A minimaj-preserving crystal structure on ordered multiset partitions

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

Research output: Contribution to conferencePaperpeer-review

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 languageEnglish
StatePublished - 2018
Event30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States
Duration: Jul 16 2018Jul 20 2018

Conference

Conference30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018
Country/TerritoryUnited States
CityHanover
Period7/16/187/20/18

Bibliographical note

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

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

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