Related papers: Combinatorial aspects of boundary loop models

A class of deformations of the q-quantum gravity loop algebra is shown to be incompatible with the combinatorics of Temperley-Lieb recoupling theory with deformation parameter at a root of unity. This incompatibility appears to extend to…

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these…

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as…

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the…

We diagonalise the Hamiltonian of the Temperley-Lieb loop model with open boundaries using a coordinate Bethe Ansatz calculation. We find that in the groundstate sector of the loop Hamiltonian, but not in other sectors, a certain constraint…

We study the holomorphic twist of 3d ${\cal N}=2$ gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral…

Round-based models are very common message-passing models; combinatorial topology applied to distributed computing provides sweeping results like general lower bounds. We combine both to study the computability of k-set agreement. Among all…

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…

We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for all of the scaling operators exist. We demonstrate the role played by the…

This thesis is concerned with aspects of the integrable Temperley--Lieb loop (TL($n$)) model on a vertically infinite lattice with two non-trivial boundaries. When $n=1$ the ground state eigenvector of the transfer matrix of this model can…

We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The…

Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra $\Lambda$ is left or right glued if and only if $\Lambda$ is…

We investigate a class of combinatory algebras, called ribbon combinatory algebras, in which we can interpret both the braided untyped linear lambda calculus and framed oriented tangles. Any reflexive object in a ribbon category gives rise…

We consider GL$_m$-dimers of triangulations of regular convex $n$-gons, which give rise to a dimer model with boundary $Q$ and a dimer algebra $\Lambda_Q$. Let $e_b$ be the sum of the idempotents of all the boundary vertices, and…

In this paper we continue the investigation of partition functions of critical systems on a rectangle initiated in [R. Bondesan et al, Nucl.Phys.B862:553-575,2012]. Here we develop a general formalism of rectangle boundary states using…

We propose a systematic method to extract conformal loop models for rational conformal field theories (CFT). Method is based on defining an ADE model for boundary primary operators by using the fusion matrices of these operators as…

We study the boundary-localized Lie algebra generated by the rank-one perturbation \(T = U + \varepsilon E\) of the unilateral shift on \(\ell^2(\mathbb{Z}_{\ge\ 0})\). While the polynomial algebra \(\langle T \rangle\) is abelian, the…

This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…

We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…

We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a…