Related papers: Centralizers in PBW Extensions

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the…

For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the…

Let $k\in \mathbb{N}\setminus\{0\}$. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\ldots,x_k]/ I $, where $I$ is the ideal generated by the set $\{x_ix_j\mid i,j=1,\ldots,k\}$.…

We investigate the differential smoothness of a certain family of skew Poincar\'e-Birkhoff-Witt extensions.

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums…

We study the problem of describing the set of real functionals on the quotient $\textrm{Sym}/(p_2-1)$ of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This…

We study Berezin-Toeplitz quantization of complex projective spaces $\mathbb{CP}^{d-1}$ and obtain full asymptotic expansions of the Berezin transformation and of products of Toeplitz operators. In each case, the remainder is controlled by…

Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$ has two distinct linear representations generated by two central elements of $B_n(r,q)$…

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an…

Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between…

We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of…

Given a smooth curve with nonzero curvature $\Sigma\subset \mathbb{R}^2$, let $E_{\Sigma}$ denote the associated Fourier extension operator. For both general compact curves and the parabola, we characterize the pairs $(p,q)\in [1,\infty]^2$…

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of…

In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ arises from…

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization…

This document is a thesis presented for the ``Habilitation \`a diriger des recherches''. The first chapter provides some background and sketch the story of the classical Schur-Weyl duality and its quantum analogue involving the Hecke…

This paper is to characterize piecewise continuous almost periodic functions as the product of Bohr almost periodic functions and sequences. As an application, the result is used to discuss piecewise continuous almost periodic solutions of…

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if $I$ is a proper ideal of the ring $R=F[t_1,\ldots,t_n]$ of polynomials over a field $F$, then…

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the…

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the…