This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic \(0\). Our most general result determines the top component in total degree, which we prove for all Shephard-Todd groups \(G(m, p, n)\) with \(m \neq p\) or \(m=1\). Our strongest result gives tight bi-degree bounds and is proven for all \(G(m, 1, n)\), which includes the Weyl groups of types \(A\) and \(B\)/\(C\). For symmetric groups (i.e. type \(A\)), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund-Remmel-Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman.

Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first paper in this sequence. In this paper we use concrete constructions including Gröbner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups \(G(m, p, n)\), which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.

Mathematics Subject Classifications: 05E16 (Primary), 20F55, 05A15 (Secondary)

Keywords: Coinvariant algebras, pseudo-reflection groups, Gröbner basis, Artin basis, differential forms, exterior derivatives

Billey-Konvalinka-Swanson studied the asymptotic distribution of the coefficients of Stanley's \(q\)-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's \(q\)-hook-content formula related to semistandard tableaux and \(q\)-hook length formulas of Björner-Wachs related to linear extensions of labeled forests. We show that, while their coefficients are "generically" asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded \(2\)-norm. The closure of these distributions in the Lévy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.

Mathematics Subject Classifications: 05A16 (Primary), 60C05, 60F05 (Secondary)

Keywords: Hook length, \(q\)-analogues, major index, semistandard tableaux, plane partitions, forests, asymptotic normality, limit laws, Irwin-Hall distribution

We introduce fluctuating tableaux, which subsume many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, standard, and transpose semistandard tableaux. Our main contribution is the introduction of promotion permutations and promotion matrices, which are new even for standard tableaux. We provide characterizations in terms of Bender-Knuth involutions, jeu de taquin, and crystals. We prove key properties in the rectangular case about the behavior of promotion permutations under promotion and evacuation. We also give a full development of the basic combinatorics and representation theory of fluctuating tableaux.

Our motivation comes from our companion paper, where we use these results in the development of a new rotation-invariant \(\operatorname{SL}_4\)-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. These planar graphs are indexed by fluctuating tableaux, tableau promotion corresponds to graph rotation, and promotion permutations correspond to key graphical information.

Mathematics Subject Classifications: 05E10, 05E18

Keywords: Tableaux, promotion, jeu de taquin, Bender-Knuth involutions, growth diagrams, crystals