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 yields a bijective proof of the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.
Original language | English |
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State | Published - 2018 |
Event | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States Duration: Jul 16 2018 → Jul 20 2018 |
Conference
Conference | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 |
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Country/Territory | United States |
City | Hanover |
Period | 7/16/18 → 7/20/18 |
Bibliographical note
Publisher Copyright:© FPSAC 2018 - 30th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.
Funding
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]. ∗[email protected] †[email protected] ‡[email protected]. P. E. Harris was partially supported by NSF grant DMS–1620202. §[email protected]. R. Orellana was partially supported by NSF grant DMS–1700058. ¶[email protected]. G. Panova was partially supported by NSF grant DMS–1500834. ‖[email protected]. A. Schilling was partially supported by NSF grant DMS–1500050. ∗∗[email protected]. M. Yip was partially supported by Simons Collaboration grant 429920.
Funders | Funder number |
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Mathematical Sciences Research Institute (MSRI) | |
Simons Collaboration | 429920 |
National Science Foundation (NSF) | DMS–1500834, DMS–1500050, DMS–1700058, DMS–1620202 |
Research Institute for Mathematical Sciences |
Keywords
- Crystal bases
- Delta Conjecture
- Equidistribution of statistics
- Minimaj statistic
- Ordered multiset partitions
ASJC Scopus subject areas
- Algebra and Number Theory