A Characterization of Finite EI Categories with Hereditary Category Algebras

In this paper we give an explicit algorithm to construct the ordinary quiver of a finite EI category for which the endomorphism groups of all objects have orders invertible in the field k. We classify all finite EI categories with hereditary category algebras, characterizing them as free EI categories (in a sense which we define) for which all endomorphism groups of objects have invertible orders. Some applications on the representation types of finite EI categories are derived.

💡 Research Summary

This paper investigates finite EI categories—categories in which every endomorphism is an isomorphism—under the arithmetic condition that the order of every automorphism group of an object is invertible in a fixed field k. The authors first develop an explicit, algorithmic construction of the ordinary quiver (the Gabriel quiver) associated with such a category. The algorithm proceeds by selecting a set of representatives for each automorphism group, then for every non‑isomorphic morphism f : X → Y computes the double‑coset orbit Aut(Y) · f · Aut(X). Each orbit contributes a single arrow from the vertex representing the isomorphism class of X to that of Y. Because the group orders are units in k, the orbit counting is purely combinatorial and avoids homological calculations, making the procedure amenable to computer implementation.

Having a concrete quiver, the authors turn to the classification problem for hereditary category algebras. A k‑category algebra k𝒞 is hereditary precisely when its quiver has no oriented cycles and its relations are trivial. The paper introduces the notion of a “free EI category”: a finite EI category in which the morphism sets between distinct objects are free Aut‑sets (i.e., they consist of a disjoint union of copies of the double‑coset action) and there are no non‑trivial compositional relations. The main theorem states that a finite EI category 𝒞 has a hereditary category algebra k𝒞 if and only if 𝒞 is a free EI category and every Aut(X) has order invertible in k. The forward direction is proved by showing that heredity forces the quiver to be a disjoint union of trees, which in turn forces the morphism sets to be freely generated and eliminates any relations. The converse direction verifies directly that a free EI category with invertible group orders yields a path algebra of a tree‑shaped quiver, which is well‑known to be hereditary.

The authors also explore representation‑type consequences. When the underlying quiver of a free EI category is a tree, the category algebra is of finite representation type; otherwise, the presence of multiple arrows or branching yields infinite (often wild) representation type. Several examples illustrate how the invertibility condition prevents the appearance of “non‑invertible” relations that would otherwise destroy heredity.

Finally, the paper discusses limitations and future work. The current classification relies crucially on the invertibility hypothesis; categories where some automorphism groups have orders divisible by the characteristic of k remain outside the scope of the present theory. Extending the framework to partially free EI categories or to categories with controlled relations is identified as a promising direction for further research.

In summary, the work provides a clear algorithm for constructing quivers of finite EI categories with invertible automorphism orders, establishes a precise equivalence between hereditary category algebras and free EI categories under this arithmetic condition, and derives immediate applications to the representation theory of such categories.