Distribution of major index for standard tableaux and asymptotic normality

We consider the distribution of the major index statistic on standard tableaux of arbitrary straight shape and certain skew shapes. We first use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield–Janson–Zeilberger, Chen–Wang–Wang, and others. This leads to a series of questions concerning locations of zero coefficients, unimodality, and asymptotic estimates for the major index generating functions over all standard tableaux of a fixed representations of the symmetric group appear in which homogeneous components of the corresponding coinvariant algebra, strengthening a recent result of the third author for the modular major index. We give conjectured answers concerning unimodality and asymptotic estimates.