  In this paper we give a combinatorial characterization of tight fusion frame
(TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case
has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our
characterization does not have this limitation. We also develop some methods
for generating TFF sequences. The basic technique is a majorization principle
for TFF sequences combined with spatial and Naimark dualities. We use these
methods and our characterization to give necessary and sufficient conditions
which are satisfied by the first three highest ranks. We also give a
combinatorial interpretation of spatial and Naimark dualities in terms of
Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences
which have unique maximal elements with respect to majorization partial order.
Finally, we give several examples illustrating our techniques including an
example of tight fusion frame which can not be constructed by the existing
spectral tetris techniques. We end the paper by giving a complete list of
maximal TFF sequences in dimensions less than ten.

Functional Analysis;Combinatorics