arXiv subjects

Michael Shapiro

Publications and source records attributed to Michael Shapiro.

At least 37 records · Page 2Linked to original sources

arXiv2019

Cluster algebras with Grassmann variables

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step of understanding the notion of cluster superalgebra

math.CO↗

arXiv2019

Upper cluster algebras and choice of ground ring

We initiate a study of the dependence on the choice of ground ring on the question of whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller's theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.

math.AC↗

arXiv2019

Plethora of cluster structures on $GL_n$

We continue the study of multiple cluster structures in the rings of regular functions on $GL_n$, $SL_n$ and $\operatorname{Mat}_n$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $\mathcal G$ corresponds to a cluster structure in $\mathcal O(\mathcal G)$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $A_n$ type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $SL_n$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $SL_n$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications.

math.QA↗

arXiv2017

Secant degeneracy index of the standard strata in the space of binary forms

The space $Pol_d\simeq \bC P^d$ of all complex-valued binary forms of degree $d$ (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition $μ\vdash d$. For each such stratum $S_μ,$ we introduce its secant degeneracy index $\ell_μ$ which is the minimal number of projectively dependent pairwise distinct points on $S_μ$, i.e., points whose projective span has dimension smaller than $\ell_μ-1$. In what follows, we discuss the secant degeneracy index $\ell_μ$ and the secant degeneracy index $\ell_{\bar μ}$ of the closure $\bar S_μ$.

math.AG↗

arXiv2017

On moments of a polytope

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.

math.MG↗

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

Drinfeld double of $GL_n$ and generalized cluster structures

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 Poisson bracket on the dual Poisson--Lie group.

math.QA↗

arXiv2017

Growth in higher Baumslag-Solitar groups

We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism $g\mapsto g^3$, and prove that such groups have rational growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the group has rational growth. In the appendix, the subgroup growth series has been computed for all $m \leq 10$.

math.GR↗

arXiv2015

Generalized cluster structure on the Drinfeld double of $GL_n$

math.QA↗

arXiv2015

Kolmogorov's axioms for probabilities with values in hyperbolic numbers

We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes' theorem. Our probability may take values which are zero--divisors and we discuss carefully this peculiarity.

math.PR↗

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

Rational growth in the Heisenberg group

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all finite generating sets, and we settle this question affirmatively. We also establish almost-convexity for all finite generating sets. Previously, both of these properties were known to hold for hyperbolic groups and virtually abelian groups, and there were no further examples in either case. Our main method is a close description of the relationship between word metrics and associated Carnot-Caratheodory Finsler metrics on the ambient Lie group. We provide (non-regular) languages in any word metric that suffice to represent all group elements.

math.GR↗

arXiv2014

Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{π/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.

math-ph↗

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↗

arXiv2014

Cluster algebras and triangulated orbifolds

We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, provide positivity of Laurent expansions of cluster variables, and prove sign-coherence of c-vectors.

math.CO↗

arXiv2014

Growth rate of cluster algebras

We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.

math.CO↗

arXiv2014

Equations in nilpotent groups

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

math.GR↗

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↗