  The study of permutation and partition statistics is a classical topic in
enumerative combinatorics. The major index statistic on permutations was
introduced a century ago by Percy MacMahon in his seminal works. In this
extended abstract, we study the well-known generalization of the major index to
standard Young tableaux. We present several new results. In one direction, we
introduce and study two partial orders on the standard Young tableaux of a
given partition shape, in analogy with the strong and weak Bruhat orders on
permutations. 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, and has
representation-theoretic consequences, both for the symmetric group and for
Shephard-Todd groups. In a different direction, we consider the distribution of
the major index on standard tableaux of arbitrary straight shape and certain
skew shapes. We 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. We also
study unimodality, log-concavity, and local limit properties.
