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math.RT — Representation Theory

Representation embeddings and the third Brauer--Thrall conjecture

Contributed by Haruhisa Enomoto

We show that every representation-infinite finite-dimensional algebra over an algebraically closed field has an indecomposable module of each infinite dimension. This settles, over algebraically closed fields, Simson's conjecture, also called the third Brauer--Thrall conjecture, that such an algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend a representation embedding of Bongartz from finite-dimensional Kronecker modules without a simple injective summand to a representation embedding of such Kronecker modules of arbitrary dimension, which commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by G\"obel and Simson gives the required modules.

Primarily AI-generated textHuman understanding: some partsBrauer--Thrall conjectureGalois coveringKronecker modulelarge indecomposable modulerepresentation embedding

Oct 10, 2026PDF ↗View work →

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Representation embeddings and the third Brauer--Thrall conjecture

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