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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.20492 |
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| _version_ | 1866929600006717440 |
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| author | Grinberg, Darij Roby, Tom Wagner, Stephan Yin, Mei |
| author_facet | Grinberg, Darij Roby, Tom Wagner, Stephan Yin, Mei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monomial identities in the Weyl algebra Grinberg, Darij Roby, Tom Wagner, Stephan Yin, Mei Combinatorics Rings and Algebras 12H05, 16S32, 05A15, 68R15 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. |
| title | Monomial identities in the Weyl algebra |
| topic | Combinatorics Rings and Algebras 12H05, 16S32, 05A15, 68R15 |
| url | https://arxiv.org/abs/2405.20492 |