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A minimaj-preserving crystal on ordered multiset partitions. (English) Zbl 1379.05015

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.

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
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References:

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