The strong Lefschetz property for Artinian algebras with non-standard grading

We define the strong Lefschetz property for finite graded modules over graded Artinian algebras whose grading is not necessarily standard. We show that most results which have been obtained for Artinian algebras with standard grading can be extended for non-standard grading. Our results on the strong Lefschetz property for non-standard grading can be used to prove that certain Artinian complete intersections with standard grading have the strong Lefschetz property.Comment: 24 pages, To appear in Journal of Algebr

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian algebra over a field of characteristic zero and let z be a linear form in A. We define the central simple modules for each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has the strong Lefschetz property if and only if there exists a linear form z in A such that all central simple modules of the pair (A,z) have the strong Lefschetz property. In the course of proof we need to extend the definition of the strong Lefschetz property to finite graded modules over graded Artinian algebra, which previously was defined only for standard graded Artinian algebras.Comment: 20 pages, To be published in Journal of Pure and Applied Algebr

Theory of Metal-Insulator Transition in PrRu4P12 and PrFe4P12

All symmetry allowed couplings between the 4f^2-electron ground state doublet of trivalent praseodymium in PrRu4P12 and PrFe4P12 and displacements of the phosphorus, iron or ruthenium ions are considered. Two types of displacements can change the crystal lattice from body-centred cubic to simple orthorhombic or to simple cubic. The first type lowers the point group symmetry from tetrahedral to orthorhombic, while the second type leaves it unchanged, with corresponding space group reductions Im3 --> Pmmm and Im3 --> Pm3 respectively. In former case, the lower point-group symmetry splits the degeneracy of the 4f^2 doublet into states with opposite quadrupole moment, which then leads to anti-quadrupolar ordering, as in PrFe4P12. Either kind of displacement may conspire with nesting of the Fermi surface to cause the metal-insulator or partial metal-insulator transition observed in PrFe4P12 and PrRu4P12. We investigate this scenario using band-structure calculations, and it is found that displacements of the phosphorus ions in PrRu4P12 (with space group reduction Im3 --> Pm3) open a gap everywhere on the Fermi surface.Comment: 6 page

Multipole Fluctuations in Filled Skutterudites

In order to clarify exotic multipole properties of filled skutterudites, we evaluate multipole susceptibility for $n$=1$\sim$5, where $n$ is the local $f$-electron number, on the basis of a multiorbital Anderson model constructed using the $j$-$j$ coupling scheme. For $n$=1, magnetic fluctuations dominate over low-temperature electronic properties, while for $n$=2 and 4, electronic states are dominated by both magnetic and quadrupole fluctuations. For $n$=3 and 5, octupole fluctuations are found to be significant, depending on the crystalline electric field potential. We discuss possible relevance of the results to actual materials.Comment: 4 pages, 3 figures, JPSJ2.cls. To be published in J. Phys. Soc. Jp