Monomial identities in the Weyl algebra
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 whic...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.2405.20492 |