A note on the theoretical approach to Grassmannians and Pl\"ucker coordinates for additive skew-symmetric pairwise comparisons matrices

TL;DR

This paper introduces a geometric framework embedding additive skew-symmetric pairwise comparison matrices into Grassmannian manifolds, revealing their algebraic consistency as geometric properties and providing a new theoretical foundation for decision-making models.

Contribution

It presents a novel geometric interpretation of pairwise comparison matrices using Grassmannians and Pl"ucker coordinates, connecting algebraic consistency to geometric conditions.

Findings

Algebraic consistency corresponds to geometric consistency in Grassmannian $G(2, n)$.

PC matrices can be interpreted as differential 2-forms, linking consistency to closedness.

Provides a new theoretical foundation for understanding pairwise comparisons through geometry.

Abstract

Symmetry and antisymmetry are fundamental concepts in many strict sciences. Pairwise comparisons (PC) matrices are fundamental tools for representing pairwise relations in decision making. In this theoretical study, we present a novel framework that embeds additive skew-symmetric PC matrices into the Grassmannian manifold $G(2, n)$. This framework leverages Pl\"ucker coordinates to provide a rigorous geometric interpretation of their structure. Our key result demonstrates that the algebraic consistency condition $a_{ij} + a_{jk} - a_{ki} = 0$ is equivalent to the geometric consistency of $2$-planes in $G(2, n)$, satisfying the Pl\"ucker relations. This connection reveals that the algebraic properties of PC matrices can be naturally understood through their geometric representation. Additionally, we extend this framework by interpreting PC matrices as differential $2$-forms, providing a new perspective on their consistency as a closedness condition. Our framework of linear algebra, differential geometry, and algebraic geometry, placing PC matrices in a broader mathematical context. Rather than proposing a practical alternative to existing methods, our study aims to offer a theoretical foundation for future research by exploring new insights into higher-dimensional geometry and the geometric modeling of pairwise comparisons.