  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.
