Combinatorics, Representations, and Catalan Theory

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Symmetric function theory is an area of mathematical research which has produced an abundance of intricate (and sometimes conjectural) identities, many of which are very difficult to prove. Symmetric functions have ties to geometry, representation theory, and physics, and there is a deep interplay between these areas: symmetric functions give a computational model for difficult matters in these other fields, and symmetric function identities are often best understood as shadows of deeper concepts from other areas. This project studies the algebraic and geometric underpinnings of a symmetric function problem called the Delta Conjecture. The project provides research training opportunities for undergraduate and graduate students.Quotients of the polynomial ring in n variables have classical ties to the combinatorics of permutations and the geometry of the flag variety. This project studies analogous quotients (and subspaces) of a newer ring called 'superspace' which has n commuting generators (corresponding to bosons) and n anticommuting generators (corresponding to fermions). These new algebraic objects are linked to the combinatorics of ordered set partitions (as studied by Haglund, Shimozono, and the PI) and the geometry of a variety of spanning line configurations introduced by Pawlowski and the PI. A deep conjecture of Zabrocki, and a related conjecture of Wilson and the PI, assert that (a variant of) superspace gives a representation-theoretic model for the Delta Conjecture. The PI will use the algebra and geometry of superspace to tie other areas of mathematics to the Delta Conjecture, which could lead to an illuminating proof thereof.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.