Submaximal factorizations of a Coxeter element in complex reflection groups
When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2011-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2955/pdf |
| _version_ | 1797270336092241920 |
|---|---|
| author | Vivien Ripoll |
| author_facet | Vivien Ripoll |
| author_sort | Vivien Ripoll |
| collection | DOAJ |
| description | When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. |
| first_indexed | 2024-04-25T02:02:39Z |
| format | Article |
| id | doaj.art-2c5affc77bf0499d95f11820bf18992d |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:39Z |
| publishDate | 2011-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-2c5affc77bf0499d95f11820bf18992d2024-03-07T14:49:33ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502011-01-01DMTCS Proceedings vol. AO,...Proceedings10.46298/dmtcs.29552955Submaximal factorizations of a Coxeter element in complex reflection groupsVivien Ripoll0Laboratoire de combinatoire et d'informatique mathématique [Montréal]When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$.https://dmtcs.episciences.org/2955/pdfcomplex reflection groupsgeneralized noncrossing partition latticegeneralized fuss-catalan numbersfactorizations of a coxeter element[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Vivien Ripoll Submaximal factorizations of a Coxeter element in complex reflection groups Discrete Mathematics & Theoretical Computer Science complex reflection groups generalized noncrossing partition lattice generalized fuss-catalan numbers factorizations of a coxeter element [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Submaximal factorizations of a Coxeter element in complex reflection groups |
| title_full | Submaximal factorizations of a Coxeter element in complex reflection groups |
| title_fullStr | Submaximal factorizations of a Coxeter element in complex reflection groups |
| title_full_unstemmed | Submaximal factorizations of a Coxeter element in complex reflection groups |
| title_short | Submaximal factorizations of a Coxeter element in complex reflection groups |
| title_sort | submaximal factorizations of a coxeter element in complex reflection groups |
| topic | complex reflection groups generalized noncrossing partition lattice generalized fuss-catalan numbers factorizations of a coxeter element [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2955/pdf |
| work_keys_str_mv | AT vivienripoll submaximalfactorizationsofacoxeterelementincomplexreflectiongroups |