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arXiv · 2405.20492

Monomial identities in the Weyl algebra

Darij Grinberg·Tom Roby·Stephan Wagner·Mei Yin
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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.

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Darij Grinberg, Tom Roby, Stephan Wagner, Mei Yin. 2024-05-30. Monomial identities in the Weyl algebra. https://arxiv.org/abs/2405.20492

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