Connections Between Frames with Rational Eigensteps and Semistandard Young Tableaux

TL;DR

This paper establishes a novel correspondence between frames with rational eigensteps and semistandard Young tableaux, providing new combinatorial methods for constructing and analyzing frames and their properties.

Contribution

It introduces a new combinatorial framework linking frames and tableaux, offering alternative construction methods and insights into frame properties and complements.

Findings

Correspondence between frame eigensteps and SSYT established

New tableau-based method for constructing frame eigensteps

Extension of Naimark complement via combinatorial techniques

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.