Benkart, Georgia; Colmenarejo, Laura; Harris, Pamela E.; Orellana, Rosa; Panova, Greta; Schilling, Anne; Yip, Martha A minimaj-preserving crystal on ordered multiset partitions. (English) Zbl 1379.05015 Adv. Appl. Math. 95, 96-115 (2018). Summary: 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 J. Haglund et al. [“The delta conjecture”, Preprint, arXiv:1509.07058] 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 J. Haglund et al. [“Ordered set partitions, generalized coinvariant algebras, and the delta conjecture”, Preprint, arXiv:1609.07575] 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. Cited in 2 ReviewsCited in 10 Documents MSC: 05A19 Combinatorial identities, bijective combinatorics 05A18 Partitions of sets 05E05 Symmetric functions and generalizations 05E10 Combinatorial aspects of representation theory 20G42 Quantum groups (quantized function algebras) and their representations 17B37 Quantum groups (quantized enveloping algebras) and related deformations Keywords:delta conjecture; ordered multiset partitions; minimaj statistic; crystal bases; equidistribution of statistics Software:SageMath; Sage-Combinat PDFBibTeX XMLCite \textit{G. Benkart} et al., Adv. Appl. Math. 95, 96--115 (2018; Zbl 1379.05015) Full Text: DOI arXiv References: [1] Bump, Daniel; Schilling, Anne, Crystal Bases. Representations and Combinatorics (2017), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1440.17001 [2] Carlsson, Erik; Mellit, Anton, A proof of the shuffle conjecture (2015), preprint · Zbl 1387.05265 [3] Haglund, James; Haiman, Mark; Loehr, Nicholas; Remmel, Jeffrey B.; Ulyanov, Alexander P., A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., 126, 2, 195-232 (2005) · Zbl 1069.05077 [4] Haglund, James; Rhoades, Brendon; Shimozono, Mark, Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture (2016), preprint · Zbl 1384.05043 [5] Haglund, James; Remmel, Jeff; Wilson, Andrew, The Delta Conjecture (2015), preprint [6] Kashiwara, Masaki, Crystalizing the \(q\)-analogue of universal enveloping algebras, Comm. Math. Phys., 133, 2, 249-260 (1990) · Zbl 0724.17009 [7] Kashiwara, Masaki, On crystal bases of the \(Q\)-analogue of universal enveloping algebras, Duke Math. J., 63, 2, 465-516 (1991) · Zbl 0739.17005 [8] Rhoades, Brendon, Ordered set partition statistics and the Delta Conjecture, J. Combin. Theory Ser. A, 154, 172-217 (2018) · Zbl 1373.05006 [9] The Sage Developers (2017), Sage Mathematics Software (Version 8.0) [10] The Sage-Combinat community (2008), Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics [11] Stembridge, John R., A local characterization of simply-laced crystals, Trans. Amer. Math. Soc., 355, 12, 4807-4823 (2003) · Zbl 1047.17007 [12] Wilson, Andrew Timothy, An extension of MacMahon’s equidistribution theorem to ordered multiset partitions, Electron. J. Combin., 23, 1, 21 (2016), Paper 1.5 · Zbl 1329.05030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.