Formula calculates higher moments of Siegel-Veech transform over Hecke triangle groups.
problem Computing higher moments of Siegel-Veech transform over specific groups.
method Geometric results and linear algebra to create integration formulas.
result Explicit integration formulas for densities of vector orbits.
Introduces a new algebra from loop braid groups with properties similar to Hecke algebras.
problem Developing new algebraic structures inspired by loop braid groups.
method Introducing a generalization of Hecke algebras from loop braid groups and proving properties of the quotient algebra.
result The quotient algebra SPn has a structure independent of the parameter t except for specific cases. Solves a problem posed by Jones linking Hecke groups to subfactor indices.
problem Relation between Hecke groups and subfactor indices in von Neumann algebras.
method Using cluster C*-algebras.
result Solved the problem posed by V. F. R. Jones.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups Γp for p≥3. result Generalizes Sarnak's result on reciprocal elements in the modular group.
The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of S3, and one is led …
Hecke's theorem on different generalized to 3-manifolds.
problem Understanding the arithmetic of branched covers of 3-manifolds.
method Analog of Hecke's theorem for 3-manifolds.
result Branch divisor is a square in the first homology group.
Researchers create projective representations of Hecke groups using TQFT.
problem Constructing projective representations of Hecke groups.
method Using Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces.
result The representation's image group is infinite at low levels in genus 2.
The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.
problem Understanding centralisers and Hecke algebras in representation theory.
method Review of classical and quantum Schur-Weyl duality, discussion of quantum groups and their centralisers, and application to knot theory and physics.
result New insights into centralisers and Hecke algebras, leading to solutions for the Yang-Baxter equation and link invariants.
Maps from buildings to spaces study K-theory of Hecke algebras.
problem Understanding K-theory of Hecke algebras.
method Constructing maps from buildings to related spaces.
result Maps help in studying K-theory of Hecke algebras.
We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.
Fast algorithm for braid group Hecke representation, applied to knot invariants.
problem Computing topological invariants of knots efficiently.
method Representation-theoretic approach to braid group, leveraging quantum topology.
result Fast algorithm for Hecke representation of braid group, finding non-trivial braids.
The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.
problem Analyzing the growth of reciprocal classes in Hecke groups.
method Utilizes the free product structure of Hecke groups, combinatorial counting, and recurrence relations.
result Proves an asymptotic formula for the number of reciprocal classes in Hecke groups.
New basis and Schur-Weyl duality for loop Hecke algebra defined.
problem Define a new basis for the loop Hecke algebra.
method Use higher linear rewriting theory and combinatorics of Dyck paths.
result Yields a conjecture of Damiani-Martin-Rowell and provides a representation theoretic interpretation.
We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the p--adic framed braids and the p--adic Yokonuma--Hecke algebras in…
Defines braids with double lines for links in a surface times circle and connects it to the affine Hecke algebra.
problem Presenting links in a surface times circle using braids with double lines.
method Defines braids with double lines, proves Alexander and Markov theorems, and connects Hecke algebra to affine Hecke algebra.
result The Hecke algebra of braids with double lines is isomorphic to the affine Hecke algebra.
We prove that the completed cohomology groups of SL_N(Z) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable cohomology is trivial, in a precisely defined sense.
We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of…
In this paper we study the kernel of the homomorphism Bg,n→Bn of the braid group Bg,n in the handlebody Hg to the braid group Bn. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra Hg,n(q), which is some analog of the Hecke algebra $H_n(q…
New geometric model for knot homology using monodromic Hecke category.
problem Geometric description of Khovanov-Rozansky knot homology.
method Monodromic model based on Soergel bimodules and Hecke category.
result Geometric description of Khovanov-Rozansky knot homology.
Representations of the Iwahori-Hecke algebra of type A_{n-1} are equivalent to representations of the braid group B_n for which the generators satisfy a certain quadratic relation. We show how to construct such representations from the natural action of B_n on the homology of configuration spaces of the punctured disk.…
New method shows any triangle group generating pair is related to special coverings.
problem Understanding generating pairs of triangle groups.
method Special almost orbifold coverings.
result Any generating pair of a triangle group is represented by a special covering.
Explains fusion for Yang-Baxter equation and braid group.
problem Understanding algebras related to braid group, Yang-Baxter, and quantum groups.
method Introduces fusion procedure for Yang-Baxter equation.
result New examples of algebras: fused Hecke algebras.
We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebra…
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
Formula for Lefschetz numbers on locally symmetric spaces.
problem Calculating Lefschetz numbers on locally symmetric spaces.
method Introduced a new group C∗-algebra and used Hecke correspondences to study Lefschetz numbers. result Explicit formula for Lefschetz numbers of Hecke operators.
We prove that the quotients of the group algebra of the braid group introduced by L. Funar in Comm. Math. Phys., 1995, collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in ad…
In this paper we classify all Markov traces on Iwahori-Hecke algebras associated with the finite Coxeter groups of type B. We then use these traces for constructing Jones-type invariants for oriented knots inside a solid torus, and finally we give skein-theoretical interpretations.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…
Triangle Artin groups split as graphs of free groups under specific conditions.
problem Conditions for triangle Artin groups to split as graphs of free groups.
method Graph of free groups analysis and poly-free property proof.
result Triangle Artin groups split as graphs of free groups if and only if labels are greater than 5 and even.
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
Classifies complex hyperbolic triangle groups by types.
problem Classifying complex hyperbolic triangle groups.
method By types defined by the ellipticity of two short words.
result Improves Schwartz conjecture.
A formula for Rademacher symbols in triangle groups is provided.
problem No specific problem stated; focuses on a mathematical formula.
method Presentation of an explicit formula for Rademacher symbols.
result Generalizes Ghys' proof of modular knot linking numbers.
Triangle groups show rigidity in hyperbolic spaces.
problem Local rigidity of triangle groups generated by reflections.
method Geometric representation and diagonal embeddings in PGL(2,R) and PSp±(2n,R). result Triangle groups are locally rigid in hyperbolic spaces.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
A mathematical isomorphism connects Floer homology to DAHA representations.
problem Connecting Floer homology to DAHA representations.
method Isomorphism between Floer homology and DAHA module structure.
result Establishes equivalence between DAHA polynomial representation and Floer homology module structure.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT Study on discrete complex hyperbolic triangle groups of specific type.
problem Discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2]. method Analysis of groups generated by complex reflections with specific orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadrat…
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
Study framizations of algebras using Schur--Weyl duality and tied braids.
problem Understanding framizations of algebras and their connections to quantum groups.
method Developing a general setting for framizations of algebras, including Yokonuma--Hecke and tied braids.
result Obtained Schur--Weyl duality for various algebras, including new framizations.
We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the c…
We define the singular Hecke algebra H(SBn) as the quotient of the singular braid monoid algebra C(q)[SBn] by the Hecke relations σk2=(q−1)σk+q, 1≤k≤n−1, and define the Markov traces on the sequence {H(SBn)}n=1+∞ in the same way as for the Marko…