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Michael Gekhtman

Publications and source records attributed to Michael Gekhtman.

At least 19 recordsLinked to original sources
arXiv2025

Orthogonality with Respect to the Hermite Product, KP Wave Functions, and the Bispectral Involution

It is well known that for any wave function $\psi(x,z)$ of the KP hierarchy, there is another wave function called its ''adjoint'' such that the path integral of their product with respect to $z$ around any sufficiently large closed path is zero. For the wave functions in the adelic Grassmannian ${\rm Gr}^{\rm ad}$, the bispectral involution which exchanges the role of $x$ and $z$ also implies the existence of an ''$x$-adjoint wave function'' $\psi^{\star}(x,z)$ so that the product of the wave function, the $x$-adjoint, and the Hermite weight ${\rm e}^{-x^2/2}$ has no residue. Utilizing this, we show that the sequences of coefficient functions in the power series expansion of any KP wave function in ${\rm Gr}^{\rm ad}$ and its image under the bispectral involution at $t_2=-\frac{1}{2}$ are always ''almost bi-orthogonal'' with respect to the Hermite product. Whether the sequences have the stronger properties of being (almost) orthogonal can easily be determined in terms of KP flows and the bispectral involution. As a special case, the exceptional Hermite orthogonal polynomials can be recovered in this way. This provides both a generalization of and an explanation of the fact that the generating functions of the exceptional Hermites are certain special wave functions of the KP hierarchy. In addition, one new surprise is that the same KP wave function which generates the sequences of functions is also a generating function for the norms when evaluated at $t_1=1$ and $t_2=0$. The main results are proved using Calogero-Moser matrices satisfying a rank one condition. The same results also apply in the case of ''spin-generalized'' Calogero-Moser matrices, which produce instances of matrix orthogonality.

nlin.SI↗
arXiv2024

On double brackets for marked surfaces

We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of the surface $S$ such that on every boundary component there is at least one point of $P$. We show that this double bracket is a noncommutative generalization of the well-known Goldman bracket, defined on the space of free homotopy classes of loops on $S$. For an algebra $A$ without polynomial identities, we construct a double bracket on the space of decorated twisted $\mathrm{GL}_n(A)$-, symplectic and indefinite orthogonal local systems.

math.DG↗
arXiv2024

Multiplicative Inequalities In Cluster Algebras Of Finite Type

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as a real-valued function on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u-variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr(2, n), Gr(3, 6), Gr(3, 7), Gr(3, 8) that every bounded Laurent monomial in Plücker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr(4, 8) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Plücker coordinates.

math.CO↗
arXiv2024

Integrable systems and cluster algebras

We review several constructions of integrable systems with an underlying cluster algebra structure, in particular the Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on perfect networks and the Goncharov-Kenyon approach based on the dimer model. We also discuss results of Galashin and Pylyavskyy on integrability of T-systems.

nlin.SI↗
arXiv2024

Associahedra as moment polytopes

Generalized associahedra are a well-studied family of polytopes associated to a finite-type cluster algebra and choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani-Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster A-variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction.

math.CO↗
arXiv2023

Generalized cluster structures related to Poisson duals of $\mathrm{SL}_n$

We study Poisson varieties $(\mathrm{SL}_n,\pi_{\bar{\mathbf{\Gamma}}}^{\dagger})$ parameterized by Belavin--Drinfeld quadruples $\bar{\mathbf{\Gamma}}:=(\mathbf{\Gamma},r_0)$ of type $A_{n-1}$ along with generalized cluster structures $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$ in $\mathbb{C}[\mathrm{SL}_n]$ compatible with $\pi_{\bar{\mathbf{\Gamma}}}^{\dagger}$. The Poisson structure $\pi_{\bar{\mathbf{\Gamma}}}^{\dagger}$ is a pushforward of the Poisson structure $\pi_{\bar{\mathbf{\Gamma}}}^*$ of the Poisson dual $\mathrm{SL}_n^*$ of $(\mathrm{SL}_n,\pi_{\bar{\mathbf{\Gamma}}})$. We prove that the generalized upper cluster algebra of $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$ is naturally isomorphic to $\mathbb{C}[\mathrm{SL}_n]$. Moreover, for any connected reductive complex group $G$ and a BD quadruple $(\mathbf{\Gamma},r_0)$, we produce a Poisson birational map $\mathcal{Q}:(G,\pi_{(\mathbf{\Gamma}_{\text{std}},r_0)}^{\dagger})\dashrightarrow(G,\pi_{(\mathbf{\Gamma},r_0)}^{\dagger})$, and when $G \in \{\mathrm{SL}_n,\mathrm{GL}_n\}$, we show that $\mathcal{Q}$ is a birational quasi-isomorphism between $\mathcal{GC}^\dagger(\mathbf{\Gamma}_{\text{std}})$ and $\mathcal{GC}^\dagger(\mathbf{\Gamma})$. Lastly, for any pair of BD triples $\tilde{\mathbf{\Gamma}} \prec \mathbf{\Gamma}$ of type $A_{n-1}$ comparable in the natural order, we use the map $\mathcal{Q}$ to construct a birational quasi-isomorphism between $\mathcal{GC}^{\dagger}(\tilde{\mathbf{\Gamma}})$ and $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$.

