Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in a series of three dealing with groups of rank up to eight (and in particular, all exceptional Coxeter groups), we prove the conjecture for finite Coxeter groups of rank five and six, further developing the algorithmic tools described in the previous article. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik-Solomon characters of the groups considered.Comment: Final Version. 17 page

The reducible Specht modules for the Hecke algebra HC,−1(Sn)\mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})

The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})$ have been classified except when $q = -1$. We prove one half of a conjecture which we believe classifies the reducible Specht modules when $q = -1$

Discovery of 7‑Tetrahydropyran-2-yl Chromans: β‑Site Amyloid Precursor Protein Cleaving Enzyme 1 (BACE1) Inhibitors That Reduce Amyloid β‑Protein (Aβ) in the Central Nervous System

In an attempt to increase selectivity vs Cathepsin D (CatD) in our BACE1 program, a series of 1,3,4,4a,10,10a-hexahydropyrano­[4,3-b]­chromene analogues was developed. Three different Asp-binding moieties were examined: spirocyclic acyl guanidines, aminooxazolines, and aminothiazolines in order to modulate potency, selectivity, efflux, and permeability. Using structure-based design, substitutions to improve binding to both the S3 and S2′ sites of BACE1 were explored. An acyl guanidine moiety provided the most potent analogues. These compounds demonstrated 10–420 fold selectivity for BACE1 vs CatD, and were highly potent in a cell assay measuring Aβ_1–40 production (5–99 nM). They also suffered from high efflux. Despite this undesirable property, two of the acyl guanidines achieved free brain concentrations (C_free,brain) in a guinea pig PD model sufficient to cover their cell IC₅₀s. Moreover, a significant reduction of Aβ_1–40 in guinea pig, rat, and cyno CSF (58%, 53%, and 63%, respectively) was observed for compound <b>62</b>