A minimaj-preserving crystal structure on ordered multiset partitions

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

A minimaj-preserving crystal structure on ordered multiset partitions

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

Mathematics

Producción científica: Paper › revisión exhaustiva

Resumen

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 yields a bijective proof of the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Idioma original	English
Estado	Published - 2018
Evento	30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States Duración: jul 16 2018 → jul 20 2018

Conference

Conference	30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018
País/Territorio	United States
Ciudad	Hanover
Período	7/16/18 → 7/20/18

Nota bibliográfica

Publisher Copyright:
© FPSAC 2018 - 30th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.

Financiación

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.

Financiadores	Número del financiador
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

ASJC Scopus subject areas

Algebra and Number Theory

Otros archivos y enlaces

Citar esto

@conference{8af2ec48ac3748ec95926f0b4c972f0b,

title = "A minimaj-preserving crystal structure 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 yields a bijective proof of 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",

note = "Publisher Copyright: {\textcopyright} FPSAC 2018 - 30th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.; 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 ; Conference date: 16-07-2018 Through 20-07-2018",

year = "2018",

language = "English",

}

TY - CONF

T1 - A minimaj-preserving crystal structure on ordered multiset partitions

AU - Benkart, Georgia

AU - Colmenarejo, Laura

AU - Harris, Pamela E.

AU - Orellana, Rosa

AU - Panova, Greta

AU - Schilling, Anne

AU - Yip, Martha

PY - 2018

Y1 - 2018

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 yields a bijective proof of 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 yields a bijective proof of 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

UR - https://www.scopus.com/pages/publications/85089336457

UR - https://www.scopus.com/pages/publications/85089336457#tab=citedBy

M3 - Paper

AN - SCOPUS:85089336457

T2 - 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018

Y2 - 16 July 2018 through 20 July 2018

ER -

A minimaj-preserving crystal structure on ordered multiset partitions

Resumen

Conference

Nota bibliográfica

Financiación

ASJC Scopus subject areas

Otros archivos y enlaces

Huella

Citar esto