A combinatorial characterization of tight fusion frames

TL;DR

This paper provides a comprehensive combinatorial characterization of tight fusion frame sequences using Littlewood-Richardson skew tableaux, extending previous results to unequal rank cases and introducing new generation methods.

Contribution

It introduces a new combinatorial framework for characterizing and generating tight fusion frame sequences beyond the equal rank case, utilizing majorization and duality principles.

Findings

Characterization of TFF sequences via Littlewood-Richardson tableaux

Development of methods for generating TFF sequences

Complete classification of maximal TFF sequences in low dimensions

Abstract

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.