The Four Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint
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Abstract
Exactly four 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), the bound-oscillation (0, 0, 1, 1), and (1, 1, 1, 2). The involution relations are γ₁² = e_L, γ₂² = e_L, γ₃² = e_T. The dimension bound (combining the convergence constraint with the involution relations and indecomposability) restricts V_L and V_S to dimension at most 1 and V_C to dimension at most 2, giving twelve candidate dimension vectors with d_T = 1, d_L, d_S ∈ {0, 1}, and d_C ∈ {0, 1, 2}. The enumeration proceeds by 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, six candidates are eliminated by involution failure, two are eliminated because every representation carrying them decomposes, and four survive: (1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), and (1, 1, 1, 2). 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 candidate (1, 1, 1, 2) survives because the two maps leaving C, to L and to T, can be independent functionals on a two-dimensional V_C, and a representation carrying it has endomorphism ring k. The four 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
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BibTeX
@misc{stewart2026four,
title = {{The Four Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint}},
author = {Arthur Stewart},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jul,
doi = {10.5281/zenodo.21644859},
url = {https://doi.org/10.5281/zenodo.21644859},
note = {CC BY-NC-ND},
}
RIS
TY - UNPB
TI - The Four Indecomposable Representations of a Four-Node Quiver Under a Convergence Constraint
AU - Arthur Stewart
PY - 2026
DA - 2026/07/28
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21644859
UR - https://doi.org/10.5281/zenodo.21644859
AB - Exactly four 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), the bound-oscillation (0, 0, 1, 1), and (1, 1, 1, 2). The involution relations are γ₁² = e_L, γ₂² = e_L, γ₃² = e_T. The dimension bound (combining the convergence constraint with the involution relations and indecomposability) restricts V_L and V_S to dimension at most 1 and V_C to dimension at most 2, giving twelve candidate dimension vectors with d_T = 1, d_L, d_S ∈ {0, 1}, and d_C ∈ {0, 1, 2}. The enumeration proceeds by 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, six candidates are eliminated by involution failure, two are eliminated because every representation carrying them decomposes, and four survive: (1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), and (1, 1, 1, 2). 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 candidate (1, 1, 1, 2) survives because the two maps leaving C, to L and to T, can be independent functionals on a two-dimensional V_C, and a representation carrying it has endomorphism ring k. The four 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
ER -