Brick polytopes, lattices and Hopf algebras
Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and w...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6401/pdf |
| _version_ | 1797270221716717568 |
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| author | Vincent Pilaud |
| author_facet | Vincent Pilaud |
| author_sort | Vincent Pilaud |
| collection | DOAJ |
| description | Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope |
| first_indexed | 2024-04-25T02:00:50Z |
| format | Article |
| id | doaj.art-9ef45b1f4a774c59921bebbb3d4bfde8 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:50Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-9ef45b1f4a774c59921bebbb3d4bfde82024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.64016401Brick polytopes, lattices and Hopf algebrasVincent Pilaud0https://orcid.org/0000-0002-2070-9223Laboratoire d'informatique de l'École polytechnique [Palaiseau]Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytopehttps://dmtcs.episciences.org/6401/pdfcombinatorics[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Vincent Pilaud Brick polytopes, lattices and Hopf algebras Discrete Mathematics & Theoretical Computer Science combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Brick polytopes, lattices and Hopf algebras |
| title_full | Brick polytopes, lattices and Hopf algebras |
| title_fullStr | Brick polytopes, lattices and Hopf algebras |
| title_full_unstemmed | Brick polytopes, lattices and Hopf algebras |
| title_short | Brick polytopes, lattices and Hopf algebras |
| title_sort | brick polytopes lattices and hopf algebras |
| topic | combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6401/pdf |
| work_keys_str_mv | AT vincentpilaud brickpolytopeslatticesandhopfalgebras |