Coherent Configurations: Foundations of Algebraic Combinatorics

Explore the fundamental mathematical structures that bridge algebraic combinatorics, graph theory, and permutation groups. This text-based course guides you through coherent configurations and association schemes, providing the foundational tools required to analyze symmetry and discrete structures. You will build a solid understanding of how combinatorial objects interact with matrix algebras, structural invariants, and modern computational graph isomorphism concepts like the Weisfeiler-Leman procedure. What you'll learn: - Understand the core definitions, axioms, and terminology of coherent configurations - Analyze adjacency matrices, intersection numbers, and the basis matrix algebra - Connect coherent configurations to association schemes and strongly regular graphs - Explore the relationship between permutation groups, orbits, and centralizer rings - Study the foundational mechanics of color refinement and Weisfeiler-Leman algorithms - Apply combinatorial and algebraic techniques through step-by-step written exercises The course begins with fundamental terminology and set-theoretic definitions before progressing to algebraic properties, group-theoretic connections, and structural classification. Designed for motivated learners, students, and practitioners in discrete mathematics and theoretical computer science seeking a structured, accessible introduction without requiring advanced prior knowledge. Start reading today to deepen your grasp of algebraic combinatorics and structural symmetry.