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Award ID contains: 1700058

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  1. (, Journal of combinatorial theory. Series A)
    In \cite{OZ}, the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products. In addition, we identify some applications and instances where special cases of these products have occurred elsewhere in the mathematical literature. 
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    Accepted Manuscript Full Text Available
  2. (, Advances in applied mathematics)
    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 enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions. 
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    Accepted Manuscript Full Text Available