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In this note, a new class of representations of the braid groups $B_{N}$ is constructed. It is proved that those representations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and an…
We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over $\R$ and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the…
This article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This…
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 consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of…
We classify irreducible representations of connected compact Lie groups whose orbit space is isometric to the orbit space of a representation of a finite extension of (positive dimensional) toric group. They turn out to be exactly the…
We use binary trees to study the Bratteli diagram of Sylow 2-subgroups of symmetric groups. We show that it is simple, has a recursive structure, and self-similarities at all scales. We contrast its subgraph of one-dimensional…
We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger…
The article presents the results of experiments in computation of statistical values related to Young diagrams, including the estimates on maximum and average (by Plancherel distribution) dimension of irreducible representation of symmetric…
We calculate extensions between certain irreducible admissible representations of p-adic groups.
We study the restriction to the symmetric group, $\mc{S}_n$ of the adjoint representation of $\mt{GL}_n(\C)$. We determine the irreducible constituents of the space of symmetric as well as the space of skew-symmetric $n\times n$ matrices as…
Motivated by physical and topological applications, we study representations of the group $\mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $\mathbb{R}^3$. Our point of view is to regard the three strand braid group…
We show the irreducibility of some unitary representations of the group of symplectomorphisms and the group of contactomorphisms.
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 determine the finite groups whose real irreducible representations have different degrees.
We determine the irreducible representations of alternating and symmetric groups and their universal central extensions that contain a non-scalar element with all but one eigenvalues of multiplicity 1. The ground field is algebraically…
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…
We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and…
Small representations of a group bring us to large symmetries in a representation space. Analysis on minimal representations utilises large symmetries in their geometric models, and serves as a driving force in creating new interesting…
We determine precisely when the branching coefficients arising from the restriction of irreducible representations of the symmetric group $S_n$ to the dihedral subgroup $D_n$ are nonzero, and we establish uniform linear lower bounds outside…