相关论文: Combinatorial interpretations for rank-two cluster…
We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with…
That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of…
We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We…
We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\'ery-like recurrence relation: these include Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and…
We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in $\mathbb Q[x]$, and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.
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…
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…
We generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials.
In 1983, Feingold and Frenkel discovered a relation between Siegel modular forms of genus two and a rank-three hyperbolic Kac--Moody algebra extending the affine Lie algebra of type $A_1$. It inspires a problem to explore more general…
Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a {\em frieze};…
Using certain results for the vertex operator algebras associated with affine Lie algebras we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we…
We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for…
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure…
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…
This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure…
We completely determine the structure constants between real root vectors in a rank 2 Kac--Moody algebra $\mathfrak{g}$. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the…
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…
We review the notion of (finitary) filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with…
Two doubly indexed families of polynomials in several indeterminates are considered. They are related to the falling and rising factorials in a similar way as the potential polynomials (introduced by L. Comtet) are related to the ordinary…
This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the…