Rooted Tree Modules in Zero-Relation Algebras

• RTMs are modules defined by a finite rooted tree and a bound quiver morphism that maps paths outside the zero-relations, bridging combinatorics and algebra.
• The construction employs a push-down functor on a one-dimensional tree representation, ensuring the module’s dimensions correspond to the preimage sizes of the target quiver.
• Indecomposability is characterized by the nonexistence of non-identity idempotent self-maps on the tree, providing a clear combinatorial criterion for module folding.

A rooted tree module (RTM) is, in the formal sense introduced for zero-relation algebras, a module $M:=M(T,F)$ determined by a rooted tree $T$ and a quiver morphism $F:T\to Q$, where $\Lambda=\mathcal KQ/\langle \rho\rangle$ is a zero-relation algebra and $F$ sends paths in $T$ to paths in $Q$ not lying in $\langle \rho\rangle$ (Mishra et al., 10 Aug 2025). The construction converts module-theoretic questions into rooted-tree combinatorics: the tree provides the support, the morphism records how that support sits over the bound quiver, and indecomposability can be characterized by the absence of certain idempotent self-maps of the rooted tree when $\operatorname{char}(\mathcal K)\neq 2$ (Mishra et al., 10 Aug 2025).

1. Definition and algebraic setting

The ambient algebra is a zero-relation algebra

$\Lambda=\mathcal KQ/\langle \rho\rangle,$

where $T$0 is a quiver and $T$1 is a set of paths of length at least $T$2 (Mishra et al., 10 Aug 2025). The paper assumes that $T$3 is locally bound, and it uses the standard equivalence between finite-dimensional $T$4-modules and finite-dimensional $T$5-representations of the bound quiver $T$6 (Mishra et al., 10 Aug 2025).

A rooted tree is a finite quiver

$T$7

whose underlying undirected graph is simply connected and which has a unique sink or a unique source, denoted $T$8 (Mishra et al., 10 Aug 2025). The theory is developed in parallel for the sink case and the source case. A quiver morphism

$T$9

is a pair of maps $F:T\to Q$0, $F:T\to Q$1, preserving sources and targets: $F:T\to Q$2 It is a bound quiver morphism when there is no path in $F:T\to Q$3 such that $F:T\to Q$4 (Mishra et al., 10 Aug 2025). Equivalently, every path in $F:T\to Q$5 maps to a path in $F:T\to Q$6 that is not killed by the zero relations.

With these data, $F:T\to Q$7 is a rooted tree over the locally bound quiver $F:T\to Q$8, and the corresponding module $F:T\to Q$9 is called a rooted tree module (Mishra et al., 10 Aug 2025). The definition is specific to this representation-theoretic setting. A common misconception is to equate RTMs with earlier “tree modules” in the hereditary-quiver sense; that older notion is different and is based on coefficient quivers rather than on a rooted tree equipped with a quiver morphism (Weist, 2010).

2. Construction of $\Lambda=\mathcal KQ/\langle \rho\rangle$0

The module is obtained by pushing down a canonical representation of the rooted tree. First one forms the $\Lambda=\mathcal KQ/\langle \rho\rangle$1-module $\Lambda=\mathcal KQ/\langle \rho\rangle$2, which places the one-dimensional vector space $\Lambda=\mathcal KQ/\langle \rho\rangle$3 at every vertex $\Lambda=\mathcal KQ/\langle \rho\rangle$4 and the identity map $\Lambda=\mathcal KQ/\langle \rho\rangle$5 on every arrow $\Lambda=\mathcal KQ/\langle \rho\rangle$6 (Mishra et al., 10 Aug 2025). Thus

$\Lambda=\mathcal KQ/\langle \rho\rangle$7

The bound quiver morphism $\Lambda=\mathcal KQ/\langle \rho\rangle$8 induces a push-down functor

$\Lambda=\mathcal KQ/\langle \rho\rangle$9

For a representation $F$0, the push-down is defined by

$F$1

The rooted tree module is then

$F$2

This is also called a generalized tree module when $F$3 is not explicitly emphasized as rooted; in the rooted case the paper uses the term RTM (Mishra et al., 10 Aug 2025).

The resulting representation has a particularly concrete form. At a vertex $F$4,

$F$5

so

$F$6

If $F$7 denotes the basis vector corresponding to $F$8, then the basis of $F$9 is given by all $T$0 with $T$1 (Mishra et al., 10 Aug 2025). For an arrow $T$2, the linear map

$T$3

is the sum of identity maps over all tree arrows mapping to $T$4: $T$5 In the sink case, if $T$6 and $T$7, then

$T$8

The module therefore arises by collapsing the one-dimensional tree representation along the fibers of $T$9 (Mishra et al., 10 Aug 2025).

