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arXiv · 1809.07386

Tableau posets and the fake degrees of coinvariant algebras

Sara C. Billey·Matjaž Konvalinka·Joshua P. Swanson
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Abstract

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.

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Sara C. Billey, Matjaž Konvalinka, Joshua P. Swanson. 2018-09-13. Tableau posets and the fake degrees of coinvariant algebras. https://arxiv.org/abs/1809.07386

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