ON THE REPRESENTATION TYPES OF CATEGORY ALGEBRAS OF FINITE EI CATEGORIES

LIPING LI

Abstract. A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They include finite groups, finite posets and free categories of finite quivers as special cases. In this paper we consider the representation types of finite EI categories with two or three objects, describe some criteria for finite representation type, and use them to classify the representation types of the category algebras of several classes of finite EI categories with extra properties.

1. Introduction

Finite EI categories are small categories with finitely many morphisms such that every endomorphism is an isomorphism. They are studied in different areas as topology ([10, 19]), representation theory ([11, 12, 15, 16, 24, 25, 27]), homological algebra ([23, 26, 28]), and fusion systems ([2, 17]). Interesting examples include finite groups (categories with one object), finite posets, and free categories of finite quivers.

For a fixed algebraically closed field k, a finite EI category C determines a finitedimensional k-algebra kC called the category algebra. It is said to be of finite representation type if kC-mod has only finitely many indecomposable objects up to isomorphism. Otherwise, it is said to be of infinite representation type. In this paper we mainly study the representation types of category algebras kC, where C has only two or three objects. This problem has been considered in Dietrich’s thesis ([11, 12]) and some useful criteria of infinite representation type have been acquired.

Let C be a connected skeletal finite EI category with two objects. In the case that the automorphism groups of both objects have orders invertible in k, we can construct the ordinary quiver Q (which is an acyclic finite quiver) of kC using the algorithm described in [15], and kC is Morita equivalent to the hereditary algebra kQ. Therefore, its representation type can be determined by Gabriel’s theorem. For the situation that the automorphism groups of both objects are p-groups, where p = char(k), the classification has been obtained in [12] by applying Bongartz’s list described in [4, 6].

2000 Mathematics Subject Classification. 16G10, 16G20, 16G60.

Key words and phrases. finite EI categories, category algebras, representation types.

The author would like to thank his thesis advisor, Professor Peter Webb, for the proposal to study this problem, and for many invaluable suggestions in numerous discussions. He wants to express appreciation to Professor Edward Green, Professor Markus Schmidmeier and Dr. Ryan Kinser, from whom the author learned many useful techniques applied in this paper. He also thanks the referee whose suggestions make this paper much more concise.

When both automorphism groups in C are arbitrary, the situation is much more complicated. We introduce some notation here. Let C be pictured as below, where G = C(x, x) and H = C(y, y). Using some reduction we can show that if kC is of finite representation type, then C(x, y) has only one orbit as an (H, G)-biset. Thus we let α be a fixed morphism in C(x, y). Define G_{0} = Stab_{G}(α), G_{1} = Stab_{G}(Hα), H_{0} = Stab_{H}(α), and H_{1} = Stab_{H}(αG). Then G_{0} ◁ G_{1} ⩽ G and H_{0} ◁ G_{1} ⩽ H. Moreover, G_{1}/G_{0} ≅ H_{1}/H_{0} as groups.

[figure: C :; x; G; HαG; ...; y; H].

We characterize representations of C explicitly by linear maps satisfying some property related to representations of the above groups. Therefore, many techniques from group representation theory (for example, induction and restriction, permutation modules, block theory, Brauer graphs, and etc.) can be applied to exploit the representation type of kC. We get the following criteria:

Theorem 1.1. Let C be as above and p = char(k) ⩾ 0. If kC is of finite representation type, then the following conditions must be satisfied:

(1) Both G and H have cyclic Sylow p-subgroups.

(2) Either G or H acts transitively on C(x, y), i.e., C(x, y) equals Hα or αG.

(3) If p ⩾ 5, then either O^{p′}G (the subgroup generated by all Sylow p-subgroups in G) or O^{p′}H acts trivially on C(x, y). Moreover, for every p-subgroup P ⩽ G (resp., Q ⩽ H), P ∩ G_{0} is either 1 or P (resp., Q ∩ H_{0} is either 1 or Q).

Consequently, in many cases the representation type of kC can be determined from two pieces of information: the transitivity of actions by G and H, and the triviality of actions by Sylow p-subgroups in G and H. Indeed, the combination of conditions (a-c)

(a) Both G and H act transitively.

(b) One of G and H acts transitively, and the other one does not.

(c) Neither G nor H acts transitively.

and conditions (1-3)

(1) Both O^{p′}G and O^{p′}H act trivially.

(2) One of O^{p′}G and O^{p′}H acts trivially, and the other one does not.

