Related papers: Bowling ball representations of braid groups
We consider a mathematical model of the breakout game where $\mathbb{R}^2$ is covered by unit square bricks everywhere except in one place. We introduce elementary blocks (particular sets of bricks) in order to describe a family of periodic…
We define a commuting family of operators $T_0,T_1,...,T_n$ in the Temperley--Lieb algebra $\mathcal{A}_n(x)$ of type $A_{n-1}$. Using an appropriate analogue to Murphy basis of the Iwahori--Hecke algebra of the symmetric group, we describe…
The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and…
Let $\Omega$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary…
We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties…
We briefly review the computational strategy we have recently introduced for computing glueball masses and matrix elements, which achieves an exponential reduction of statistical errors compared to standard techniques. The global symmetries…
The braid group $B_4$ naturally acts on the rational projective plane $\mathbb{P}^2(\mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of…
We discuss in this paper combinatorial aspects of boundary loop models, that is models of self-avoiding loops on a strip where loops get different weights depending on whether they touch the left, the right, both or no boundary. These…
We settle a question of Bressoud concerning the existence of an explicit bijection from a class of oriented square-ice graphs to a class of tournaments. We give an algorithm constructing such a bijection.
We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented)…
Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. In this paper we extend the techniques to also include projective representations. As our main application we…
While much is known about the faithfulness of the Burau representation, the problem remains open for the Gassner representation for every $B_n$ with $n\geq 4$. We first find the definition of the Colored-Burau representation of Ainshel,…
With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of…
Let $G$ be a connected reductive algebraic group over a field of positive characteristic $p$ and denote by $\mathcal T$ the category of tilting modules for $G$. The higher Jones algebras are the endomorphism algebras of objects in the…
We present a unified framework for representing commutative rings through affine algebraic theories and Boolean rings through hyperaffine algebraic theories. This yields categorical equivalences between these theories and, respectively,…
Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of…
A new type of algebras that represent a generalization of both quantum groups and braided groups is defined. These algebras are given by a pair of solutions of the Yang--Baxter equation that satisfy some additional conditions. Several…
The OU matrix of a braid diagram is a square matrix that represents the number of over/under crossings of each pair of strands. In this paper, the OU matrix of a pure braid diagram is characterized for up to 5 strands. As an application,…
Bonded knots arise naturally in topological protein modeling, where intramolecular interactions such as disulfide bridges stabilize folded configurations. These structures extend classical knot theory by incorporating embedded graphs, and…
We introduce a family of automorphisms on the bosonic extension of arbitrary type and show that they satisfy the braid relations. They preserve the global basis and the crystal basis. Using this braid group action, we define a subalgebra…