PC Seminar: Monomial identities in the Weyl algebra

Abstract: The Weyl algebra has two generators D and U and the defining relation DU - UD = 1. One possible interpretation is that D is the differentiation operator, and U is multiplication by x, both acting on polynomials or power series in x.

It is possible that distinct monomials in the Weyl algebra are equal, for example UDDU = DUUD. This caused Richard Stanley to put forward several questions and conjectures about equivalence classes of monomials. In this talk, the solutions to these questions will be discussed: specifically, a combinatorial characterization of equivalence classes of monomials and several enumerative results on these equivalence classes.

Joint work with Darij Grinberg, Tom Roby, and Mei Yin.

This is a seminar in our seminar series on Probability and Combinatorics (PC).