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A. Varchenko

Publications and source records attributed to A. Varchenko.

At least 55 records · Page 3Linked to original sources
arXiv2008

Spaces of quasi-exponentials and representations of gl_N

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{λ^{(s)}})_λ, the weight subspace of weight $λ$ of the tensor product of k polynomial irreducible gl_N-modules with highest weights λ^{(1)},...,λ^{(k)}, respectively. The Bethe algebra depends on N complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are distinct, we prove that the image of B_K in the endomorphisms of (\otimes_{s=1}^k L_{λ^{(s)}})_λis isomorphic to the algebra of functions on the intersection of k suitable Schubert cycles in the Grassmannian of N-dimensional spaces of quasi-exponentials with exponents K. We also prove that the B_K-module (\otimes_{s=1}^k L_{λ^{(s)}})_λis isomorphic to the coregular representation of that algebra of functions. We present a Bethe ansatz construction identifying the eigenvectors of the Bethe algebra with points of that intersection of Schubert cycles.

math.QA↗
arXiv2007

On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model

In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the gl_N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [MTV4] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.

math.QA↗
arXiv2007

Schubert calculus and representations of general linear group

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection. In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.

math.QA↗
arXiv2007

Bethe algebra and algebra of functions on the space of differential operators of order two with polynomial solutions

We show that the following two algebras are isomorphic. The first is the algebra $A_P$ of functions on the scheme of monic linear second-order differential operators on $\C$ with prescribed regular singular points at $z_1,..., z_n, \infty$, prescribed exponents $\La^{(1)}, ..., \La^{(n)}, \La^{(\infty)}$ at the singular points, and having the kernel consisting of polynomials only. The second is the Bethe algebra of commuting linear operators, acting on the vector space $\Sing L_{\La^{(1)}} \otimes ... \otimes L_{\La^{(n)}}[\La^{(\infty)}]$ of singular vectors of weight $\La^{(\infty)}$ in the tensor product of finite dimensional polynomial $gl_2$-modules with highest weights $\La^{(1)},..., \La^{(n)}$.

math.QA↗
arXiv2007

Differential equation for Jacobi-Pineiro polynomials

For $r\in \Z_{\geq 0}$, we present a linear differential operator %$(\di)^{r+1}+ a_1(x)(\di)^{r}+...+a_{r+1}(x)$ of order $r+1$ with rational coefficients and depending on parameters. This operator annihilates the $r$-multiple Jacobi-Piñeiro polynomial. For integer values of parameters satisfying suitable inequalities, it is the unique Fuchsian operator with kernel consisting of polynomials only and having three singular points at $x=0, 1, \infty$ with arbitrary non-negative integer exponents $0, m_1+1, >..., m_1+...+m_r+r$ at $x=0$, special exponents $0, k+1, k+2,..., k+r$ at $x=1$ and arbitrary exponents at $x=\infty$.

math.QA↗
arXiv2007

Bethe eigenvectors of higher transfer matrices

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $gl_N$. The models are defined on a tensor product irreducible $gl_N$-modules. For each model, there exist $N$ one-parameter families of commuting operators on the tensor product, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

math.QA↗
arXiv2006

Resonance relations, holomorphic trace functions and hypergeometric solutions to qKZB and Macdonald-Ruijsenaars equations

The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weyl anti-invariant hypergeometric qKZB solutions with values in a tensor product of finite-dimensional modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of the quantum group invariants, and is computed by the Verlinde algebra when q is a root of unity.

math.QA↗
arXiv2006

Quiver $D$-Modules and Homology of Local Systems over an Arrangement of Hyperplanes

Let $C$ be an arrangement of affine hyperplanes in a complex affine space $X$, $D$ the ring of algebraic differential operators on $X$. We define a category of quivers associated with $C$. A quiver is a collection of vector spaces, attached to strata of the arrangement, and suitable linear maps between the vector spaces . To a quiver we assign a $D$-module on $X$, called the quiver $D$-module. We describe basic operations for $D$-modules in terms of linear algebra of quivers. We give an explicit construction of a free resolution of a quive $D$-module and use the construction to describe the associated perverse sheaf. As an application, we calculate the cohomology of $X$ with coefficients in the quiver perverse sheaf (under certain assumptions).

