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.

中文翻译：

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Tableau 偏序集和共变代数的假度数

我们在给定分区形状的标准 Young 表上引入了两个新的偏序，类似于排列上的强和弱 Bruhat 顺序。两个偏序集都按主索引统计量排序，偏移固定偏移。这种排序偏序结构的存在使我们能够对任意直线形状和某些倾斜形状的标准表的可实现主要指标统计进行分类。根据 Lusztig--Stanley 的定理，这种分类可以解释为确定对称群的哪些不可约表示存在于其中对应的共变代数的齐次分量，加强了第三作者最近关于模主指数的结果。我们的方法是识别标准画面中的模式，允许以受控方式改变下降集。