Monomial Identities in the Weyl Algebra - A Combinatorial and Computational Approach

Explore a mathematical lecture that delves into the intricacies of monomial identities in the Weyl algebra, also known as the Heisenberg-Weyl algebra. Learn about the fundamental properties of this free algebra with two generators D and U, connected by the relation DU - UD = 1. Discover how this relation leads to equivalent monomials, such as DU U D and U DDU, and understand the various ways to characterize these equalities in a field of characteristic 0. Examine the operational approach through combinatorial equivalence relations, computational methods using lattice path invariants, and connections to rook theory. Study the enumeration of equivalence classes and their variants, while considering potential extensions to other algebras. This collaborative research presentation by Darij Grinberg from Drexel University, conducted with Tom Roby, Stephan Wagner, and Mei Yin, was inspired by Richard P. Stanley's question.