Combinatorial Objects and Actions in Higher Dimensional Settings

This project is jointly funded by the Combinatorics program and the Established Program to Stimulate Competitive Research (EPSCoR). Dynamics is the study of how systems change and is fundamental to mathematics and applications, while combinatorics explores beautiful mathematical objects composed of discrete structures. Often, combinatorics and dynamics provide insight into the structure of mathematical and physical objects, revealing hidden symmetries and connections. There has been much success in the discovery of mathematical objects and actions that display extremely elegant properties, but many of these objects are two-dimensional. The theme of this proposal is to advance such findings in higher dimensional realms of mathematics and statistical physics. This study will also include mentoring students and early career researchers, especially those of underrepresented groups in mathematics. This work is comprised of several projects involving intricate bijections and actions, building upon results in dynamical algebraic combinatorics and symmetric functions. One project uses a new bijection between certain types of labelled posets that correlates the actions of promotion and rowmotion. Another involves alternating sign matrices, which are multidimensional Catalan analogues related to the square ice model of statistical physics. It has been a long-standing open problem to construct an explicit bijection between these and certain plane partitions. This project aims to make progress on this problem by interpreting these objects as different types of pipe dreams, objects with connections to Schubert polynomials. Finally, web bases for irreducible representations of symmetric groups have applications to quantum link invariants, cluster algebras, and positive geometries. This work searches for such bases in higher dimensional cases, with surprising connections to alternating sign matrices. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.