Related papers: Pseudo-reflection groups and essential dimension

We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known…

We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…

We determine the finite groups whose real irreducible representations have different degrees.

In this paper, we compute the essential $p$-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the…

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…

We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In…

Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive…

To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear…

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct…

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential…

We discuss the structure of finite groups for which the projective indecomposable modules have special given dimensions. In particular, we prove the converse of Fong's dimension formula for $p$-solvable groups. Furthermore, we characterize…

We study the essential dimension of a finite group G over a field K. A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159-179, Theorem 5.3) is obtained. We also get lower bounds of…

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.

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results…

Let $P$ be a finite set of points in $\mathbb{R}^d$ or $\mathbb{C}^d$. We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by $P$ is at least the number of $(d-2)$-flats spanned by $P$. In answering…

In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to…

We propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in characteristic $p$, for finite commutative unipotent group schemes. And we show some evidence and some consequences of this new…

In this paper, we compute the essential $l$-dimension of the finite groups of classical Lie type for odd primes $l$ not equal to the defining prime, specifically the general linear groups, the symplectic groups, the orthogonal groups, and…

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at…