A combinatorial characterization of tight fusion frames

Marcin Bownik, Kurt Luoto and Edward Richmond

Vol. 275 (2015), No. 2, 257–294

DOI: 10.2140/pjm.2015.275.257

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 $\le 9$.

Keywords

tight fusion frame, majorization, orthogonal projection, partition, Schur function, Littlewood–Richardson coefficient, Schubert calculus, symmetric functions

Mathematical Subject Classification 2010

Primary: 05E05, 15A57, 42C15

Secondary: 14N15, 14M15

Milestones

Received: 7 October 2013

Revised: 27 September 2014

Accepted: 18 October 2014

Published: 15 May 2015