Zur Reziprozität quadratischer Charaktersummen in algebraischen Zahlkörpern

Ohne Zusammenfassung     

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF ₄, andH _r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH _R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF ₄ over the ringA = ℤ[u ^1/2,u ^-1/2] of Laurent polynomials in an indeterminateu ^1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.     

Strukturänderungen bei der Reduktion von 9-(Diphenylmethyliden)fluoren mit Natrium-Metall zu Schichten (R2O)Na⊕-verknüpfter Kohlenwasserstoff-Dianionen

Structural Changes on Reduction of 9-(Diphenylmethylidene)fluorene by Sodium Metal to Sheets of (R₂O)Na⊕- Connected Hydrocarbon Dianions The single-crystal structure of 9-(diphenylmethylidene)fluorene (monoclinic, space group P2₁/n, Z = 4) proves a C?C bond length of 135 pm and a twisting of the two Ph rings out of the methylidene-fluorene plane by 70° and 83°. Its reduction in Et₂O solution by a Na mirror under Ar yields dark green crystals with a metallic luster. Their structure, determined in a N₂ flow cooled to 223 K (monoclinic, space group C2/c, Z = 4), reveals sheets of hydrocarbon dianions, connected by sodium(diethylether) links. The Na centers are sixfold coordinated as half-sandwiches to the fluorene benzo rings and threefold allylic to the Ph rings. In the dianion of 9-(diphenyl-methylidene)fluorene, the central C?C bond is stretched by 13 pm to 148 pm length, and the two molecular cyanine-like halves are twisted by 51° relative to each other. Both the contact-ion multiple structures as well as the structural changes on twofold reduction are discussed by way of comparison with literature examples and based on semiempirical calculations.