A note on embedding and generating dual polar spaces.
The notion of a transfer (or T -transform) is central in the theory of majorization. For instance, it lies behind the characterization of majorization in terms of doubly stochastic matrices. We introduce a new type of majorization transfer called L-transforms and prove some of its properties. Moreover, we discuss how L-transforms give a new perspective on Ryser’s algorithm for constructing (0; 1)-matrices with given row and column sums.
We prove that a rank Dowling geometry of a group is partition representable if and only if is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.
It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + mkl ≤ max{mik + mjl, mil + mjk} for all distinct i,j,k,l.
After describing a (general and special) coordinatization of -nets there are found algebraic equivalents for the validity of certain quadrangle configuration conditions in -nets with small degree .