Tanisaki witness relations for harmonic differential forms

Inspired by a series of conjectures related to higher coinvariant algebras, we present two families of relations involving harmonic differential forms of the symmetric group. Our relations, together with a novel bijection, are sufficient to give a filtration of the $1$-forms suggested by work of Haglund--Rhoades--Shimozono with composition factors given by Tanisaki quotients. These are "almost all" of the necessary relations in a certain asymptotic sense we make precise.Comment: 30 page

On the existence of tableaux with given modular major index

We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted to classifying which partitions $\lambda \vdash n$ have $f^\lambda \leq n^d$ for fixed $d$. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov (1995) for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.Comment: Incorporated referee comment

Promotion permutations for tableaux

In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux. In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux.Comment: 42 pages, 8 figure

Current and Emerging Uses of Statins in Clinical Therapeutics: A Review

Statins, a class of cholesterol-lowering medications that inhibit 3-hydroxy-3-methyl-glutaryl-coenzyme A reductase, are commonly administered to treat atherosclerotic cardiovascular disease. Statin use may expand considerably given its potential for treating an array of cholesterol-independent diseases. However, the lack of conclusive evidence supporting these emerging therapeutic uses of statins brings to the fore a number of unanswered questions including uncertainties regarding patient-to-patient variability in response to statins, the most appropriate statin to be used for the desired effect, and the efficacy of statins in treating cholesterol-independent diseases. In this review, the adverse effects, costs, and drug–drug and drug–food interactions associated with statin use are presented. Furthermore, we discuss the pleiotropic effects associated with statins with regard to the onset and progression of autoimmune and inflammatory diseases, cancer, neurodegenerative disorders, strokes, bacterial infections, and human immunodeficiency virus. Understanding these issues will improve the prognosis of patients who are administered statins and potentially expand our ability to treat a wide variety of diseases