A minimaj-preserving crystal on ordered multiset partitions

Georgia Benkart; Laura Colmenarejo; Pamela E. Harris; Rosa Orellana; Greta Panova; Anne Schilling; Martha Yip

doi:10.1016/j.aam.2017.11.006

A minimaj-preserving crystal on ordered multiset partitions

Georgia Benkart, Laura Colmenarejo, Pamela E. Harris, Rosa Orellana, Greta Panova, Anne Schilling, Martha Yip

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization R_n,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra R_n. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Original language	English
Pages (from-to)	96-115
Number of pages	20
Journal	Advances in Applied Mathematics
Volume	95
DOIs	https://doi.org/10.1016/j.aam.2017.11.006
State	Published - Apr 2018

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Funding

Our work on this group project began at the workshop Algebraic Combinatorixx 2 at the Banff International Research Station (BIRS) in May 2017. “Team Schilling,” as our group of authors is known, would like to extend thanks to the organizers of ACxx2, to BIRS for hosting this workshop, and to the Mathematical Sciences Research Institute (MSRI) for sponsoring a follow-up meeting of some of the group members at MSRI in July 2017 supported by the National Science Foundation under Grant No. DMS-1440140 . We would like to thank Meesue Yoo for early collaboration and Jim Haglund, Brendon Rhoades and Andrew Wilson for fruitful discussions. This work benefited from computations and experimentations in Sage [10,9] . P. E. Harris was partially supported by NSF grant DMS-1620202 . R. Orellana was partially supported by NSF grant DMS-1700058 . G. Panova was partially supported by NSF grant DMS-1500834 . A. Schilling was partially supported by NSF grant DMS-1500050 . M. Yip was partially supported by Simons Collaboration grant 429920 .

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-1440140
Research Institute for Mathematical Sciences
Simons Collaboration	429920
National Science Foundation Arctic Social Science Program	DMS-1620202, DMS-1700058, DMS-1500834, DMS-1500050

Keywords

Crystal bases
Delta Conjecture
Equidistribution of statistics
Minimaj statistic
Ordered multiset partitions

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aam.2017.11.006

Cite this

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title = "A minimaj-preserving crystal on ordered multiset partitions",

abstract = "We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.",

keywords = "Crystal bases, Delta Conjecture, Equidistribution of statistics, Minimaj statistic, Ordered multiset partitions",

author = "Georgia Benkart and Laura Colmenarejo and Harris, {Pamela E.} and Rosa Orellana and Greta Panova and Anne Schilling and Martha Yip",

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doi = "10.1016/j.aam.2017.11.006",

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AU - Benkart, Georgia

AU - Colmenarejo, Laura

AU - Harris, Pamela E.

AU - Orellana, Rosa

AU - Panova, Greta

AU - Schilling, Anne

AU - Yip, Martha

PY - 2018/4

Y1 - 2018/4

N2 - We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

AB - We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

KW - Crystal bases

KW - Delta Conjecture

KW - Equidistribution of statistics

KW - Minimaj statistic

KW - Ordered multiset partitions

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JO - Advances in Applied Mathematics

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ER -

A minimaj-preserving crystal on ordered multiset partitions

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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