Inverse semigroup spectral analysis for partially ranked data

TL;DR

This paper introduces a novel spectral analysis framework for the rook monoid, extending symmetric group analysis to partially ranked data, with applications in voting data and potential generalizations.

Contribution

It develops the first spectral analysis for non-group semigroups, specifically the rook monoid, linking it to symmetric group analysis and applying it to voting data.

Findings

Spectral analysis on rook monoid characterized in terms of symmetric group analysis

Application demonstrated on real-world partially ranked voting data

Framework generalizable to arbitrary finite inverse semigroups

Abstract

Motivated by the notion of symmetric group spectral analysis developed by Diaconis, we introduce the notion of spectral analysis on the rook monoid (also called the symmetric inverse semigroup), characterize its output in terms of symmetric group spectral analysis, and provide an application to the statistical analysis of partially ranked (voting) data. We also discuss generalizations to arbitrary finite inverse semigroups. This paper marks the first non-group semigroup development of spectral analysis.