Principal Components along Quiver Representations

TL;DR

This paper introduces an algorithm to compute principal components within quiver representations, enabling analysis of vector space sections through constrained optimization and eigenvector solutions.

Contribution

It provides a novel algorithm for calculating sections of quiver representations and defining principal components in this context, bridging representation theory and data analysis.

Findings

Algorithm successfully computes sections of quiver representations

Principal components are characterized as eigenvectors of a matrix pencil

Method enables constrained optimization over representation spaces

Abstract

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil.