We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We 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. These results can be interpreted as giving a very precise description of the distribution of irreducible representations in different degrees of coinvariant algebras of certain complex reflection groups. We conclude with some conjectures concerning unimodality, log-concavity, and local limit theorems.

Reiner-Stanton-White [RSW04] defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content.There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a "universal" sieving statistic on words, flex.A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf [WW94].

We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index r mod n, for all r. Our result generalizes the r = 1 case due essentially to Klyachko [Kly74] and proves a recent conjecture due to Sundaram [Sun17] for the r = 0 case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted to classifying which partitions λ ⊢ n have f λ ≤ n d for fixed d. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov [FL95] for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.

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We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes. By a theorem of Lusztig-Stanley, this classification can be interpreted as determining which irreducible representations of the symmetric group exist in which homogeneous components of the corresponding coinvariant algebra, strengthening a recent result of the third author for the modular major index. Our approach is to identify patterns in standard tableaux that allow one to mutate descent sets in a controlled manner. By work of Lusztig and Stembridge, the arguments extend to a classification of all nonzero fake degrees of coinvariant algebras for finite complex reflection groups in the infinite family of Shephard-Todd groups.