Noncrossing sets and a Graßmannian associahedron
We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by t...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2014-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2427/pdf |
| _version_ | 1797270290258984960 |
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| author | Francisco Santos Christian Stump Volkmar Welker |
| author_facet | Francisco Santos Christian Stump Volkmar Welker |
| author_sort | Francisco Santos |
| collection | DOAJ |
| description | We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc. |
| first_indexed | 2024-04-25T02:01:55Z |
| format | Article |
| id | doaj.art-807152eec157450ba82a9538419406cb |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:55Z |
| publishDate | 2014-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-807152eec157450ba82a9538419406cb2024-03-07T14:53:18ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502014-01-01DMTCS Proceedings vol. AT,...Proceedings10.46298/dmtcs.24272427Noncrossing sets and a Graßmannian associahedronFrancisco Santos0Christian Stump1Volkmar Welker2Departamento de Matemáticas, Estadística y ComputaciónInstitut für MathematikFachbereich Mathematik und Informatik [Marburg] [Dept. of Math and Computer Science]We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc.https://dmtcs.episciences.org/2427/pdfgraßmannianassociahedroncrossingorder polytopetriangulation[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Francisco Santos Christian Stump Volkmar Welker Noncrossing sets and a Graßmannian associahedron Discrete Mathematics & Theoretical Computer Science graßmannian associahedron crossing order polytope triangulation [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Noncrossing sets and a Graßmannian associahedron |
| title_full | Noncrossing sets and a Graßmannian associahedron |
| title_fullStr | Noncrossing sets and a Graßmannian associahedron |
| title_full_unstemmed | Noncrossing sets and a Graßmannian associahedron |
| title_short | Noncrossing sets and a Graßmannian associahedron |
| title_sort | noncrossing sets and a grassmannian associahedron |
| topic | graßmannian associahedron crossing order polytope triangulation [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/2427/pdf |
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