Dimension towers of SICs. II. Some constructions

A SIC is a maximal equiangular tight frame in finite dimensional Hilbert space. Given dimension $d$, there good evidence that always exists an aligned $d(d-2)$, having predictable symmetries and smaller frames embedded them. We provide recipe for how to calculate sets of vectors $d(d-2)$ share these properties. They consist maximally entangled certain subspaces defined by the numbers entering $d$ SIC. However, construction contains free parameters we have not proven they can be chosen so one give some worked examples that, hope, may suggest reader our improved. For simplicity restrict ourselves case odd dimensions.