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We introduce two new partial orders on the standard Young tableaux of a givenpartition shape, in analogy with the strong and weak Bruhat orders onpermutations. Both posets are ranked by the major index statistic offset by afixed shift. The existence of such ranked poset structures allows us toclassify the realizable major index statistics on standard tableaux ofarbitrary straight shape and certain skew shapes. By a theorem ofLusztig--Stanley, this classification can be interpreted as determining whichirreducible representations of the symmetric group exist in which homogeneouscomponents of the corresponding coinvariant algebra, strengthening a recentresult of the third author for the modular major index. Our approach is toidentify patterns in standard tableaux that allow one to mutate descent sets ina controlled manner. By work of Lusztig and Stembridge, the arguments extend toa classification of all nonzero fake degrees of coinvariant algebras for finitecomplex reflection groups in the infinite family of Shephard--Todd groups.

Tableau posets and the fake degrees of coinvariant algebras