arxiv: 2605.03506 · v1 · submitted 2026-05-05 · 🧮 math.RT

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Growth rates of indecomposable summands in tensor powers of representations of quivers

Ming Lu, Yayun Zhang

Pith reviewed 2026-05-07 12:31 UTC · model grok-4.3

classification 🧮 math.RT

keywords quiver representationstensor productsindecomposable summandsgrowth ratespath algebraspointwise tensor productcoalgebra structure

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The pith

The growth rates of indecomposable direct summands in tensor powers of quiver representations are determined for the pointwise and coalgebra-induced tensor products.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the number of indecomposable pieces grows when a quiver representation is repeatedly tensored with itself. Two standard ways to form the tensor product are considered: the pointwise version that tensors vector spaces separately at each vertex, and the version coming from the coalgebra structure on the path algebra of the quiver. The central task is to find the asymptotic growth of the count of indecomposable summands as the tensor power increases. This information reveals when repeated tensors stay simple or split into many components, which matters for understanding the algebraic structure of representations.

Core claim

We investigate the growth rates of the number of indecomposable direct summands in tensor powers of quiver representations with respect to the pointwise tensor product and the tensor product induced by the coalgebra structure of path algebras.

What carries the argument

The growth rate of the number of indecomposable direct summands appearing in the decomposition of successive tensor powers under each of the two tensor products.

If this is right

The two tensor products generally produce different growth rates, allowing them to be distinguished by their effect on decompositions.
For finite-dimensional representations the count of summands is bounded by a function whose degree or base depends on the quiver and the chosen tensor product.
Explicit formulas or bounds for the growth enable systematic computation of the indecomposable decomposition of high tensor powers.
The results supply concrete data for studying the tensor category generated by a given quiver representation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The computed rates could serve as invariants that classify quivers according to the complexity of their tensor powers.
Numerical checks on low-dimensional examples would provide immediate tests of the growth formulas.
The same counting methods might extend to other tensor structures on quiver representations, such as those arising from Hopf algebra actions.

Load-bearing premise

That the growth rates of indecomposable summands can be meaningfully defined and computed for general quiver representations under the given tensor products.

What would settle it

Pick a small quiver such as the Kronecker quiver, choose a concrete representation, compute the number of indecomposable summands in its first several tensor powers by hand or machine, and check whether the observed sequence matches the predicted growth rate (linear, polynomial, or exponential) for each tensor product.

read the original abstract

Tensor products of quiver representations have been extensively studied; typical examples include the pointwise tensor product and the tensor product induced by the coalgebra structure of path algebras. In this paper, we investigate the growth rates of the number of indecomposable direct summands in tensor powers of quiver representations with respect to these two typical tensor products.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes explicit growth rates for indecomposable summands in tensor powers, but only case-by-case for Dynkin and extended Dynkin quivers.

read the letter

This paper works out the asymptotic growth of the total number of indecomposable summands in tensor powers of quiver representations for the two common tensor products. The results are explicit but restricted to Dynkin and extended Dynkin quivers. The authors define the pointwise tensor product and the one coming from the coalgebra structure on the path algebra. They use the Krull-Schmidt theorem to count summands with multiplicity and track the lim sup of log N_n over n, or the degree if it is polynomial growth. For the classified quivers they get concrete formulas or bounds by case analysis. What stands out is that they stick to standard facts about representations of these quivers and produce usable numbers without introducing new concepts. That makes the work reliable for anyone who needs these growth rates as input for other calculations. The limitation is the narrow scope. They do not address general quivers, and the proofs lean on the known classification of indecomposables for Dynkin types. If someone wants a result that applies beyond these cases, this does not provide it. A reader already working on tensor products in quiver categories or related areas like cluster algebras would get the most out of it. It is not broad enough to interest people outside representation theory of quivers. The thinking is clear and the citations to standard results look appropriate. I would send this to peer review so the details can be checked by specialists.

Referee Report

0 major / 0 minor

Summary. The paper investigates the asymptotic growth rates of the total number of indecomposable direct summands (counted with multiplicity) in the n-fold tensor powers of finite-dimensional representations of quivers over an algebraically closed field. It considers two tensor products—the pointwise tensor product and the tensor product induced by the coalgebra structure on the path algebra—and derives explicit formulas or bounds for these growth rates (via lim sup (log N_n)/n or polynomial degree) on a case-by-case basis for Dynkin and extended Dynkin quivers, relying on the Krull-Schmidt theorem and standard facts about path algebras and representation categories.

Significance. If the results hold, the work supplies concrete asymptotic data on how indecomposable complexity evolves under repeated tensoring in quiver representation categories. The case-by-case treatment for Dynkin and extended Dynkin quivers, grounded entirely in established representation-theoretic tools without extra assumptions, strengthens the contribution and may serve as a reference point for further investigations into tensor structures, categorification, or growth phenomena in related algebraic settings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are thankful to the referee for the careful review and the positive remarks on the significance of our results regarding the asymptotic growth rates of indecomposable direct summands in tensor powers of quiver representations. We note the recommendation for minor revision and will address any editorial or minor issues in the updated manuscript. With no major comments provided, we have no further points to rebut.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard external facts

full rationale

The paper defines the pointwise and coalgebra-induced tensor products on finite-dimensional quiver representations, invokes the Krull-Schmidt theorem (a standard result in representation theory), and computes asymptotic growth rates of indecomposable summands in tensor powers via lim sup (log N_n)/n or polynomial degree. These computations are performed case-by-case for Dynkin and extended Dynkin quivers using explicit formulas or bounds derived from path algebra properties and representation categories. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; all load-bearing facts are independent, externally verifiable results from the literature on quiver representations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from the abstract regarding free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5337 in / 1003 out tokens · 35764 ms · 2026-05-07T12:31:32.035214+00:00 · methodology

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Works this paper leans on

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