Semi-invariant Matrices over Finite Groups

The semi-center of an artinian semisimple module-algebra over a finite group G can be described using the projective representations of G. In particular, the semi-center of the endomorphism ring of an irreducible projective representation over an algebraically closed field has a structure of a twisted group algebra. The following group-theoretic result is deduced: the center of a group of central type embeds into the group of its linear characters.     

Minimal characters of the finite classical groups

Let G(q) be a finite simple group of Lie type over a finite field of order q and d(G(q)) the minimal degree of faithful projective complex representations of G(q). For the case G(q) is a classical group we deter-mine the number of projective complex characters of G(q) of degree d(G(q)). In several cases we also determine the projective complex characters of the second and the third lowest degrees. As a corollary of these results we deduce the classification of quasi-simple irreducible complex linear groups of degree at most 2r r a prime divisor of the group order.     

Affinely Prime Dynamical Systems(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic‐symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation‐equivalent to the Hilbert scheme of length‐3 subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two‐torsion. © 2011 Wiley Periodicals, Inc.     

Fine Control over the Projective Characters of the Centre of a Group

In this article we will study the restriction of the irreducible projective characters of a finite group to a central subgroup. We will also consider under what conditions on the restriction of such irreducible projective characters to an abelian normal subgroup that we can deduce that the subgroup is central. Finally, we will investigate the relationship between the inertia subgroup and the absolute centralizer of a central subgroup relative to a fixed 2-cocycle of the group.     

On Multiplicity-Free Skew Characters and the Schubert Calculus

In this paper we introduce a partial order on the set of skew characters of the symmetric group which we use to classify the multiplicity-free skew characters. Furthermore, we give a short and easy proof that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the ‘inverse’ partitions (Theorem 4.3).     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

Gaussian Limit for Projective Characters of Large Symmetric Groups

In 1993, S. Kerov obtained a central limit theorem for the Plancherel measure on Young diagrams. The Plancherel measure is a natural probability measure on the set of irreducible characters of the symmetric group S _n. Kerov's theorem states that, as n, the values of irreducible characters at simple cycles, appropriately normalized and considered as random variables, are asymptotically independent and converge to Gaussian random variables. In the present work we obtain an analog of this theorem for projective representations of the symmetric group. Bibliography: 27 titles.     

Characters of wreath products and some applications to representation theory and combinatorics

The salient point arising out of a consideration of some seemingly independent topics in representation theory, combinatorics and the theory of numerical polynomials turns out to be a result involving characters of representations of wreath products. The topics are: symmetrized inner products of representations, irreducible characters of wreath products, Frobenius' formula for the irreducible ordinary characters of symmetric groups, the Pólya-Redfield theory of enumeration under group action in combinatorics and results of Rudvalis and Snapper that certain polynomials arising from generalized cycleindices of permutation groups are numerical.     

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.     

THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS

1. Notations and Basic ResultsLet G be a finite nonabelian group. Then frs(G/G') is an abelian group under themultiplication of characters and acts on the set of non-linear irreducible characters of G viathe multiplication of characters. The purpose of this paper is to investigate this action. Asan application of our theoryl in the end of Section 3 we give the classification of groupshaving exactly three non-linear irreducible caracters.All groups in the paper are finite. For a factor grou…     

The design of linear algebra and geometry

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.