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Kyungyong Lee

Publications and source records attributed to Kyungyong Lee.

At least 55 records · Page 3Linked to original sources
arXiv2014

Greedy bases in rank 2 quantum cluster algebras

We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.

math.QA↗
arXiv2013

Limits of Modified Higher (q,t)-Catalan Numbers

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher $q,t$-Catalan numbers and show that these limits equal the generating function for integer partitions. We also identify certain coefficients of the higher $q,t$-Catalan numbers as enumerating suitable integer partitions, and we make some conjectures on the homological significance of the Bergeron-Garsia nabla operator.

math.CO↗
arXiv2013

Combinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula

The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This paper proves the equivalence of the two definitions for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and all $n\leq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_n(q,t)$ for all $m\geq 1$ and all $n\leq 4$. We give analogous results for certain rational-slope $q,t$-Catalan polynomials.

math.CO↗
arXiv2013

Positivity and tameness in rank 2 cluster algebras

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.

math.CO↗
arXiv2012

Greedy elements in rank 2 cluster algebras

A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was developed several years ago by P.Sherman - A.Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

math.RA↗
arXiv2012

On natural maps from strata of quiver Grassmannians to ordinary Grassmannians

Caldero and Zelevinsky studied the geometry of quiver Grassmannians for the Kronecker quiver and computed their Euler characteristics by examining natural stratification of quiver Grassmannians. We consider generalized Kronecker quivers and compute virtual Poincare polynomials of certain varieties which are the images under projections from strata of quiver Grassmannians to ordinary Grassmannians. In contrast to the Kronecker quiver case, these polynomials do not necessarily have positive coefficients. The key ingredient is the explicit formula for noncommutative cluster variables given by Ralf Schiffler and the first author.

math.RA↗
arXiv2011

A step towards the cluster positivity conjecture

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.

math.QA↗
arXiv2011

A Combinatorial Formula for Rank 2 Cluster Variables

Let $r$ be any positive integer, and let $x_1, x_2$ be indeterminates. We consider the sequence $\{x_n\}$ defined by the recursive relation $$ x_{n+1} =(x_n^r +1)/{x_{n-1}} $$ for any integer $n$. Finding a combinatorial expression for $x_n$ as a rational function of $x_1$ and $x_2$ has been an open problem since 2001. We give a direct elementary formula for $x_n$ in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.

math.CO↗
arXiv2010

On cluster variables of rank two acyclic cluster algebras

In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix $\left(\begin{array}{cc} 0 & c -c & 0 \end{array}\right)$, and show that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincaré characteristics of the corresponding quiver Grassmannians.

math.CO↗
arXiv2010

Strong sign-coherency of certain symmetric polynomials, with application to cluster algebras

For each positive integer n, we define a polynomial in the variables z_1,...,z_n with coefficients in the ring $\mathbb{Q}[q,t,r]$ of polynomial functions of three parameters q, t, r. These polynomials naturally arise in the context of cluster algebras. We conjecture that they are symmetric polynomials in z_1,...,z_n, and that their expansions in terms of monomial, Schur, complete homogeneous, elementary and power sum symmetric polynomials are sign-coherent.

math.CO↗
arXiv2010

Virtual Private Overlays: Secure Group Commounication in NAT-Constrained Environments

Structured P2P overlays provide a framework for building distributed applications that are self-configuring, scalable, and resilient to node failures. Such systems have been successfully adopted in large-scale Internet services such as content delivery networks and file sharing; however, widespread adoption in small/medium scales has been limited due in part to security concerns and difficulty bootstrapping in NAT-constrained environments. Nonetheless, P2P systems can be designed to provide guaranteed lookup times, NAT traversal, point-to-point overlay security, and distributed data stores. In this paper we propose a novel way of creating overlays that are both secure and private and a method to bootstrap them using a public overlay. Private overlay nodes use the public overlay's distributed data store to discover each other, and the public overlay's connections to assist with NAT hole punching and as relays providing STUN and TURN NAT traversal techniques. The security framework utilizes groups, which are created and managed by users through a web based user interface. Each group acts as a Public Key Infrastructure (PKI) relying on the use of a centrally-managed web site providing an automated Certificate Authority (CA). We present a reference implementation which has been used in a P2P VPN (Virtual Private Network). To evaluate our contributions, we apply our techniques to an overlay network modeler, event-driven simulations using simulated time delays, and deployment in the PlanetLab wide-area testbed.

cs.NI↗