Abstract: We construct distributive lattices refining the weak order on permutations and refined by the Bruhat order, generalizing the middle order defined by M. Bouvel, L. Ferrari and B. E. Tenner. These lattices, which we also call middle orders, are defined using a direct bijection between permutations and lower sets of a certain poset. We study combinatorial properties of these lattices, and show they are the only distributive lattices between the weak and Bruhat orders. We also consider generalizations of middle orders to other finite Coxeter groups.

Abstract: We consider quotients of the lattice of ASMs which are isomorphic to the lattice of Dyck paths. Using the geometric structures of their irreducibles posets, we show these quotients define basis of the Temperley-Lieb algebra, generalizing a recent result of Nantel Bergeron and Lucas Gagnon. We introduce a family of Gelfand-Tsetlin triangles encoding these quotients, which can be ordered to obtain a lattice very similar to the lattice of ASMs. Joint work with Florent Hivert and Vincent Pilaud.

Abstract: Gog and Magog triangles are simple combinatorial objects which are equienumerated. Howewer, the problem of finding an explicit bijection between these has been an open problem since the 80’s. These are related to other interesting objects such as alternating sign matrices, plane partitions or aztec diamond tillings. All these objects can be ordered in such a way that the obtained posets are distributive lattices. We will present Gog and Magog triangles under a lattice-theoretic point of view, giving new explanations of the link between alternating sign matrices and aztec diamond tillings, or between the lattice of Gog triangles, the Bruhat and weak orders on permutations.