FRIEZES, STRINGS AND CLUSTER VARIABLES

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an explicit normalising monomial factor.

Polyhedral Oxaruthenaborane Chemistry. Characterisation of a [(η6-C6Me6)RuOB9H13] Species of arachno Eleven-Vertex Cluster Character and Other Aspects of Oxaborane Chemistry

[(η6-C6Me6)RuOB9H13], obtained in trace yield from the reaction between [{RuCl2(η6-C6Me6)}2] and the [arachno-B6H11]- anion, has been shown by DFT calculations of structure and of boron-atom nuclear shielding to be of an arachno eleven-vertex cluster geometry that is derived by the removal of adjacent six-connected and five-connected vertices from a closed thirteen-vertex deltahedral 1:6:5:1 stack. The results lead to further considerations in oxaborane chemistry, which are explored and consolidated by calculation. Additional products from the reaction include [4-(η6-C6Me6)-n-arachno-4-RuB8H14], also substantiated by DFT calculations, and several four-, five-, six-, nine- and ten-vertex ruthenaboranes that are either known compounds or simple variants of known compounds.