(3) Neither O^{p′}G nor O^{p′}H acts trivially.

gives us 8 situations (the combination (a)+(2) cannot happen). By Theorem 1.1, if char(k) = p ≠ 2, 3, we get infinite representation type of kC for five cases: (c)+(1), (c)+(2), (c)+(3), (a)+(3), (b)+(3). The following theorem tells us the finite representation type for case (a)+(1).

Theorem 1.2. Let C be as in the previous theorem, and suppose that both G and H act transitively on C(x, y). If char(k) = p ≠ 2, 3, then kC is of finite representation type if and only if both O^{p′}G and O^{p′}H act trivially on C(x, y).

Although cases (b)+(1) and (b)+(2) are not completely resolved here, by considering the structure of certain permutation modules we are able to obtain the following criteria.

Theorem 1.3. Let C be as in Theorem 1.1 and suppose without loss of generality that H acts transitively on C(x, y). If kC is of finite representation type, then the following conditions must be satisfied:

(1) dim_{k} End_{kH}(k ↑_{H_{1}}^{H}) = |H_{1}\H/H_{1}| ⩽ 3;

(2) all summands of Top(k ↑_{H_{1}}^{H}) are not isomorphic;

(3) If M, N are indecomposable summands of k ↑_{H_{1}}^{H}, then Hom_{kH}(M, N) = 0.

We classify the representation type of kC for p ⩾ 5 when both G and H are abelian.

Theorem 1.4. Let C be as in Theorem 1.1 and char(k) = p ≠ 2, 3. Without loss of generality assume that H acts transitively on C(x, y). Let s = |O^{p′}G|, t = |O^{p′}H|, and n = |H : H_{1}|. If both G and H are abelian, then kC is of finite representation type if and only if both O^{p′}G and O^{p′}H act trivially on C(x, y), and one of the following conditions holds:

(1) n = 1 for s, t ⩾ p, i.e., H = H_{1};

(2) n ⩽ 2 for s = 1, t ⩾ p or t = 1, s ⩾ p;

(3) n ⩽ 3 for t = s = 1.

This paper is organized as follows. Some background knowledge and preliminary reductions on finite EI categories and their representations are included in the second section. In Section 3 we study in details decomposition of bisets and actions of groups on these bisets, and develop criteria contained in Theorems 1.1 and 1.3. Using these results, we determine the representation types of category algebras of most finite EI categories with two objects in Section 4. In the last section we consider some special finite EI categories with three objects.

Throughout this paper k is an algebraically closed field and k-vec is the category of finite-dimensional k-vector spaces. For a finite EI category C, by Ob C and Mor C we denote sets of objects and morphisms in C respectively; C(x, y) denotes the set of morphisms from x to y; and by C-rep (or kC-mod) we denote the category of representations of C. For a module M, Top(M) and Soc(M) mean the top and the socle of M respectively. All modules we consider are finitely generated left modules. Composition of morphisms and maps is from right to left. To deal with all characteristics in a uniform way, when p = 0, we view the trivial group 1 as the Sylow 0-subgroup of a finite group.

2. Preliminaries

For the reader’s convenience, we include in this section some background on the representation theory of finite EI categories. Please refer to [24, 25, 26, 27] for more details.

A finite EI category C is a small category with finitely many morphisms such that every endomorphism in C is an isomorphism. Examples of finite EI categories include finite groups (viewed as categories with one object), finite posets (all endomorphisms are identities), and orbit categories (see [25]). The category C is connected if for any two distinct objects x and y, there is a list of objects x = x_{0}, x_{1}, …, x_{n} = y such that either C(x_{i}, x_{i+1}) or C(x_{i+1}, x_{i}) is not empty, 0 ⩽ i ⩽ n − 1. Every finite EI category is a disjoint union of connected components, and each component as a full subcategory is again a finite EI category.

A representation of C is a covariant functor R : C → k-vec. The functor R assigns a vector space R(x) to each object x in C, and a linear transformation R(α) : R(x) → R(y) to each morphism α : x → y such that all composition relations of morphisms in C are preserved under R. A homomorphism φ : R_{1} → R_{2} of two representations is a natural transformation of functors.

A finite EI category C determines a finite-dimensional k-algebra kC called the category algebra. It has a basis constituted of all morphisms in C, and the multiplication is defined by the composition of morphisms (the composite is 0 when two morphisms cannot be composed) and bilinearity. By Theorem 7.1 of [21], a representation of C is equivalent to a kC-module. Thus we do not distinguish these two concepts throughout this paper.

