![]() Although this definition is quite simple, there are very few results known for Artin groups in general. María Cumplido Cabello, University of SevilleĪrtin (or Artin-Tits) groups are generalizations of braid groups that are defined using a finite set of generators $S$ and relations $abab\cdots=baba\cdots$, where both words of the equality have the same length.""twisted"" setting as introduced by Dyer, and asks the question ofĭetermining which braids have a minimal braid complex which is perverse. On the perversity of minimal Rouquier complexes of positive simple braids to a Shown for finite Weyl groups by Dyer and Lehrer, can be generalized toĪrbitrary Coxeter systems by adapting a result of Elias and Williamson The positivity of the KL expansion of Mikado braids, Kazhdan-Lustig expansion, together with the fact that simple dual braidsĪre Mikado braids. Lusztig's inverse positivity, which predicts that certain elements ofĪrtin-Tits groups, which we call ""Mikado braids"", have a positive Obtained in spherical type using a generalization of Kazhdan and ![]() We will explain how positivity of images of simple dual braids can be This is especially interesting in type A, as simple dualīraids yield a basis of the Temperley-Lieb quotient of the Hecke Images of these elements in the Hecke algebra still have a positive KLĮxpansion or not. Introduced for spherical type Artin-Tits groups. AnĪlternative Garside structure, called dual Garside structure, was So-called simple elements of the classical Garside structure. Positive lifts of the elements of the Coxeter group in the Artin-Tits group are the In the case where the Coxeter group is finite, the Elements of the standardīasis have a positive expansion in one of Kazhdan and Lusztig'sĬanonical bases, i.e., have coefficients which are Laurent polynomials with nonnegative coefficients. Then their images in the HeckeĪlgebra yield the so-called standard basis of the Hecke algebra. Consider the positive lifts of the elements of theĬoxeter group in the Artin-Tits group. Symmetric group, or more generally from any Artin-Tits group to theĬorresponding Hecke algebra. Group of invertible elements of the) Iwahori-Hecke algebra of the There is a well-known homomorphism from Artin's braid group to (the In their proof, Carlsson and Mellit introduce a new interesting algebra denoted $A_$ has a solvable word problem and finite virtual cohomological dimension when $\Gamma$ is of spherical type or of affine type. The shuffle theorem gives a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. Anthony Licata, Australian National University. ![]() The overarching goals of the program are to establish and clarify the key questions driving each field, and to improve each group’s understanding of the tools, techniques, and perspectives of the others. ![]() The proposed semester program aims to bring together researchers working in diverse areas through the common thread of their interaction with braid and mapping class groups. For example, developing fast machine-based calculations of link homology invariants is a goal shared by representation theorists, low-dimensional topologists, symplectic and algebraic geometers, and string theorists. For example, in modern representation theory, important equivalences of categories are organized into 2-representations of braid groups, and these same 2-representations appear prominently in parts of geometry and mathematical physics concerned with mirror dualities in low-dimensional topology, manifolds are presented and related to each other via braids and mapping classes.Ĭomputational applications and questions about braid groups have also emerged in disparate mathematical contexts in some cases, these coalesce around the same computational problem. Braid and mapping class groups are prominent players in current mathematics not only because these groups are rich objects of study in their own right, but also because they provide organizing structures for a variety of different areas. In the last 15 years, fields with an interest in braids have independently undergone rapid development these fields include representation theory, low-dimensional topology, complex and symplectic geometry, and geometric group theory. Since then, braid groups, mapping class groups, and their generalizations have come to occupy a significant place in parts of both pure and applied mathematics. Braid groups were introduced by Emil Artin almost a century ago.
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