Graduate Student Seminar

Speaker: Kylie Schnoor
Title: Frames with Rational Eigensteps
Abstract: Frames can be seen as generalizations of bases that have many uses in both data science and machine learning applications. In this work, we explore a correspondence between frames with rational eigensteps and semistandard Young tableaux (SSYT), via the relation assigning a Gelfand-Tsetlin pattern to a frame via the frame’s eigensteps. First, we begin with an introduction to frame theory as well as introducing the relevant combinatorial background. We will identify how certain key structures in SSYTs correlate with particular frame properties. Additionally, this correspondence leads to a novel way to construct the eigensteps of a frame coming solely from tableaux. This results in a combinatorial reinterpretation of known algorithms in frame theory. Leveraging combinatorial literature, we further discuss how a frame’s Naimark complement may be represented combinatorially leading to a further discussion on the generalization of such a complement for non-tight frames. Further research points to an analysis of equiangular tight frames and their corresponding tableaux, as well as using more combinatorial operations to further analyze frames.