Connections Between Frames with Rational Eigensteps and Semistandard Young Tableaux
Emily J. King, Kylie Schnoor
TL;DR
This work builds a bridge between frame theory and combinatorics by representing frame eigensteps through Gelfand-Tsetlin patterns and semistandard Young tableaux. By establishing correspondences between outer/inner eigensteps and GT/SSYT structures, the authors derive tableau-based constructions for eigensteps and frames, introduce the notion of clearable frames with integer GT patterns via a clearing constant $\ell$, and connect Naimark complements and generalized Naimark complements to tableau involutions (e.g., boxcomp). The approach offers a combinatorial, discretized alternative to the Top Kill method and suggests far-reaching implications for analyzing and constructing frames, including potential extensions to equiangular tight frames. Overall, the paper provides a cohesive framework linking spectral frame data to tableau operations, enabling new tools for frame design and analysis with deep combinatorial underpinnings.
Abstract
In this paper, 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. We will identify how certain key structures in SSYTs correlate with particular frame properties. For example, the weight of an SSYT yields the sequence of norms of any compatible frame. Additionally, this correspondence leads to a novel way to construct the eigensteps of a frame coming solely from tableaux. This is an alternative to the Top Kill algorithm which may be viewed as a combinatorial reinterpretation of the algorithm. We further employ other combinatorial techniques such as the boxcomp method to generate a ``complement" SSYT. On the frame side, this corresponds to a tight frame's Naimark complement as well as to a generalization of the Naimark 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.