We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3. Date: Version W06v6c.tex last changed 05 November 2002.
Die \'on TREDER vorgeschlagenen Gravitationsfeldgleichungen 4. Ordnung mit ciner Linearkombination des Bachtensors und des Einsteintensors auf der linken Seite besitzen statische zentralsymmetrischc Losungen, welche in ciner Umgebung des S~inmctriczentrums analytisch uiid nicht flach sind.The fourth order field equations proposed by TREUER with a linear combination of BACH'S tensor and EINSTEIS'S tensor on the lcft-hand sidc admit static ccntrally symmetric solutions which are analytical and non-flat in some neighborhood of the criit rc o f s yiii i i i ct ry .
It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space H of the Heisenberg magnet are [5,6,7,8].We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN ], idempotents of C[SN ], discrete Fourier transforms of SN , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[SN ].Let U (r 1 ,r 2 ) µ be a subspace of eigenvectors of a a fixed eigenvalue µ of H with weight (r1, r2). If one determines the smallest symmetry class for every v ∈ U (r 1 ,r 2 ) µ then one can observe jumps of the symmetry behaviour. For "generic" v ∈ U (r 1 ,r 2 ) µ all smallest symmetry classes have the same maximal dimension d and "structure". But U (r 1 ,r 2 ) µ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely.In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.