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There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation…
In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their…
This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal…
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility…
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded…
This text brings to an end the classification of non-reduced parabolic subgroups in positive characteristic, especially two and three: they are all obtained as intersections of parabolics having maximal reduced part. We prove this result…
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional…
The irreducible representations of all of the 80 diperiodic groups, being the symmetries of the systems translationally periodical in two directions, are calculated. To this end, each of these groups is factorized as the product of a…
A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…
We construct a class of negative spin irreducible representations of the su(2) Lie algebra. These representations are infinite-dimensional and have an indefinite inner product. We analyze the decomposition of arbitrary products of positive…
The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried…
We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.
Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group $G$ factors through a projective representation of $G$, except for some groups of Lie…
We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group. In particular, our methods yield two simple proofs of the classical Thoma's description of the characters of the…
Over an algebraically closed field we classify all minimal representation-infinite algebras where the lattice of two-sided ideals is not distributive. As a consequence there are only finitely many isomorphism classes of minimal…
We classify the possibilities for the fixed field of the kernel of an irreducible three-dimensional Artin representation of $\Q$ with solvable image ramified at one prime by using the classification of the finite irreducible subgroups of…
In this paper we completely characterise irreducible tensor products of representations of alternating groups in characteristic 2 of a basic spin module with an irreducible module. This completes the classification of irreducible tensor…
The symmetry group structures of two dimensional coupled nonlinear Shr\"{o}dinger equations are considered. We first show that the equations admit infinite dimensional symmetry algebra as well as the corresponding symmetry group depending…
We prove a characterization of monomial projective representations of finitely generated nilpotent groups. We also characterize polycyclic groups whose projective representations are finite dimensional.
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…