The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group…
In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where…
Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected…
The classical construction of representations of quivers enables us to consider linear maps between several vector spaces. The mixed representations of quivers helps us to work with linear maps as well as bilinear forms on several vector…
Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of…
Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated
We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.
A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected…
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex…
The McKay correspondence has had much success in studying resolutions of 3-fold quotient singularities through a wide range of tools coming from geometry, combinatorics, and representation theory. We develop a computational perspective in…
Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite…
We prove a Mackey formula for representations of finite groups of Lie type, in the case where the groups come from disconnected reductive groups.
This paper deals with the representation theory of a locally finite quiver in which the number of paths between any two given vertices is finite. We first study some properties of the finitely presented or co-presented representations, and…
We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $\Omega$ is any connected quiver, the category of locally finite-dimensional representations of…