2025/08/10 by Mishra, Suraj, Amit Kuber, Kuber, Amit
Mathematics · #16G20 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2508.07435
openalex publication_date 2025/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A rooted tree module (RTM) M:=M(T,F) over a zero-relation algebra Λ:=\mathcal KQ/⟨ρ⟩ over a field \mathcal K is given by the data of a quiver morphism F:T→ Q from a rooted tree T (either with a source or a sink) taking paths in T to paths in Q not lying in ⟨ρ⟩. When char(\mathcal K)≠2, we provide a checkable combinatorial characterization of the indecomposability of the RTM M in terms of non-existence of idempotent quiver morphisms ι:T→ T satisfying F∘ι=F and ι≠ 1T. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs.