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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| Vydané v: | Discrete mathematics and theoretical computer science Ročník DMTCS Proceedings, 28th... |
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| Hlavný autor: | |
| Médium: | Journal Article Konferenčný príspevok.. |
| Jazyk: | English |
| Vydavateľské údaje: |
DMTCS
22.04.2020
Discrete Mathematics & Theoretical Computer Science |
| Predmet: | |
| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
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| Shrnutí: | 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 |
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| ISSN: | 1365-8050 1462-7264 1365-8050 |
| DOI: | 10.46298/dmtcs.6401 |