The Three Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint

Exactly three dimension vectors carry representations of the quotient algebra A = kQ/I that satisfy the convergence constraint dim(V_T) = 1 and are indecomposable in Rep(A): the full-cycle (1, 1, 1, 1), the contextualized (0, 1, 1, 1), and the bound-oscillation (0, 0, 1, 1). The involution relations are γ₁² = e_L, γ₂² = e_L, γ₃² = e_T. The dimension bound (combining the convergence constraint with the involution relations) restricts every vertex space to dimension at most 1, giving eight candidate dimension vectors with d_T = 1 and d_i ∈ {0, 1}. The enumeration turns on one principle, stated here as Lemma 1.2: a cycle relation γ² = e_v is vacuous when V_v = 0 and binding when dim V_v = 1, because the identity on a zero space is the zero map. Applying it, five candidates are eliminated by involution failure and three survive: (1, 1, 1, 1), (0, 1, 1, 1), and (0, 0, 1, 1). The candidate (1, 0, 1, 1) does not survive, because V_S = 0 forces γ₁ to act as the zero map on a one-dimensional V_L, where the relation γ₁² = e_L is binding and fails. The three are indecomposable in the sense that matters: decomposability is tested in the full category Rep(A) rather than in the constrained class, because the class of representations satisfying dim(V_T) = 1 is not closed under direct sums and therefore admits no internal notion of decomposition. **Keywords:** quiver representation, indecomposable module, dimension vector, path algebra, bound quiver algebra, involution, cycle relations, Krull-Schmidt