Standard monomial theory for wonderful varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation Theory. The final publication is available at Springer via http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in Section 3 and in Section

Cyclotrimerization reactions catalyzed by rRhodium(I) half-sandwich complexes: A mechanistic density functional study

We propose and examine a comprehensive mechanism of the [(eta(5)-C5H5)Rh]-catalyzed [2+2+2] cycloadditions of acetylene to benzene and of acetylene and acetonitrile to 2-methylpyridine, based on an extensive and detailed exploration of the potential energy surfaces using density functional theory. Both processes involve the formation of a coordinatively unsaturated 16-electron metallacycle, occurring after the replacement of the ancillary ligands L of the catalyst precursor of general formula [(eta(5)-C5H5)RhL2] (typically L = C2H4, CO, PH3 or L-2 = 1,5-cyclooctadiene) by two acetylene molecules. The facile coordination of a third acetylene molecule, and its subsequent addition to the pi electron system of the rhodacycle, leads to the formation of an intermediate, which is characterized by a six-membered arene ring coordinated to the metal in eta(4) fashion. The release of benzene occurs by stepwise addition of two acetylene molecules, which regenerates the catalyst. In the presence of acetonitrile, a nitrile molecule coordinates to the rhodacycle, and different stages are outlined for the process, leading to the eventual release of 2-methylpyridine. The steric and electronic effects of the pi ligand coordinated to the metal are also included in our exploration by addressing the whole mechanism of the [(eta(5)-C9H7)Rh]-catalyzed alkyne self-trimerization to benzene. The kinetic parameters, i.e., the energies in vacuum and in different solvents, and the geometries of the intermediates and of the transition states are analyzed in detail