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: We define affine alternating sign matrices as an infinite and periodic version of classical alternating sign matrices. We describe a natural distributive lattice structure on affine ASMs, containing the Bruhat order on affine permutations as a subposet. For period 1, this lattice coincides with the Young lattice. For general period n, we give a bijection between affine ASMs and tuples of integer partitions satisfying cyclic compatibility conditions. Comparison in the affine ASM lattice translates to componentwise containment of the corresponding integer partitions, which generalizes the description of the Bruhat order on affine permutations given by Rostam. We also describe an affine analog of tilings of the Aztec diamond, which, as in the finite case, are strongly related to affine ASMs. Joint work with Clément Chenevière.

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.