Abstract:
We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta functions of arithmetic groups $\Gamma \subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form of $\mathsf{SL}_3$: assuming that $\Gamma$ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of $\Gamma$. Our results are based on explicit, uniform formulae for the representation zeta functions of the $p$-adic analytic groups $\mathsf{SL}_3(\mathfrak{o})$ and $\mathsf{SU}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$. These formulae build on our classification of similarity classes of integral $\mathfrak{p}$-adic $3\times3$ matrices in $\mathfrak{gl}_3(\mathfrak{o})$ and $\mathfrak{gu}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form $\mathsf{SL}_3(\mathfrak{o})$, $\mathsf{SU}_3(\mathfrak{o})$, $\mathsf{GL}_3(\mathfrak{o})$, and $\mathsf{GU}_3(\mathfrak{o})$, arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of $\mathfrak{o}$ is either $0$ or sufficiently large. Analysis of some of these formulae leads us to observe $p$-adic analogues of `Ennola duality'.

Abstract:
The determination of the density functions for products of random elements from specified classes of matrices is a basic problem in random matrix theory and is also of interest in theoretical physics. For connected simple Lie groups of $2\times 2$ matrices and conjugacy and spherical classes a complete solution is given here. The problem/solution can be re-stated in terms of the structure of certain Hecke algebras attached to groups of $2\times 2$ matrices.

Abstract:
Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically represented elements in terms of their permutation representation. In particular, we represent elements of the group $U_3(3) : 2$, where $U_3(3)$ is the projective special unitary group of $3 \times 3$ matrices with entries in the field $\mathbb{F}_3$, as a permutation on fourteen letters from the projective general linear group $PGL_2(7)$, followed by a word of length at most two in the fourteen involutory symmetric generators.

Abstract:
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.

Abstract:
Irreducible representations of quantum groups $SL_q(2)$ (in Woronowicz' approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the~case of $q$ being an~odd root of unity. Here we find the~irreducible representations for all roots of unity (also of an~even degree), as well as describe "the~diagonal part" of tensor product of any two irreducible representations. An~example of not completely reducible representation is given. Non--existence of Haar functional is proved. The~corresponding representations of universal enveloping algebras of Jimbo and Lusztig are provided. We also recall the~case of general~$q$. Our computations are done in explicit way.

Abstract:
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \cite{VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups.

Abstract:
For a knot K in S^3 we construct according to Casson--or more precisely taking into account Lin and Heusener's further works--a volume form on the SU(2)-representation space of the group of K. We prove that this volume form is a topological knot invariant and explore some of its properties.

Abstract:
The new approach to the theory of complex representrations of the finite symmetric groups which based on the notions of Coxeter generators., Gelfand-Zetlin algebras, Hecke algebra, Young-Jucys-Murphi generators and which hardly used inductive structure - is systematically developed. The appearence of Young diagrams, tables is naturally explained - the set of content vectors of all Young tables is the spectra of Gel'fand-Zetlin algebra. The first steps of the theory (the list of the irrediucible representatinos, branching rule, Young orthopgonal form, Murnagan-Nakayama rule etc.)are established.This appraoch canbe used for other classical series, wreath products with symmetric groups. This paper is the revcised version of the paper of the same authors which was published in Selecta MAth. (New Series) V.2(1996).

Abstract:
In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles.

Abstract:
By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions $\lambda$ and $\mu$ are equal if and only if $\lambda$ and $\mu$ are conjugate. An analogous result is proved for Hecke algebras. 2. A Specht module for the symmetric group $S_n$, defined over an algebraically closed field of odd characteristic, is decomposable on restriction to the alternating group $A_n$ if and only if it is simple, and the labelling partition is self-conjugate. This result is generalised to an arbitrary field of odd characteristic.