By Proposition 2.2 in [24], if C and D are equivalent finite EI categories, kC-mod is Morita equivalent to kD-mod. Moreover, if C = ⨆_{i=1}^{m} C_{i} is a disjoint union of several full subcategories, then kC = kC_{1} ⊕ … ⊕ kC_{m} as algebras. Thus it is sufficient to study the representation types of connected, skeletal finite EI categories. We make the following convention:

Convention: All finite EI categories in this paper are connected and skeletal. Thus endomorphisms, isomorphisms and automorphisms in a finite EI category coincide.

Under this hypothesis, if x and y are two distinct objects in C with C(x, y) ≠ ∅, then C(y, x) is empty. Indeed, if this is not true, we can take α ∈ C(y, x) and β ∈ C(x, y). The composite βα is an endomorphism of y, hence an automorphism. Similarly, the composite αβ is an automorphism of x. Thus both α and β are isomorphisms, so x is isomorphic to y. But this is impossible since C is skeletal and x ≠ y.

We give here some illustrative examples showing that the representation theory of finite EI categories has applications in relevant fields such as representation theory of finite groups.

Example 2.1. Let C be the following finite EI category such that: C(x, x) ≅ C(y, y) ≅ G which is a finite group; C(x, y) ≅ G on which C(x, x) acts by multiplication from the right side and C(y, y) acts by multiplication from the left side.

[figure: C :; x; G; _{G}G_{G}; ...; y; G]

Then a representation C is nothing but a kG-module homomorphism φ : M → N.

Define another finite EI category D as below, where: D(x, x) ≅ D(y, y) ≅ ⟨g⟩ is a cyclic group of order p = char(k); D(x, y) = {α} on which both automorphism groups act trivially.

[figure: D :; x; ⟨g⟩; α; y; ⟨g⟩]

A representation of D is a k⟨g⟩-module homomorphism φ : M → N such that φ(Top(M)) ⊆ Soc(N).

Let F : D → C be a functor between two finite EI categories. Then F induces a functor Res_{F} : C-rep → D-rep. Explicitly, for M ∈ C-rep, Res_{F} (M) = M ∘ F is a functor from D to k-vec. This functor Res_{F} is called the restriction along F (Definition 3.2.1 in [26]). In the case that F is injective on Ob D, it induces an algebra homomorphism φ_{F} : kD → kC by sending α ∈ Mor D to F (α) ∈ Mor C (Proposition 3.1, [24]). In that case, the restriction functor Res_{F} is precisely the usual restriction functor from kC-mod to kD-mod induced by the algebra homomorphism φ_{F}.

An important case is that D is a subcategory of C and F is the inclusion functor. Then kD is a subalgebra of kC and φ_{F} is the inclusion. In this case we define the induction functor ↑_{D}^{C} mapping N ∈ kD-mod to kC ⊗_{kD} N ∈ kC-mod. The co-induction functor ⇑_{D}^{C} sends N to N ⇑_{D}^{C}= Hom_{kC}(kC, N).

In his thesis Xu has developed an induction-restriction theory for finite EI categories. We do not repeat his results here and suggest the reader to look at [26, 27] for a detailed description. Although many results in the representation theory of finite groups generalize to finite EI categories, there exist differences, as shown by the following example.

Example 2.2. Let C be the following EI category such that g has order 2 and permutes α and β. The category algebra kC has finite representation type.

[figure: C :; x; 1; α; β; y; ⟨g⟩]

Let D be the subcategory formed by deleting the morphism g. Actually, D viewed as a quiver is the Kronecker quiver. Therefore, kD is of infinite representation type.

Suppose that char(k) = 0. Then R : \xymatrix{k \ar[r]^{1} \ar[r]_{3} & k} is a representation of D. Clearly,

dim_{k} R = 2. Let {u, v} be a basis of R such that α · u = v and β · u = 3v. Consider the induced module R̃ : kC ⊗_{kD} R. We have

g ⊗ v = g ⊗ α · u = gα ⊗ u = β ⊗ u = 1_{y} ⊗ β · u = 1_{y} ⊗ 3v = 3(1_{y} ⊗ v).

Therefore,

1_{y} ⊗ v = g^{2} ⊗ v = g(g ⊗ v) = g(3(1_{y} ⊗ v)) = 3(g ⊗ v) = 9(1_{y} ⊗ v).

