Tableau posets and the fake degrees of coinvariant algebras

TL;DR

This paper introduces new partial orders on standard Young tableaux, classifies major index statistics, and links these to the representation theory of symmetric groups and coinvariant algebras, extending to complex reflection groups.

Contribution

It develops two new poset structures on tableaux, classifies realizable major index statistics, and connects these to fake degrees of coinvariant algebras for complex reflection groups.

Findings

Classified realizable major index statistics on tableaux.

Extended classification of fake degrees to complex reflection groups.

Linked tableau posets to representation theory of symmetric groups.

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.