Title: On Structural Decomposition of Finite Frames

Speaker: Sivaram Narayan (Central Michigan University)

In this talk we discuss the combinatorial structure of frames and their decomposition into tight or scalable subsets using partially-ordered sets (posets). We define factor poset of a frame {fi}i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J ⊆ I is in the factor poset if and only if {fj}j∈J is a tight frame for Hn. A similar definition is given for the scalability poset of a frame. We discuss conditions which factor posets satisfy and present the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We mention a necessary condition for solving the inverse factor poset problem in Hn which is also sufficient for H2. We describe how factor poset structure of frames is preserved under orthogonal projections. We present results regarding when a frame can be scaled to have a given factor poset.