Since char(k) = 0, we get 1_{y} ⊗ v = 0. Consequently,

α ⊗ u = 1_{y} ⊗ α · u = 1_{y} ⊗ v = 0

and similarly β ⊗ u = 0. Therefore, R̃ is spanned by 1_{x} ⊗ u, so is one-dimensional! Actually, it has the form R̃(x) ≅ k → R̃(y) = 0.

The following result relates the representation type of kC to that of the category algebra of a subcategory.

Proposition 2.3. Let D be a subcategory of a finite EI category C.

(1) If M | M ↓_{D}^{C}↑_{D}^{C} for all M ∈ kC-mod, then kC is of finite representation type whenever so is kD.

(2) Dually, if N | N ↑_{D}^{C}↓_{D}^{C} for all N ∈ kD-mod, then kC is of infinite representation type whenever so is kD.

(3) If kD | (_{kD}kC_{kD}), then N | N ↑_{D}^{C}↓_{D}^{C} for every N ∈ kD-mod. In particular, if kD is of infinite representation type, so is kC.

Proof. Note that kD is a subalgebra of kC. The conclusions follows from well known results of induction and restriction procedure. □

If furthermore D is a full subcategory of C, then kD = 1_{D}kC1_{D}, and the restriction functor Res_{F} sends M ∈ kC-mod to M ↓_{D}^{C}= 1_{D}M. It has a left adjoint functor ↑_{D}^{C} and a right adjoint functor ⇑_{D}^{C}. In this case, for every kD-module N, we have N ↑_{D}^{C}↓_{D}^{C}≅ N, see [26]. This immediately implies:

Proposition 2.4. Let D be a full subcategory of a finite EI category C. If kD is of infinite representation type, so is kC.

Now we consider another type of important functors G : C → D between two finite EI categories, called quotient functors. That is, G is a full functor and is bijective restricted to objects. Correspondingly, we call D a quotient category of C. By Proposition 3.1 in [24], G induces an algebra homomorphism φ_{G} : kC → kD. Moreover, this homomorphism is surjective, so kD is a quotient algebra of kC, and hence its infinite representation type implies the infinite representation type of kC. Namely we proved

Proposition 2.5. Let D be a quotient category of a finite EI category C. If kC is of finite representation type, so is kD.

Applying these preliminary results, we obtain the following elementary criteria for finite representation type. Recall the underlying poset P of a finite EI category C is defined as follows: elements in P coincide with objects in C; for two objects x, y, we define x ⩽ y if C(x, y) ≠ ∅. The reader can check that P is indeed a poset when C is skeletal.

Proposition 2.6. Let C be a finite EI category such that kC is of finite representation type. Then the following conditions must be true:

(1) The group C(x, x) is of finite representation type for each x ∈ Ob C.

(2) The underlying poset P of C is of finite representation type.

(3) For x ≠ y ∈ Ob C, C(x, y) has at most one orbit as a (C(y, y), C(x, x))-biset.

(4) If x, y, z ∈ Ob(C) are distinct such that C(x, y) and C(y, z) are non-empty, then C(x, z) = C(y, z) ∘ C(x, y) = {β ∘ α | α ∈ C(x, y), β ∈ C(y, z)}.

Proof. (1): Apply Proposition 2.4 to the full subcategory formed by x.

(2): There is a quotient functor F : C → P. It is the identity map restricted to Ob C, and sends every morphism (if exists) in C(x, y) to the unique morphism P(x, y) for x, y ∈ Ob C. The conclusion follows from Proposition 2.5.

(3): Consider the full subcategory D with objects x and y. Let O_{1}, O_{2}, …, O_{n} be the (D(y, y), D(x, x))-orbits of D(x, y) = C(x, y). Take a representative α_{i} for each orbit O_{i}, 1 ⩽ i ⩽ n. Let E be the following category: Ob E = {x, y} and Mor E = {1_{x}, 1_{y}, α_{1}, …, α_{n}}. We claim that E is a quotient category of D.

Indeed, define G : D → E as follows. Restricted to Ob D, G is the identity map; G(δ) = 1_{x} for δ ∈ D(x, x); G(ρ) = 1_{y} for ρ ∈ D(y, y); and G(α) = α_{i} for α ∈ O_{i} ⊆ D(x, y), 1 ⩽ i ⩽ n. We check that G is well defined. Therefore E is a quotient category of D, so kE must be of finite representation type. But this happens if and only if n ⩽ 1.

