Spectral Decomposition of Involution Overlap Matrices in Group Actions

We study the spectral properties of overlap matrices Φ = (| Fix(j)∩Fix(k)|)j,k∈I arising from finite group actions on sets, where I is a set of involutions. We show that Φ generates a Bose-Mesner algebra, establishing a connection with association scheme theory. For structures with nonempty center CI = T j∈I Fix(j), we prove that the principal eigenvector of Φ over-represents involutions with many fixed points.We provide explicit examples illustrating the spectral decomposition and discuss applications to the study of symmetric structures. Reading Notes for the Complementary Structures Series 1.- Classification of Faithful Actions of G = Z₂ × Z₂ on Sets of 4 Points (10.5281/zenodo.18507173)Establishes foundational examples of group actions and involutions. Essential first step to understand concrete cases of complementary structures. 2.- Finite Complementary Structures: A Framework for Classification, Existence and Parametric Analysis (10.5281/zenodo.18506394)General framework for defining and classifying finite complementary structures. Builds on Paper 1 and sets the stage for parametric and algebraic analyses. 3.- Catalog of Algebraic-Topological Invariants for Complementary Structures (10.5281/zenodo.18507004)Systematic compilation of invariants used to classify and compare complementary structures. Relies on the framework from Paper 2. 4.- Parametric Rigidity Theorem for Complementary Structures: Uniform Bounding of Degrees of Freedom (10.5281/zenodo.18506930)Develops rigidity results and bounds degrees of freedom using invariants from Paper 3. Illustrates constraints on structural variability. 5.- An Algebraic Framework for Complementary Structures: Commuting Operators and Spectral Properties (10.5281/zenodo.18509398)Introduces the algebraic perspective with commuting operators and spectral methods. Provides the mathematical tools for analyzing structural symmetries and eigenvectors. 6.- Invariants and Exploratory Study of Complementary Structures for Groups of Order 8 (10.5281/zenodo.18509498)Applies the invariants and algebraic framework from Papers 3–5 to concrete groups of order 8. Demonstrates practical computation and structural insights. 7.- Convex Optimization with Involutions-Induced Symmetries (10.5281/zenodo.18509603)Explores optimization problems constrained by involution symmetries. Builds directly on algebraic and spectral results from Papers 5–6. 8.- Spectral Decomposition of Involution Overlap Matrices in Group Actions (10.5281/zenodo.18509686)Consolidates spectral decomposition results and principal axes analogues. Recommended last to integrate theory, examples, and applications from all previous papers. Suggested Reading Flow: 1 → 2 → 3 → 4 → 5 → 6 → 7 → 8. This order moves from concrete examples and definitions to invariants, algebraic frameworks, applications, and spectral consolidation.