This description shows why the zero-relation condition is essential. If some path in $Q$0 mapped into $Q$1, the induced action would fail to factor through $Q$2. The admissibility of $Q$3 is therefore part of the definition, not merely a technical convenience (Mishra et al., 10 Aug 2025).

3. Indecomposability and combinatorial idempotents

The principal structural result is an indecomposability criterion stated in explicitly combinatorial terms. When $Q$4, the paper proves that for a rooted tree $Q$5 over $Q$6, the following are equivalent: $Q$7 is indecomposable, and there is no non-identity idempotent quiver morphism

$Q$8

satisfying

$Q$9

(Mishra et al., 10 Aug 2025). Thus indecomposability is detected by the non-existence of a rooted-tree self-folding that is invisible after projection to the bound quiver.

In the sink case, the theorem is strengthened by an equivalent formulation in terms of generalized graph maps (GGMs). There is no GGM $\langle \rho\rangle$0 with $\langle \rho\rangle$1 satisfying

$\langle \rho\rangle$2

This condition says that two distinct branches with the same parent and the same root-arrow image under $\langle \rho\rangle$3 must not become identified by the GGM apparatus (Mishra et al., 10 Aug 2025).

The proof passes through pullback networks $\langle \rho\rangle$4, their signed $\langle \rho\rangle$5-covering networks $\langle \rho\rangle$6, completeness and $\langle \rho\rangle$7-freeness conditions, and the notion of ghost-freeness. For pairs of RTMs with sinks, and dually for pairs with sources, the paper proves ghost-freeness and then obtains

$\langle \rho\rangle$8

(Mishra et al., 10 Aug 2025). This identifies homomorphisms with combinatorial correspondences between rooted trees.

The idempotent criterion is sharp because an idempotent quiver morphism produces an idempotent module endomorphism. In the sink case,

$\langle \rho\rangle$9

defines an idempotent in $\operatorname{char}(\mathcal K)\neq 2$0; in the source case the induced endomorphism is

$\operatorname{char}(\mathcal K)\neq 2$1

Hence a nontrivial idempotent $\operatorname{char}(\mathcal K)\neq 2$2 forces decomposability, exactly as in the standard criterion that a finite-dimensional module is indecomposable if and only if its endomorphism ring has no idempotents except $\operatorname{char}(\mathcal K)\neq 2$3 and $\operatorname{char}(\mathcal K)\neq 2$4 (Mishra et al., 10 Aug 2025).

4. Recursive decomposition and recursive construction

The criterion immediately yields a recursive decomposition procedure. In the sink case, if a non-identity idempotent $\operatorname{char}(\mathcal K)\neq 2$5 with $\operatorname{char}(\mathcal K)\neq 2$6 exists, the paper defines the fixed-point set

$\operatorname{char}(\mathcal K)\neq 2$7

This is a rooted subtree. If $\operatorname{char}(\mathcal K)\neq 2$8 are the connected components of $\operatorname{char}(\mathcal K)\neq 2$9, then

$\Lambda=\mathcal KQ/\langle \rho\rangle,$0

In the source case the analogous role is played by

$\Lambda=\mathcal KQ/\langle \rho\rangle,$1

and the decomposition is obtained from the connected components of $\Lambda=\mathcal KQ/\langle \rho\rangle,$2 (Mishra et al., 10 Aug 2025). Iterating this process decomposes an RTM into indecomposable RTMs.

The paper also gives a branchwise criterion well suited to recursive construction. In the sink case, let $\Lambda=\mathcal KQ/\langle \rho\rangle,$3 be the vertices adjacent to the root, so $\Lambda=\mathcal KQ/\langle \rho\rangle,$4, and set

$\Lambda=\mathcal KQ/\langle \rho\rangle,$5

Assume each $\Lambda=\mathcal KQ/\langle \rho\rangle,$6 is indecomposable. Then $\Lambda=\mathcal KQ/\langle \rho\rangle,$7 is decomposable if and only if there exist $\Lambda=\mathcal KQ/\langle \rho\rangle,$8 such that

$\Lambda=\mathcal KQ/\langle \rho\rangle,$9

and there is a quiver morphism

$T$00

with

$T$01

The source version is dual (Mishra et al., 10 Aug 2025).

