The Four-Gon Quiver: Path Algebra, Cycle Relations, and Representation Theory

This paper presents the formal mathematics of the four-gon quiver Q, the quiver on four nodes with six directed edges. Section 1 defines Q and its adjacency matrix. Section 2 derives Q's graph-theoretic properties (strong connectivity, first Betti number β₁ = 3, the three independent directed cycles). Section 3 constructs the path algebra kQ, computes the first five powers of the adjacency matrix, and reports the graded dimensions 4, 6, 8, 12, 18 at lengths 0 through 4. Section 4 specifies the cycle-relation ideal I = ⟨r₁, r₂, r₃⟩ and shows the quotient algebra A = kQ/I is well-defined. Section 5 computes the graded dimensions of A at lengths 0 through 4 and identifies the formalism's transition to representation theory at the convergence vertex T. Section 6 computes the canonical Tits form of Q on the cycle-generator basis, shows it is indefinite with one isotropic generator (γ₃), and shows the braid relation (γ₁γ₂)³ = e_L fails as an identity in A. Section 7 specifies the representation category Rep(A, κ) under the convergence constraint dim(V_T) = 1, proves its indecomposable representations have bounded vertex-space dimensions (Theorem 5.5), and states the classification of its κ-satisfying indecomposables by four dimension vectors with the enumeration argument cited to the companion paper *The Three Indecomposable Representations of the Four-Gon Quiver Under a Convergence Constraint* (Stewart, 2026ae). Section 8 proves the iterated-four-gon (Hamiltonian) count 4ⁿ at depth n, a count of basepoint sequences rather than of walks. Section 9 reports the graded dimensions of kQ and A, proves that dim(kQ_n) is even for every n by exhibiting the period-7 orbit of the all-ones vector under the adjacency matrix modulo 2, derives the rational generating function of the reduced path count R(n) from an order-13 integer recurrence, computes the asymptotic growth rate λ ≈ 1.3163 against the walk-count growth μ ≈ 1.4656, shows Q is of infinite mutation type, proves that the exchange matrix does not distinguish Q from the quiver obtained by deleting both digon arrows, and names three open computations the algebraic program extends to. All results in this paper are stated as definitions, lemmas, theorems, and proofs. The quiver and its algebra are self-contained mathematical objects. A bilinear form B on H₁(Q; ℝ) (Section 6) distinct from the canonical Tits form, with Cartan matrix of A₂ × A₁ type, is constructed by the axiomatic route in *The Endomorphic Collapse Traverses the Foundations of Mathematics* (Stewart, 2026e) and by the pure-math route in *The Four-Gon Quiver, Whose Bilinear Invariants Produce A₂ × A₁* (Stewart, 2026ba), with the two routes compared in *Two Routes to the Bilinear Form on H₁(Q; ℝ)* (Stewart, 2026au). The rank-4 extension of B to a form B̃ via an additional generator γ_τ is constructed in *The Temporal Generator* (Stewart, 2026q). Related representation-theoretic structure and research connections are developed in *The Path Algebra of the Endomorphism* (Stewart, 2026f). **Keywords:** quiver, path algebra, directed graph, bound quiver algebra, representation theory, cycle space, Betti number, strongly connected digraph, iterated Hamiltonian, Tits form, mutation type