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Preprint
Monomial identities in the Weyl algebra
arXiv.org
30 May 2024
Abstract
Motivated by a question and some enumerative conjectures of Richard Stanley,
we explore the equivalence classes of words in the Weyl algebra, $\mathbf{k}
\langle D,U\rangle/(DU-UD=1)$. 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.
Armed with this we deduce a number of enumerative results about the number of
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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Details
- Title
- Monomial identities in the Weyl algebra
- Creators
- Darij GrinbergTom RobyStephan WagnerMei Yin
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021881382604721