相关论文: Functorial properties of the hypergeometric map

We construct Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation for $U_{q}(\widehat{\mathfrak{sl}}_{2})$ on the parabolic Kazhdan--Lusztig bases. They are characterized by non-symmetric Koornwinder…

This paper studied what shall be called the Long equation: that is the system of nonlinear equation $R^{12}R^{13}=R^{13}R^{12}$ and $R^{12}R^{23}= R^{23}R^{12}$. Any solution of this system supplies us a solution for the integrability…

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…

This is a survey on Zariski equisingularity. We recall its definition, main properties, and a variety of applications in Algebraic Geometry and Singularity Theory. In the first part of this survey, we consider Zariski equisingular families…

We discuss the correspondence between the Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Calogero model in the case when $n$ is not necessarily equal to $N$. This can be viewed as a natural…

We give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra $gl_{N+1}$. Asymptotic solutions to qKZ are constructed. The leading term of an asymptotic solution is…

Hypergeometric solutions to the q-Painlev\'e equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric…

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez-Etingof cyclotomic Knizhnik-Zamolodchikov (KZ) equations…

In this article, we define an algebraic version of the Knizhnik--Zamolodchikov functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by…

In this paper, we completely classify the rational weights $k$ for which the Kaneko-Zagier (KZ) differential equation admits a fundamental system of solutions consisting of modular forms for a principal congruence subgroup $\Gamma(N)$. By…

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational…

For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(z) solving the difference equation f(z+1)=R(z, f(z)) is finite and bounded just in terms of the degrees of R in the…

We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent…

We prove general Dwork-type congruences for Hasse--Witt matrices attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions…

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes…

We write the integral formula of Tarasov-Varchenko type for the solutions to the quantum Knizhnik-Zamolodchikov associated with a tensor product the of vector representations of sl_n. We consider the case where the deformation parameter q…

We introduce, for every $\mathbb{Z}$-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of…

We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial…

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space…