math.QA↗
arXiv2023

On bounded ratios of minors of totally positive matrices

We provide several examples of bounded Laurent monomials of minors of a totally positive matrix, which can not be factored into a product of so called primitive ratios, thus showing that the conjecture about factorization of bounded ratios stated in [3] by Fallat, Gekhtman, and Johnson does not hold. However, all found examples satisfy subtraction-free conjecture stated also in [3]. In addition, we show that the set of all bounded ratios form a polyhedral cone of dimension $\binom{2n}{n}-2n$.

math.RA↗
arXiv2022

Validity of tests for time-to-event endpoints in studies with the Pocock and Simon covariate-adaptive randomization

In the presence of prognostic covariates, inference about the treatment effect with time-to-event endpoints is mostly conducted via the stratified log-rank test or the score test based on the Cox proportional hazards model. In their ground-breaking work Ye and Shao (2020) have demonstrated theoretically that when the model is misspecified, the robust score test (Wei and Lin, 1989) as well as the unstratified log-rank test are conservative in trials with stratified randomization. This fact, however, was not established for the Pocock and Simon covariate-adaptive allocation other than through simulations. In this paper, we expand the results of Ye and Shao to a more general class of randomization procedures and show, in part theoretically, in part through simulations, that the Pocock and Simon covariate-adaptive allocation belongs to this class. We also advance the search for the correlation structure of the normalized within-stratum imbalances with minimization by describing the asymptotic correlation matrix for the case of equal prevalence of all strata. We expand the robust tests proposed by Ye and Shao for stratified randomization to minimization and examine their performance trough simulations.

math.ST↗
arXiv2019

Asymptotic sign coherence conjecture

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.

math.CO↗
arXiv2017

Exotic cluster structures on $SL_n$: the Cremmer-Gervais case

This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds to a cluster structure in $Ø(\G)$. We have shown before that this conjecture holds for any $\G$ in the case of the standard Poisson-Lie structure and for all Belavin-Drinfeld classes in $SL_n$, $n<5$. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on $SL_n$, which is the least similar to the standard one.

math.QA↗
arXiv2017

Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

We introduce and study a Hamiltonian formalism of mutations in cluster algebras using canonical variables, where the Hamiltonian is given by the Euler dilogarithm. The corresponding Lagrangian, restricted to a certain subspace of the phase space, coincides with the Rogers dilogarithm. As an application, we show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from the Hamiltonian/Lagrangian point of view.

math.RA↗
arXiv2015

Generalized cluster structure on the Drinfeld double of $GL_n$

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the dual Poisson--Lie bracket.

math.QA↗
arXiv2015

On the properties of the exchange graph of a cluster algebra

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra $\A$ in two cases: when $\A$ is of geometric type and when $\A$ is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of $\A$. Both conjectures were formulated recently by Fomin and Zelevinsky.

math.CO↗
arXiv2014

Integrable cluster dynamics of directed networks and pentagram maps

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. This paper expands on our research announcement arXiv:1110.0472

math.DS↗
arXiv2013

Cremmer--Gervais cluster structure on $SL_n$

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds to a cluster structure in $Ø(\G)$. We have shown before that this conjecture holds for any $\G$ in the case of the standard Poisson--Lie structure and for all Belavin-Drinfeld classes in $SL_n$, $n<5$. In this paper we establish it for the Cremmer-Gervais Poisson-Lie structure on $SL_n$, which is the least similar to the standard one. Besides, we prove that on $SL_3$ the cluster algebra and the upper cluster algebra corresponding to the Cremmer-Gervais cluster structure do not coincide, unlike the case of the standard cluster structure. Finally, we show that the positive locus with respect to the Cremmer-Gervais cluster structure is contained in the set of totally positive matrices.

math.QA↗
arXiv2012

Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model

We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.

nlin.SI↗
arXiv2012

Higher pentagram maps, weighted directed networks, and cluster dynamics

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. In this paper, we extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces developed by M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\C$, we give a geometric interpretation in terms of circle patterns.

math.QA↗