math.QA↗
arXiv2006

A generalization of the Capelli identity

We prove a generalization of the Capelli identity. As an application we obtain an isomorphism of the Bethe subalgebras actions under the (gl(N),gl(M)) duality.

math.QA↗
arXiv2006

Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential

Let $V = < x^{λ_i}p_{ij}(x), i=1,...,n, j=1, ..., N_i > $ be a space of quasi-polynomials in $x$ of dimension $N=N_1+...+N_n$. The regularized fundamental differential operator of $V$ is the polynomial differential operator $\sum_{i=0}^N A_{N-i}(x)(x \frac d {dx})^i$ annihilating $V$ and such that its leading coefficient $A_0$ is a monic polynomial of the minimal possible degree. Let $U = < z_a^{u} q_{ab}(u), a=1,...,m, b=1,..., M_a >$ be a space of quasi-exponentials in $u$ of dimension $M=M_1+...+M_m$. The regularized fundamental difference operator of $U$ is the polynomial difference operator $\sum_{i=0}^M B_{M-i}(u)(τ_u)^i$ annihilating $U$ and such that its leading coefficient $B_0$ is a monic polynomial of the minimal possible degree. Here $(τ_uf)(u)=f(u+1)$. Having a space $V$ of quasi-polynomials with the regularized fundamental differential operator $D$, we construct a space of quasi-exponentials $U = $ whose regularized fundamental difference operator is the difference operator $\sum_{i=0}^N u^i A_{N-i}(τ_u)$. The space $U$ is constructed from $V$ by a suitable integral transform. Similarly, having $U$ we can recover $V$ by a suitable integral transform. Our integral transforms are analogs of the bispectral involution on the space of rational solutions to the KP hierarchy \cite{W}. As a corollary of the properties of the integral transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two $(gl_N, gl_M)$ dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model.

math.QA↗
arXiv2006

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This in particular implies the following result: If all ramification points of a parametrized rational curve $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has a real spectrum. In Appendix we discuss properties of differential operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians.

math.AG↗
arXiv2006

Higher Lame Equations and Critical Points of Master Functions

Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power $U(\n_-)^{\otimes (n-1)}$, where $n$ is the number of singular points in $\C$ and $U(\n_-)$ is the enveloping algebra of the nilpotent subalgebra of $\glg_{r+1}$.

math.CA↗
arXiv2006

Determinants of the hypergeometric period matrices of an arrangement and its dual

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.

math.AG↗
arXiv2005

Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl_{r+1} irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight. We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Piñeiro multiple orthogonal polynomial and others are given by Wronskian type determinants of Jacobi-Piñeiro polynomials. As a byproduct we obtain a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.

math.QA↗
arXiv2005

Bispectral and $(\glN,\glM)$ Dualities

Let $V = < p_{ij}(x)e^{\la_ix}, i=1,...,n, j=1, ..., N_i >$ be a space of quasi-polynomials of dimension $N=N_1+...+N_n$. Define the regularized fundamental operator of $V$ as the polynomial differential operator $D = \sum_{i=0}^N A_{N-i}(x)\p^i$ annihilating $V$ and such that its leading coefficient $A_0$ is a polynomial of the minimal possible degree. We construct a space of quasi-polynomials $U = < q_{ab}(u)e^{z_au} >$ whose regularized fundamental operator is the differential operator $\sum_{i=0}^N u^i A_{N-i}(\partial_u)$. The space $U$ is constructed from $V$ by a suitable integral transform. Our integral transform corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) to the KP hierarchy, see \cite{W}. As a corollary of the properties of the integral transform we obtain a correspondence between critical points of the two master functions associated with the $(\glN,\glM)$ dual Gaudin models as well as between the corresponding Bethe vectors.

math.QA↗
arXiv2005

Combinatorics of rational functions and Poincare-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form

We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt basis. As a byproduct we obtain results on the combinatorics of rational functions, namely non-trivial identities are proved between certain rational functions with partial symmetries.

math.CO↗