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A. Varchenko

Publications and source records attributed to A. Varchenko.

At least 19 recordsLinked to original sources
arXiv2023

Hypergeometric integrals, hook formulas and Whittaker vectors

We determine the coefficient of proportionality between two multidimensional hypergeometric integrals. One of them is a solution of the dynamical difference equations associated with a Young diagram and the other is the vertex integral associated with the Young diagram. The coefficient of proportionality is the inverse of the product of weighted hooks of the Young diagram. It turns out that this problem is closely related to the question of describing the action of the center of the universal enveloping algebra of $\mathfrak{gl}_n$ on the space of Whittaker vectors in the tensor product of dual Verma modules with fundamental modules, for which we give an explicit basis of simultaneous eigenvectors.

math-ph↗
arXiv2020

Dynamical ${\frak{sl}}_2$ Bethe algebra and functions on pairs of quasi-polynomials

We consider the space $\text{Fun}_{\frak{sl}_2}V[0]$ of functions on the Cartan subalgebra of $\frak{sl}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional $\frak{sl}_2$-modules. We consider the algebra $\mathcal B$ of commuting differential operators on $\text{Fun}_{\frak{sl}_2}\,V[0]$, constructed by V.Rubtsov, A.Silantyev, D.Talalaev in 2009. We describe the relations between the action of $\mathcal B$ on $\text{Fun}_{\frak{sl}_2}V[0]$ and spaces of pairs of quasi-polynomials.

math-ph↗
arXiv2017

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $K$-theoretic stable envelopes and our elliptic stable envelopes. We show that the $K$-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $\frak{gl}_2$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $K$-theoretic stable envelopes.

math.AG↗
arXiv2017

Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$

We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of strata $\Omega_{\mathbf{\Lambda}}$ labeled by unordered sets $\mathbf{\Lambda}=(\lambda^{(1)},\dots,\lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{\lambda^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $\lambda^{(i)}$. We show that the closure of a stratum $\Omega_{\mathbf{\Lambda}}$ is the union of the strata $\Omega_{\mathbf\Xi}$, $\mathbf{\Xi}=(\xi^{(1)},\dots,\xi^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N} (V_{\xi^{(i)}}, \otimes_{j\in I_i}V_{\lambda^{(j)}}\big)\neq 0$ for $i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$.

math.QA↗
arXiv2015

On the Gaudin model associated to Lie algebras of classical types

We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.

math.QA↗
arXiv2015

Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians

We consider the rational dynamical quantum group $E_y(gl_2)$ and introduce an $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the equivariant cohomology algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))$ of the cotangent bundle of the Grassmannian $\Gr(k,n)$ with coefficients extended by a suitable ring of rational functions in an additional variable $\lambda$. We consider the dynamical Gelfand-Zetlin algebra which is a commutative algebra of difference operators in $\lambda$. We show that the action of the Gelfand-Zetlin algebra on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the natural action of the algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[\delta^{\pm1}]$ on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $\delta : \zeta(\lambda)\to\zeta(\lambda+y)$ is the shift operator. The $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [FTV] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ induced by the weight functions are dynamical variants of Chern-Schwartz-MacPherson classes of Schubert cells.

math.AG↗
arXiv2015

Equivariant Chern-Schwartz-MacPherson classes in partial flag varieties: interpolation and formulae

Consider the natural torus action on a partial flag manifold $Fl$. Let $\Omega_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\Omega_I)$, as well as a formula, of `localization type', for $c^{sm}(\Omega_I)$. In fact, we proved similar results for a class $\kappa_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(\Omega_I)=\kappa_I$.

math.AG↗
arXiv2014

Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

We consider the cotangent bundle $T^*F_\lambda$ of a $GL_n$ partial flag variety, $\lambda=(\lambda_1,...,\lambda_N)$, $|\lambda|=\sum_i\lambda_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_\lambda)$. We introduce K-theoretic stable envelope maps $\Stab_{\sigma}: \oplus_{|\lambda|=n} K_T((T^*F_\lambda)^T)\to\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$, where $\sigma\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_\lambda))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_\lambda))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_\lambda)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_\lambda))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_\lambda)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_\lambda)$ and with the help of the trigonometric weight functions introduced in [TV1, TV3] to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix 5 we describe the Bethe algebra of the XXZ model by generators and relations.

math.AG↗
arXiv2013

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety $S_I$ in the cotangent bundle $T^* Gr$ of the Grassmannian $Gr$ of $k$-planes in $C^n$. This conormal bundle has a fundamental class ${κ_I}$ in the equivariant cohomology $H^*_{T}(T^* Gr)$. Here $T=(C^*)^n\times C^*$. The torus $(C^*)^n$ acts on $T^* Gr$ in the standard way and the last factor $C^*$ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum $Y_I$ of the Yangian $Y(gl_2)$ weight functions $(W_J)_J$. We describe a relation of $Y_I$ with the double Schur polynomial $[S_I]$. A modified version of the $κ_I$ classes, named $κ'_I$, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold $T^* Gr$. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on $Gr$. The classes $(κ'_I)_I$ form a basis in the suitably localized equivariant cohomology $H^*_{T}(T^* Gr)$. This basis depends on the choice of the coordinate flag in $C^n$. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.

math.AG↗
arXiv2013

BGG resolutions via configuration spaces

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the sl_2 Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.

math.AG↗
arXiv2013

Critical points of master functions and the mKdV hierarchy of type A^2_2

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the affine Lie algebra $A^2_2$. We describe how the critical points of this population define rational solutions of the equations of the mKdV hierarchy associated with $A^2_2$.

math.AG↗
arXiv2013

Bethe subalgebras of the group algebra of the symmetric group

We introduce families of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra of the symmetric group. Bethe subalgebras are deformations of the Gelfand-Zetlin subalgebra. We describe various properties of Bethe subalgebras.

math.QA↗
arXiv2013

Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)

We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace $V(b)_λ\subset V(b)$ of weight $λ$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b)_λ$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b)^{sing}_λ$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b)^{sing}_λ$ with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map. These results we announced earlier in relation with a description of the quantum equivariant cohomology of the cotangent bundle of a partial flag variety and a description of commutative subalgebras of the group algebra of a symmetric group.

math.AG↗
arXiv2013

Cohomology of a flag variety as a Bethe algebra

We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in \C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (\C^N)^{\otimes n}.

math.QA↗
arXiv2013

Partial flag varieties, stable envelopes and weight functions

We consider the cotangent bundle T^*F_λof a GL_n partial flag variety, λ= (λ_1,...,λ_N), |λ|=\sum_iλ_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*_T(T^*F_λ). In [MO], a Yangian module structure was introduced on \oplus_{|λ|=n} H^*_T(T^*F_λ). We identify this Yangian module structure with the Yangian module structure introduced in [GRTV]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*F_λ), described in [MO], with the action of the dynamical Hamiltonians from [TV2, MTV1, GRTV]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [MO]. The formula is in terms of the Yangian weight functions introduced in [TV1], c.f. [TV3, TV4], in order to construct q-hypergeometric solutions of qKZ equations.

math.AG↗
arXiv2012

Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

math.QA↗
arXiv2012

Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag variety F_\lambda parametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i, as the Yangian Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) as the algebra of functions on fibers of a discrete Wronski map. In particular this gives generators and relations of that algebra. This identification also gives us hypergeometric solutions of the associated quantum differential equation. That fact manifests the Landau-Ginzburg mirror symmetry for the cotangent bundle of the flag variety.

math.AG↗