Pipe dreams, Tamari lattices, and binary trees
The theory of Schubert polynomials is an active area of research that was initiated by A. Lascoux and M.-P. Schützenberger in connection to the study of cohomology classes of Schubert cycles in flag varieties. A combinatorial understanding of Schubert polynomials was discovered by S. Fomin and A. Kirillov and N. Bergeron and S. Billey, who introduced a new family of combinatorial objects, called RC-graphs at that point, in order to compute them. These combinatorial objects were revisited in the context of Gröbner geometry of Schubert varieties by A. Knutson and E. Miller, who coined the name (reduced) pipe dreams. Since then, pipe dreams have been studied from various perspectives and connections to diverse fields have been discovered.
One remarkable connection of importance for this proposal is the relation between pipe dreams and several generalizations of the associahedron. Two examples from our work are:
- The multiassociahedron: a simplicial complex whose facets encode multi-triangulations.
- The \(\nu\)-Associahedron: a polytopal complex induced by an arrangement of tropical hyperplanes, whose vertices encode a family of binary trees called \(\nu\)-trees.
Furthermore, pipe dream complexes are special cases of the subword complexes introduced by A. Knutson and E. Miller in, which are certain simplicial complexes that encode the combinatorics of reduced expressions of an element in a word in the context of Coxeter groups. In previous work, we showed that the generalized associahedra arising from the theory of cluster algebras can be described in terms of subword complexes. This work has lead to further investigations in connection to various fields such as brick polytopes, toric varieties, Hopf algebras, cluster algebras, and their g-vector fans, among others.
Pipe dreams and subword complexes thus provide a solid framework for the study of associahedra and their generalizations, which benefits from the well established theory of Coxeter groups and opens further directions of research in connection to other fields. Our general objective in this part of the proposal is:
Objective 1
Study combinatorial and geometric properties of special families of pipe dreams and subword complexes, which will help us to settle some specific open problems in this field.
