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Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element
When W is a finite reflection group, the noncrossing partition lattice of type W is a 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 as a gen...
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| Published in: | Journal of algebraic combinatorics 2012-12, Vol.36 (4), p.649-673 |
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| Language: | English |
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| container_end_page | 673 |
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| container_title | Journal of algebraic combinatorics |
| container_volume | 36 |
| creator | Ripoll, Vivien |
| description | When
W
is a finite reflection group, the noncrossing partition lattice
of type
W
is a 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
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
as fibers of a Lyashko–Looijenga covering (
), constructed from the geometry of the discriminant hypersurface of
W
. We study algebraically the map
, describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of
W
. |
| doi_str_mv | 10.1007/s10801-012-0354-4 |
| format | article |
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W
is a finite reflection group, the noncrossing partition lattice
of type
W
is a 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
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
as fibers of a Lyashko–Looijenga covering (
), constructed from the geometry of the discriminant hypersurface of
W
. We study algebraically the map
, describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of
W
.</description><identifier>ISSN: 0925-9899</identifier><identifier>EISSN: 1572-9192</identifier><identifier>DOI: 10.1007/s10801-012-0354-4</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Combinatorics ; Computer Science ; Convex and Discrete Geometry ; Group Theory and Generalizations ; Lattices ; Mathematics ; Mathematics and Statistics ; Order ; Ordered Algebraic Structures</subject><ispartof>Journal of algebraic combinatorics, 2012-12, Vol.36 (4), p.649-673</ispartof><rights>Springer Science+Business Media, LLC 2012</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c283t-5147e4e25e1b8d02746428e426e171cb8db59d7c158cba39551932549e337b703</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Ripoll, Vivien</creatorcontrib><title>Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element</title><title>Journal of algebraic combinatorics</title><addtitle>J Algebr Comb</addtitle><description>When
W
is a finite reflection group, the noncrossing partition lattice
of type
W
is a 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
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
as fibers of a Lyashko–Looijenga covering (
), constructed from the geometry of the discriminant hypersurface of
W
. We study algebraically the map
, describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of
W
.</description><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Convex and Discrete Geometry</subject><subject>Group Theory and Generalizations</subject><subject>Lattices</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><issn>0925-9899</issn><issn>1572-9192</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KAzEUhYMoWKsP4C4vEM3NT5MspfgHA7rQdchM77RTO5OSTKF15Tv4hj6JU8a1qwOX810OHyHXwG-Ac3ObgVsOjINgXGrF1AmZgDaCOXDilEy4E5o569w5uch5zTl3FvSEvBaHkFcf8efru4ixWWO3DLSNabtqcptp6BY078o27Js2bGgdqj6m5jP0TewyjTUNdB732GOiuMEWu_6SnNVhk_HqL6fk_eH-bf7EipfH5_ldwSphZc80KIMKhUYo7YILo2ZKWFRihmCgGm6ldgtTgbZVGaTTGpwUWjmU0pSGyymB8W-VYs4Ja79Nw8Z08MD9UYkflfhBiT8q8WpgxMjkodstMfl13KVumPkP9AsLzWUb</recordid><startdate>20121201</startdate><enddate>20121201</enddate><creator>Ripoll, Vivien</creator><general>Springer US</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20121201</creationdate><title>Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element</title><author>Ripoll, Vivien</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c283t-5147e4e25e1b8d02746428e426e171cb8db59d7c158cba39551932549e337b703</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Combinatorics</topic><topic>Computer Science</topic><topic>Convex and Discrete Geometry</topic><topic>Group Theory and Generalizations</topic><topic>Lattices</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ripoll, Vivien</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of algebraic combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ripoll, Vivien</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element</atitle><jtitle>Journal of algebraic combinatorics</jtitle><stitle>J Algebr Comb</stitle><date>2012-12-01</date><risdate>2012</risdate><volume>36</volume><issue>4</issue><spage>649</spage><epage>673</epage><pages>649-673</pages><issn>0925-9899</issn><eissn>1572-9192</eissn><abstract>When
W
is a finite reflection group, the noncrossing partition lattice
of type
W
is a 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
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
as fibers of a Lyashko–Looijenga covering (
), constructed from the geometry of the discriminant hypersurface of
W
. We study algebraically the map
, describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of
W
.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10801-012-0354-4</doi><tpages>25</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0925-9899 |
| ispartof | Journal of algebraic combinatorics, 2012-12, Vol.36 (4), p.649-673 |
| issn | 0925-9899 1572-9192 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s10801_012_0354_4 |
| source | Springer Nature |
| subjects | Combinatorics Computer Science Convex and Discrete Geometry Group Theory and Generalizations Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures |
| title | Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element |
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