Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach

TL;DR

This paper develops a representation-theoretic framework for constructing Frobenius-Schur frames on the symmetric group, enabling meaningful analysis of ranked data through graph signal processing.

Contribution

It generalizes the construction of Frobenius-Schur frames from the permutahedron to arbitrary inverse-closed generating sets of the symmetric group.

Findings

Characterization of Frobenius-Schur frames compatible with generating sets

Extension of frame constructions from permutahedron to general Cayley graphs

Application to meaningful interpretation of ranked data

Abstract

An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked data analysis, as its vertices represent permutations, while the generating set formalizes a notion of distance between rankings. Taking advantage of the rich theory of representations of the symmetric group, we study a particular class of frames, called Frobenius-Schur frames, where every atom belongs to the coefficient space of only one irreducible representation of the symmetric group. We provide a characterization for all Frobenius-Schur frames on the group algebra of the symmetric group which are "compatible" with respect to the generating set. Such frames have been previously studied for the permutahedron, the Cayley graph of the symmetric group with the generating set of adjacent transpositions, and have proved to be capable of producing meaningful interpretation of the ranked data set via the analysis coefficients. Our results generalize frame constructions for the permutahedron to any inverse-closed generating set.