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We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and most of the algebras of types B and D, we also establish upper bounds. Moreover, we…

As the 3-string braid group B(3) and the modular group PSL(2,Z) are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most'…

In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group…

We present a metric condition ${\LARGE{\tau}}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations…

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over…

Let L be a Leibniz algebra of dimension n. I prove the existence of a faithful L-module of dimension less than or equal to n+1.

The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that…

In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension…

Given a finite smooth group scheme $G$ over a field of characteristic $p > 0$, we show that the essential dimension of $G$ at $p$ is $0$ when $p$ does not divide the order of $G$, and $1$ when it does.

In this paper we give an intimate connection between the characteristic zero representation theories of the Additive and Heisenberg groups, and their characteristic p >0 theories when p is much larger than the dimension a representation. In…

We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of…

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for the classifying space for proper actions of G. They are the first examples that have…

The small finitistic dimension $\fPD(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we investigate the small finitistic dimensions of four types of…

In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the…

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

An infinite graph G is minor excluded if there is a finite graph that is not a minor of G. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups.…

Attached to a singular analytic curve germ in $d$-space is a numerical semigroup: a subset $S$ of the non-negative integers which is closed under addition and whose complement isfinite. Conversely, associated to any numerical semigroup $S$…

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover.…

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is…