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 {f_i}_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 {f_j}_j∈J is a tight frame for H_n. 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 H_n 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.