A Proof of Rubey's Lattice Conjecture
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912641920794624 |
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| author | Billey, Sara C. McCausland, Connor Minnerath, Clare |
| author_facet | Billey, Sara C. McCausland, Connor Minnerath, Clare |
| contents | In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation $w$ and conjectured that the induced poset on reduced pipe dreams is a lattice. In this paper, we prove this conjecture. Our key tool is a new type of move operation $\mathcal{M}_{ij}$, defined as a composite of certain general ladder moves in Rubey's poset. We show that joins and meets exist in Rubey's poset by proving simple recursive formulas in terms of $\mathcal{M}_{ij}$ operations. In addition, we give an explicit criterion to determine if two elements of Rubey's poset are comparable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18852 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Proof of Rubey's Lattice Conjecture Billey, Sara C. McCausland, Connor Minnerath, Clare Combinatorics 05A05, 06A07 In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation $w$ and conjectured that the induced poset on reduced pipe dreams is a lattice. In this paper, we prove this conjecture. Our key tool is a new type of move operation $\mathcal{M}_{ij}$, defined as a composite of certain general ladder moves in Rubey's poset. We show that joins and meets exist in Rubey's poset by proving simple recursive formulas in terms of $\mathcal{M}_{ij}$ operations. In addition, we give an explicit criterion to determine if two elements of Rubey's poset are comparable. |
| title | A Proof of Rubey's Lattice Conjecture |
| topic | Combinatorics 05A05, 06A07 |
| url | https://arxiv.org/abs/2507.18852 |