On the complexity, tractability and group theoretic applications of the relative eigenvector problem

The Relative Eigenvector Problem (REP) generalizes the classical eigenvector problem and arises naturally in computational group theory and representation theory. It plays a central role in methods for constructing finite simple subgroups of exceptional algebraic groups, as well as in algorithms for computing tensor decompositions and systems of imprimitivity of group representations. In this paper, we investigate both the computational complexity and the practical tractability of REP. We show that a decision variant of REP is NP-complete, over fixed finite fields, via a direct reduction from CNF-SAT. We then relate REP to the MinRank problem with target rank 1 and use algebraic techniques involving determinantal ideals to derive complexity bounds and identify broad parameter regimes in which generic instances of REP can be solved efficiently. We conclude with an application to the classification of embeddings of the alternating group A6 into algebraic groups of type F4. This example demonstrates how REP-based methods may admit spurious solutions and how additional polynomial constraints can be used to refine REP instances to recover genuine embeddings.