arXiv subjects

Alexander Shapiro

Publications and source records attributed to Alexander Shapiro.

52 records · Page 3Linked to original sources

arXiv2018

On $b$-Whittaker functions

The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmüller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppäläinen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

math-ph↗

arXiv2018

Matrix completion with deterministic pattern - a geometric perspective

We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: under what condition there will be (at least locally) unique solution to the matrix completion problem, i.e., the underlying true matrix is identifiable. We answer the question from a certain point of view and outline a geometric perspective. We give an algebraically verifiable sufficient condition, which we call the well-posedness condition, for the local uniqueness of MRMC solutions. We argue that this condition is necessary for local stability of MRMC solutions, and we show that the condition is generic using the characteristic rank. We also argue that the low-rank approximation approaches are more stable than MRMC and further propose a sequential statistical testing procedure to determine the "true" rank from observed entries. Finally, we provide numerical examples aimed at verifying validity of the presented theory.

cs.LG↗

arXiv2018

Decomposability and time consistency of risk averse multistage programs

Two approaches to time consistency of risk averse multistage stochastic problems were discussed in the recent literature. In one approach certain properties of the cor-responding risk measure are postulated which imply its decomposability. The other approach deals directly with conditional optimality of solutions of the considered problem. The aim of this paper is to discuss a relation between these two approaches.

math.OC↗

arXiv2018

Convergence Analysis of Sample Average Approximation of Two-stage Stochastic Generalized Equations

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables.This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate of the convergence is shown by using the perturbed partial linearization of functions. Moreover, sufficient conditions for the existence, uniqueness, continuity and regularity of solutions of two-stage stochastic generalized equations are presented under an assumption of monotonicity of the involved functions. These theoretical results are given without assuming relatively complete recourse, and are illustrated by two-stage stochastic non-cooperative games of two players.

math.OC↗

arXiv2017

Continuous tensor categories from quantum groups I: algebraic aspects

We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_λ$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_λ\otimes \mathcal{P}_μ$ into irreducibles.

math.RT↗

arXiv2017

Quantum groups, quantum tori, and the Grothendieck-Springer resolution

We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.

math.QA↗

arXiv2016

Statistical inference and hypotheses testing of risk averse stochastic programs

We study statistical properties of the optimal value and optimal solutions of the Sample Average Approximation of risk averse stochastic problems. Central Limit Theorem type results are derived for the optimal value and optimal solutions when the stochastic program is expressed in terms of a law invariant coherent risk measure. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider non-asymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the Stochastic Mirror Descent algorithm. Numerical simulations show how to use our developments to choose among different distributions and show the superiority of the asymptotic tests on a class of risk averse stochastic programs.

math.OC↗

arXiv2015

Dual pairs of quantum moment maps and doubles of Hopf algebras

For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.

math.QA↗

arXiv2015

Poisson Geometry of Monic Matrix Polynomials

We study the Poisson geometry of the first congruence subgroup $G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of Smith Normal Forms. This classification extends known descriptions of symplectic leaves on the (thin) affine Grassmannian and the space of $SL_m$-monopoles. We show that a generic leaf is covered by open charts with Poisson transition functions, the charts being birationally isomorphic to products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms of (thick) affine Grassmannians and Zastava spaces.

math-ph↗

arXiv2014

Rational and polynomial representations of Yangians

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian.

math.RT↗

arXiv2013

Rational representations of the Yangian Y(gl_n)

We construct a series of rational representations of Y(gl_n) and intertwining operators between them. We find explicit expressions for the images of highest-weight vectors under the intertwining operators. Finally, we state a conjecture that all irreducible finite-dimensional rational Y(gl_n)-modules arise as images of the constructed intertwining operators.

math.RT↗

arXiv2013

Uniqueness of Kusuoka Representations

This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka representation is derived from this initial result. Further, stochastic order relations are employed to identify the minimal Kusuoka representation. It is shown that measures in the minimal representation are extremal with respect to the order relations. The tools are finally employed to provide the minimal representation for important practical examples. Although the Kusuoka representation is usually given only for nonatomic probability spaces, this presentation closes the gap to spaces with atoms.

math.ST↗

arXiv2010

Three realizations of quantum affine algebra $U_q(A_2^{(2)})$

In this article we establish explicit isomorphisms between three realizations of quantum twisted affine algebra $U_q(A_2^{(2)})$: the Drinfeld ("current") realization, the Chevalley realization and the so-called $RLL$ realization, investigated by Faddeev, Reshetikhin and Takhtajan.

math.QA↗

arXiv2010

Weight function for the quantum affine algebra $U_q(A_2^{(2)})$

In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra $U_q(A_2^{(2)})$. The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types.

math.QA↗

arXiv2005

Rational transformations of systems of commuting nonselfadjoint operators

The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation of algebraic curves, we define the notion of a rational transformation of a system of commuting nonselfadjoint operators.

math.AG↗

arXiv2005

Rational transformations of algebraic curves and elimination theory

Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve.

math.AG↗