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.