This gives a recursive synthesis principle. One starts with indecomposable branch RTMs and attaches them to a new root; the resulting RTM remains indecomposable precisely when no branch can be folded into another branch with the same $T$02-label at the root edge. The paper’s examples make this concrete. In one sink-rooted example, the idempotent

$T$03

satisfies $T$04 and yields

$T$05

where $T$06 comes from the fixed subtree $T$07 and $T$08 is the simple RTM on the singleton vertex $T$09 (Mishra et al., 10 Aug 2025).

5. Adjacent notions and non-equivalent usages

The expression “rooted tree module” is recent and specific. Earlier literature contains several rooted-tree-based constructions that are related in spirit but not identical in definition.

Direction	Core object	Relation to RTM
Tree modules (Weist, 2010)	A quiver representation whose coefficient quiver is a tree	Foundational background, but not a rooted tree module
Rooted tree maps (Bachmann et al., 2017)	$T$10-linear endomorphisms of $T$11 indexed by rooted forests	Reasonably interpreted as a module/action viewpoint, but not called RTM
Space of rooted tree maps (Murahara et al., 2024)	The graded vector space $T$12 with explicit basis and relations	Algebraically close to an RTM interpretation, but a different object
Principal $T$13-module of a rooted tree (Li et al., 2019)	$T$14 for the Terwilliger algebra at the root	A module-theoretic rooted-tree invariant, not the RTM $T$15
Assigned rational functions on rooted trees (Damnjanović, 2022)	Canonical rational-function summaries $T$16 of rooted subtrees	Explicitly RTM-like as a compositional rooted-tree summary, but not a module over a zero-relation algebra

The classical tree-module paper studies finite-dimensional representations of an acyclic quiver and calls a representation a tree module when some coefficient quiver $T$17 is a tree (Weist, 2010). That notion is historically important, but it does not use a rooted tree $T$18 mapping into a bound quiver.

The rooted-tree-map literature constructs operators on $T$19 indexed by rooted forests. These papers explicitly note that they do not define an object called a rooted tree module; the module viewpoint is interpretive, arising from the action of the Connes–Kreimer Hopf algebra on $T$20 (Tanaka, 2017). Later work identifies a basis for the space of rooted tree maps and proves an infinite family of relations inside that space, again without switching to the representation-theoretic RTM formalism (Murahara et al., 2024).

A different module-theoretic rooted-tree encoding appears in the Terwilliger-algebra setting. For a finite rooted tree $T$21, the principal module

$T$22

is irreducible and recognizes the rooted-tree isomorphism class; the rooted-subtree combinatorics are recovered from depth projections and subtree-type projectors in the Terwilliger algebra (Li et al., 2019). This is genuinely module-theoretic, but it is not the same as $T$23.

Finally, the spectral paper on assigned rational functions gives a bottom-up, subtree-composable summary

$T$24

and shows that whole-tree characteristic polynomials factor as products of these local summaries (Damnjanović, 2022). This is explicitly described as “very RTM-like” in the supplied analysis, yet it remains a recursive rooted-tree calculus rather than a module over $T$25.

6. Scope, assumptions, and significance

The present RTM theory is subject to several explicit assumptions. The rooted tree $T$26 is finite; the target algebra is a zero-relation algebra $T$27; the quiver $T$28 is locally bound; and $T$29 must be a bound quiver morphism, meaning that no path in $T$30 maps into $T$31 (Mishra et al., 10 Aug 2025). The root may be a unique sink or a unique source, and the two cases are handled dually.

The restriction

$T$32

is essential in the indecomposability theory. The proof uses the $T$33-covering network $T$34, whose vertices carry signs $T$35, and if $T$36, then $T$37, so $T$38 and the signed machinery collapses (Mishra et al., 10 Aug 2025). The paper does not claim that its criterion extends to characteristic $T$39.

There is also a textual subtlety in the source-version theorem: the detailed exposition notes that Section 4 prints condition (2) as the non-existence of a non-identity quiver morphism $T$40 with $T$41, whereas the abstract, introduction, and proof show that the relevant notion is again a non-identity idempotent quiver morphism (Mishra et al., 10 Aug 2025). This does not alter the conceptual content, but it matters for precise formulation.

Within its intended scope, the significance of the RTM notion is that it packages a $T$42-module into finite combinatorial data $T$43 and converts indecomposability, decomposition, and recursive construction into rooted-tree questions. The fundamental design pattern is simple: start from the canonical one-dimensional representation $T$44, push it down along $T$45, and study whether the resulting module can be folded by an idempotent self-map of the rooted support tree. In that sense, the RTM formalism makes rooted combinatorics a first-class language for the representation theory of zero-relation algebras (Mishra et al., 10 Aug 2025).