Algebraic Theory of Quasi-crystals: A Generalization of the Hypoplactic Monoid and a Littelmann Path Model - Guilherme, Ricardo Jorge Pratas (2022)
The plactic monoid was first introduced by Lascoux and Schützenberger based on work by Schensted and Knuth, resulting in a description of the plactic monoid by a presentation and by Young tableaux and an insertion algorithm. Then, due to work by Kashiwara on crystal bases, the plactic monoid was obtained by identifying vertices in the crystal graph associated to the general linear Lie algebra. This process proved to be fruitful, as it enriched the structure of the already acclaimed plactic monoid and allowed its generalization based on other crystal graphs. Analogously, the hypoplactic monoid was introduced by Krob and Thibon via a presentation and via quasi-ribbon tableaux and an insertion algorithm. Recent work by Cain and Malheiro showed a construction of the hypoplactic monoid by identifying vertices in a quasi-crystal graph derived from the crystal graph associated to the general linear Lie algebra. Although this construction is based on Kashiwara’s work, it cannot be extended to other crystal graphs, and it lacks connections to representation theory such as a Littelmann path model. In this thesis we present an algebraic theory of quasi-crystals. We associate a quasi-crystal graph to each quasi-crystal and describe a one-to-one correspondence between seminormal quasi-crystals and quasi-crystal graphs. We show that a hypoplactic congruence can be defined over any seminormal quasi-crystal, from which a general notion of hypoplactic monoid emerges. We prove that the construction of the hypoplactic monoid proposed by Cain and Malheiro can be placed in context as the hypoplactic monoid associated with the general linear Lie algebra. Based on this framework, we make a study of the hypoplactic monoid associated to the symplectic Lie algebra. Finally, We present a Littelmann path model for quasi-crystals.