Related papers: From i-boxes to signed words
The implicit signature k consists of the multiplication and the ({\omega}-1)-power. We describe a procedure to transform each {\kappa}-term over a finite alphabet A into a certain canonical form and show that different canonical forms have…
We describe a translation from a fragment of SUMO (SUMO-K) into higher-order set theory. The translation provides a formal semantics for portions of SUMO which are beyond first-order and which have previously only had an informal…
This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs (Zelevinsky),…
Generalising a recent work of Dequ\^ene et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is…
We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…
For a given marked surface $(S,M)$ and a fixed tagged triangulation $T$ of $(S,M)$, we show that each tagged triangulation $T'$ of $(S,M)$ is uniquely determined by the intersection numbers of tagged arcs of $T$ and tagged arcs of $T'$. As…
Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of…
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the…
Card-based cryptography is a research area to implement cryptographic procedures using a deck of physical cards. In recent years, it has been found to be related to finite group theory and algebraic combinatorics, and is becoming more and…
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer…
In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$,…
Let U be the quantum group associated to a Lie algebra of type A_n. The negative part U^- of U has a canonical basis B defined by Lusztig and Kashiwara, with favourable properties. We show how the spanning vectors of the cones defined by…
We establish a bijective correspondence between certain non-self-intersecting curves in an $n$-punctured disc and positive ${\mathbf c}$-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices.…
We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by…
It is known how to compute the Zariski closure of a finitely generated monoid of matrices and, more generally, of a set of matrices specified by a regular language. This result was recently used to give a procedure to compute all polynomial…
First, we prove the Kac-Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian…
Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its…
We introduce two new algebras that we call \emph{tied--boxed Hecke algebra} and \emph{tied--boxed Temperley--Lieb algebra}. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second…