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R. Rimanyi

Publications and source records attributed to R. Rimanyi.

At least 19 recordsLinked to original sources
arXiv2020

Bow varieties---geometry, combinatorics, characteristic classes

Cherkis bow varieties are believed to be the set of spaces where 3d mirror symmetry for characteristic classes can be observed. We describe geometric structures on a large class of Cherkis bow varieties by developing the necessary combinatorial presentations, including binary contingency tables and skein diagrams. We make the first steps toward the sought after statement for 3d mirror symmetry for characteristic classes by conjecturing a formula for cohomological stable envelopes. Additionally we provide an account of the full statement, with examples, for elliptic stable envelopes.

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arXiv2017

An iterated residue perspective on stable Grothendieck polynomials

Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool, similar to generating sequences, which we call the iterated residue technique. We prove new formulas on the calculus of iterated residues and use them to prove straightening laws and multiplication formulas for stable Grothendieck polynomials. As a further application of our method, we give new proofs that the $K$-Pieri rule and the expansions of Grothendieck polynomials in the Schur basis both exhibit alternating signs. As a consequence, we observe that our method implies a new combinatorial statement of the $K$-Pieri rule. Our results indicate that the iterated residue technique should be further explored as a new line of attack on open conjectures regarding positivity and stability, for example of quiver polynomials and Thom polynomials, in $K$-theory.

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arXiv2017

Chern-Schwartz-MacPherson classes of degeneracy loci

The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas---including residue generating sequences---for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (2-step) positivity properties for some relevant representations.

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arXiv2016

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 $λ$. We consider the dynamical Gelfand-Zetlin algebra which is a commutative algebra of difference operators in $λ$. 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 $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[δ^{\pm1}]$ on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $δ: ζ(λ)\toζ(λ+y)$ 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.

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

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arXiv2014

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

We consider the cotangent bundle $T^*F_λ$ of a $GL_n$ partial flag variety, $λ=(λ_1,...,λ_N)$, $|λ|=\sum_iλ_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_λ)$. We introduce K-theoretic stable envelope maps $\Stab_σ: \oplus_{|λ|=n} K_T((T^*F_λ)^T)\to\oplus_{|λ|=n}K_T(T^*F_λ)$, where $σ\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|λ|=n}K_T(T^*F_λ)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_λ))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_λ))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_λ)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_λ))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_λ)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_λ)$ 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.

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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

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_λ;\C) of the cotangent bundle of a partial flag variety F_λparametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=λ_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_λ;\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↗
arXiv2013

On the Cohomological Hall Algebra of Dynkin quivers

Consider the Cohomological Hall Algebra as defined by Kontsevich and Soibelman, associated with a Dynkin quiver. We reinterpret the geometry behind the multiplication map in the COHA, and give an iterated residue formula for it. We show natural subalgebras whose product is the whole COHA (except in the $E_8$ case). The dimension count version of this statement is an identity for quantum dilogarithm series, first proved by Reineke. We also show that natural structure constants of the COHA are universal polynomials representing degeneracy loci, a.k.a. quiver polynomials.

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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}.

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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.

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arXiv2012

Equivariant and invariant theory of nets of conics

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems---in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics---can be solved studying equivariant classes of orbit closures.

math.AG↗
arXiv2008

Equivariant classes of matrix matroid varieties

Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by $X_C$. We study the equivariant cohomology class represented by the Zariski closure $Y_C$ of this set. We show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces. We also show how to calculate these classes and present their basic properties.

math.AG↗
arXiv2008

The general quadruple point formula

Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $Σ^2Σ^0$ multisingularities.

math.AG↗
arXiv2007

Calculation of Thom polynomials for group actions

In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand we describe a way to calculate Thom polynomials: the method of restriction equations. It turned out that though the idea is quite simple the method is very powerful. We reproduced and improved earlier result in several directions (singularities, Schubert calculus, quivers). However a proper introduction to the basic theorems was missing. In this paper we try to pay this debt as well as we present the connections with obstruction theory and equivariant cohomology. We give some new results and outline possible generalizations and problems.

math.AG↗
arXiv2007

On the structure of Thom polynomials of singularities

Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.

math.AG↗
arXiv2007

The degree of the discriminant of irreducible representations

We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the highest weight of the representation. The proof uses equivariant cohomology techniques, namely, the theory of Thom polynomials, and a new method for their computation. We study the combinatorics of our formulas in various special cases.

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