Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on general linear group of finite dimensional real vector space, is exact dimensional. We also establish Ledrappier-Young type formula for its dimension. The strategy proof based argument given by Feng dimensionality self-affine measures.

As is well known, Young diagrams consisting of n boxes parameterize isomorphism classes of finite-dimensional irreducible representations of the symmetric group Sn of degree n, and moreover the structure of each irreducible representation is described in terms of tableaux on the corresponding Young diagram; namely, a basis of the representation is labeled by standard tableaux, with which the ac...

We construct a commutative ring with identity which extends the ring of characters of finite dimensional representations of sl(3). It is generated by characters with values in the group ring Z[W̃ ] of the extended affine Weyl group of ŝl(3)k at k 6∈ Q. The ‘quantised’ version at rational level k + 3 = 3/p realises the fusion rules of a WZW conformal field theory based on admissible representatio...

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X . If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X , i.e., distributions invariant under Hamiltonian flows. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X ...

Let Λ be a finite dimensional algebra over an algebraically closed field, and S a finite sequence of simple left Λ-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of Λ having sequence S of consecutive composition factors. Our principal objectives here are t...

0. Introduction 1 0.1. The Finite-Dimensional Case SU(n) 2 0.2. The Infinite-Dimensional Case LSU(n) 4 0.3. Plan of this Dissertation 13 1. Factorization and Coordinates for Finite-Dimensional Lie Groups 16 1.1. Lie Algebras 16 1.2. Lie Groups 19 1.3. Parametrizations with Sequences of Simple Roots 22 1.4. Parametrizations with Inversion Sets 26 2. Reduced Words for Infinite Elements of Affine ...