This chapter focuses on the derivation and interpretation of functional relations among commuting transfer matrices, which underpin a wide range of powerful techniques in quantum integrable systems. These include spectral equations, Baxter’s TQ-relations, and the analytic Bethe ansatz, as well as methods based on wronskian Bethe equations. Rather than isolated tools, these structures are intricately connected and reveal deep algebraic insights with significant implications, including quantum–classical duality and the completeness problem.

Further emphasis is placed on the group-theoretic structure underlying these functional relations. Connections are drawn with the classical theory of characters, where objects such as Schur polynomials and Jacobi–Trudi formulae lead to bilinear relations naturally interpreted as Hirota equations in the context of transfer matrices. This perspective offers a conceptual bridge between quantum integrable models and classical representation theory, highlighting the unifying role of functional relations in both structure and application. Special attention is given to Q-functions governed by Baxter’s TQ-relations and, more broadly, by determinant identities from the wronskian formalism. These functions encode the transfer matrix eigenvalues and serve as fundamental algebraic objects. Functional methods thus offer powerful tools for exploring integrable models, especially when conventional Bethe ansatz techniques become impractical.