arXiv subjects
Publications and source records attributed to Natalia Tsilevich.
We introduce the notion of $GL(n)$-dependence of matrices, which is a generalization of linear dependence taking into account the matrix structure. Then we prove a theorem, which generalizes, on the one hand, the fact that $n+1$ vectors in an $n$-dimensional vector space are linearly dependent and, on the other hand, the fact that the natural action of the group $GL(n,{\cal K})$ on ${\cal K}^n\setminus\{0\}$ is transitive.
Higher Lie characters form a distinguished family of symmetric group characters, which appear in many areas of algebra and combinatorics. An old open problem of Thrall is to decompose them into irreducibles. We propose a novel asymptotic approach to this problem, showing that many families of higher Lie characters tend to be proportional to the regular character. In particular, a random higher Lie character tends, in probability, to be proportional to the regular character.
We prove that there is a natural grading-preserving isomorphism of $\sl$-modules between the basic module of the affine Lie algebra $\widehat\sl$ (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group $\sinf$ with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of $\widehat\sl$ and the Virasoro algebra on the one hand, and the representation theory of $\sinf$ and the related combinatorics on the other hand.
We explicitly construct and study an isometry between the spaces of square integrable functionals of an arbitrary Levy process and a vector-valued Gaussian white noise. In particular, we obtain explicit formulas for this isometry at the level of multiplicative functionals and at the level of orthogonal decompositions, as well as find its kernel. We consider in detail the central special case: the isometry between the $L^2$ spaces over a Poisson process and the corresponding white noise. The key role in our considerations is played by the notion of measure and Hilbert factorizations and related notions of multiplicative and additive functionals and logarithm. The obtained results allow us to introduce a canonical Fock structure (an analogue of the Wiener--Ito decomposition) in the $L^2$ space over an arbitrary Levy process. An application to the representation theory of current groups is considered. An example of a non-Fock factorization is given.
We consider the simplest split-merge Markov operator $T$ on the infinite-dimensional simplex $Σ_1$ of monotone non-negative sequences with unit sum. For a sequence $x\inΣ_1$, it picks a size-biased sample (with replacement) of two elements of $x$; if these elements are distinct, it merges them and reorders the sequence, and if the same element is picked twice, it splits this element uniformly into two parts and reorders the sequence. We prove that the means along the $T$-trajectory of the $\de$-measure at the vector $(1,0,0,{...})$ converge to the Poisson--Dirichlet distribution PD(1).