Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.07435 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915438043070464 |
|---|---|
| author | Mishra, Suraj Kuber, Amit |
| author_facet | Mishra, Suraj Kuber, Amit |
| contents | A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $Λ:=\mathcal KQ/\langleρ\rangle$ over a field $\mathcal K$ is given by the data of a quiver morphism $F:T\to Q$ from a rooted tree $T$ (either with a source or a sink) taking paths in $T$ to paths in $Q$ not lying in $\langleρ\rangle$. When $\mathrm{char}(\mathcal K)\neq2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms $ι:T\to T$ satisfying $F\circι=F$ and $ι\neq 1_T$. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rooted tree modules Mishra, Suraj Kuber, Amit Representation Theory 16G20 A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $Λ:=\mathcal KQ/\langleρ\rangle$ over a field $\mathcal K$ is given by the data of a quiver morphism $F:T\to Q$ from a rooted tree $T$ (either with a source or a sink) taking paths in $T$ to paths in $Q$ not lying in $\langleρ\rangle$. When $\mathrm{char}(\mathcal K)\neq2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms $ι:T\to T$ satisfying $F\circι=F$ and $ι\neq 1_T$. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs. |
| title | Rooted tree modules |
| topic | Representation Theory 16G20 |
| url | https://arxiv.org/abs/2508.07435 |