The Yang--Baxter equation (YBE) is a fundamental relation in mathematical physics that has its roots in the investigation of many-particle interactions in two-dimensional quantum dynamical systems, where it can be interpreted as governing integrability properties. When a system presents a boundary that interacts with the objects in motion, the Reflection Equation (RE) - as first studied by Cherednik and Sklyanin in 1984 and 1988, respectively - is a condition that ensures the persistence of integrability. To determine a uniform procedure that produces special solutions, in 1990 Drinfel'd suggested studying the set-theoretical YBE, thus paving the way for the introduction of new combinatorial tools in the subject. In recent years, Caudrelier and Zhang initiated the investigation of the set-theoretic formulation of the RE, taking inspiration from the study of soliton collisions on the half-line. The aim of this talk is to provide a self-contained overview of the set-theoretic YBE with a particular attention to the role played by self-distributive structures. Moreover, we study the interplay between bijective non-degenerate set-theoretic solutions of the YBE and their reflections, with a focus on solutions of derived type. Based on a joint work with M. Mazzotta and P. Stefanelli.
Set-theoretic solutions of the Yang-Baxter equation and their reflections Sponsored by the Meyer Fund
Tue, Nov. 4 3:30pm (MATH 3…
Topology
Ben Knudsen (Colorado State University)
X
The homology groups of the unordered configuration spaces of a graph form a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. Wishing to lift this result to an analogue of representation stability, one encounters the unavoidable fact that the corresponding algebraic structure at the level of ordered configuration spaces is non-Noetherian. Thus, finite generation results are likely to be neither useful nor within easy reach. Drawing on ideas from twisted algebra and factorization homology, we circumvent this obstacle to give a complete asymptotic calculation of the Betti numbers of pure graph braid groups over any field and, in characteristic zero, of the multiplicities of many irreducible representations of the symmetric groups. This talk represents joint work with Hainaut and Wawrykow, building on joint work with An and Drummond-Cole.
Representation asymptotics in the homology of pure graph braid groups
Set-theoretic solutions of the Yang-Baxter equation and their reflections