Mathematics Colloquia and Seminars

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 This talk delves into the intricate world of quasi-crystals, an extension of the well-established theory of crystal bases in representation theory. Originating from the groundbreaking work of Drinfeld, Jimbo, and Kashiwara in the 1980s, quantum groups and crystal bases have played a pivotal role in theoretical physics and various mathematical domains. Kashiwara's development of crystal bases and crystal graphs provided a framework for studying representations of quantum groups, revealing intriguing connections with Young tableaux and the plactic monoid.    The plactic monoid, central to the theory of symmetric polynomials and the Littlewood–Richardson rule, bridges the gap between crystal structures and Schur polynomials. In parallel, the hypoplactic monoid enters the scene in the realm of quasi-symmetric functions, offering an analogue to the classical plactic monoid with applications in quasi-ribbon tableaux.    Building on the quasi-crystal structure introduced with Alan Cain, this talk explores the extension of crystal concepts to quasi-crystals. We present a set of local axioms for quasi-crystal graphs of simply-laced root systems, drawing parallels with Stembridge's work on crystals. The characterization of quasi-crystal graphs arising from the quasi-crystal of type An answers an open question, providing insight into the local structural properties that define these structures.