Groups with representations of bounded degree II

k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...

Uniformly Bounded Representations and Exact Groups

We characterize groups with Guoliang Yu’s property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation. Property A is a large scale geometric property that can be viewed as a weak counterpart of amenability. It was shown in [12], that for a finitely generated group property A implies the Novikov conjecture. It was...

QUASI-PERMUTATION REPRESENTATIONS OF METACYCLIC 2-GROUPS

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

Lecture 2: Representations of Symmetric Groups, Ii

Recall that last time we have introduced the set of weights Wt(n) ⊂ C and an equivalence relation ∼ on Wt(n). Our goal is to describe this set and this equivalence relation – the equivalence classes are in a natural bijection with Irr(Sn). The first step here is as follows. Pick a path P = V 1 → V 2 → . . . → V n in the branching graph and an integer k with 1 < k < n. We fix all vertices but V ...

Projective Representations of Reflection Groups Ii