Monomial identities in the Weyl algebra

TL;DR

This paper studies the structure and enumeration of equivalence classes of words in the Weyl algebra, providing algorithms and extending results to related combinatorial objects and algebraic structures.

Contribution

It introduces a linear-time algorithm for checking equivalence of words, characterizes classes via balanced subwords, and extends results to c-Dyck words and other algebras.

Findings

Equivalence classes generated by swapping balanced subwords.

Linear-time algorithm for equivalence checking.

Enumerative results for class sizes and extensions to c-Dyck words.

Abstract

Motivated by a question and some enumerative conjectures of Richard Stanley, we explore the equivalence classes of words in the Weyl algebra, $\mathbf{k} \left< D,U \mid DU - UD = 1 \right>$. We show that each class is generated by the swapping of adjacent *balanced subwords*, i.e., those which have the same number of $D$'s as $U$'s, and give several other characterizations, as well as a linear-time algorithm for equivalence checking. Armed with this, we deduce several enumerative results about such equivalence classes and their sizes. We extend these results to the class of $c$-Dyck words, where every prefix has at least $c$ times as many $U$'s as $D$'s. We also connect these results to previous work on bond percolation and rook theory, and generalize them to some other algebras.