(4): Clearly, C(y, z) ∘ C(x, y) ⊆ C(x, z). We show the other inclusion by contradiction. Suppose that C(y, z) ∘ C(x, y) is a proper subset of C(x, z). We can take φ, ψ ∈ C(x, z) such that φ ∈ C(y, z) ∘ C(x, y) and ψ ∉ C(y, z) ∘ C(x, y). Then the (C(z, z), C(x, x))-orbit generated by φ is contained in C(y, z) ∘ C(x, y). So φ and ψ are in different orbits. Therefore, C(x, z) as a (C(z, z), C(x, x))-biset has more than one orbits. By (3), kC is of infinite representation type. The conclusion follows from contradiction. □

In the rest of this paper we only consider finite EI categories C satisfying all conditions in Proposition 2.6 since otherwise we can immediately conclude that kC is of infinite representation type.

3. Criteria of finite representation type

Throughout this section let C be a (connected, skeletal) finite EI category with two objects x and y. Without loss of generality we assume that C(x, y) ≠ ∅, but C(y, x) = ∅. Conditions (2) and (4) in Proposition 2.6 are satisfied trivially. We suppose that conditions (1) and (3) also hold. That is, both C(x, x) and C(y, y) have cyclic Sylow p-subgroups, and C(x, y) has only one orbit as a biset.

Let G = C(x, x) and H = C(y, y). Fix a representative morphism α : x → y in the unique orbit. Then the structure of C can be pictured as below:

[figure: C :; x; G; HαG; ...; y; H]

The following notation will be used in this section: G_{0} = Stab_{G}(α), H_{0} = Stab_{H}(α), G_{1} = Stab_{G}(Hα), and H_{1} = Stab_{H}(αG). By Lemma 3.3 in [15], G_{0} ◁ G_{1} ⩽ G, H_{0} ◁ H_{1} ⩽ H, and G_{1}/G_{0} ≅ H_{1}/H_{0}. We identify these two quotient groups. Clearly, G acts transitively on C(x, y) if and only if H_{1} = H, and H acts transitively on C(x, y) if and only if G_{1} = G.

3.1. A description of representations. Let R be a representation of C. It is easy to check that R(x) (resp., R(y)) is a kG-module (resp., a kH-module). Moreover, suppose that (φ_{x}, φ_{y}) : R_{1} → R_{2} is a kC-module homomorphism, then φ_{x} : R_{1}(x) → R_{2}(x) is a kG-module homomorphism, and φ_{y} : R_{1}(y) → R_{2}(y) is a kH-module homomorphism. For an arbitrary β ∈ C(x, y), since C(x, y) has only one orbit as a biset, we can find some g ∈ G and h ∈ H such that β = hαg. Therefore, R(β) = R(h)R(α)R(g).

From the above analysis we know that R is uniquely determined by a linear map R(α) from a kG-module V to a kH-module W. Clearly, not every linear map φ : V → W can be used to define a representation of C. We characterizes in the next proposition those linear maps which indeed define representations of C. Let V^{⊥} be the kernel of φ, V̄ = V/V^{⊥}, and W^{⊤} be the image of φ. The map φ gives rise to a vector space isomorphism φ̄ : V̄ → W^{⊤} defined by φ̄(v̄) = φ(v).

Proposition 3.1. Notation as above. The linear map φ : V → W determines a representation R of C by defining R(α) = φ if and only if V̄ (resp., W^{⊤}) is a kG_{1}-module (resp., a kH_{1}-module) on which G_{0} (resp., H_{0}) acts trivially, and φ̄ is a k(G_{1}/G_{0}) ≅ k(H_{1}/H_{0})-module isomorphism.

Proof. Suppose that there exists a representation R of C such that R(α) = φ, i.e, φ(v) = αv for all v ∈ V. Take w ∈ W^{⊤}. Then we can find some v ∈ V such that αv = w. For every h ∈ H_{0}, we get hw = hαv = αv = w. Thus H_{0} acts trivially on W^{⊤}. Moreover, for h ∈ H_{1}, we can find some g ∈ G such that hα = αg. Therefore, hw = hαv = αgv = φ(gv), so hw ∈ W^{⊤}. Consequently, W^{⊤} is a kH_{1}-module.

Now consider V̄. First, for every g ∈ G_{1} and v ∈ V^{⊥}, we can find some h ∈ H such that hα = αg. Therefore, α(gv) = (hα)v = h(αv) = hφ(v) = 0 since v is in the kernel of φ. We deduce that V^{⊥} is a kG_{1}-module, so V̄ is a kG_{1}-module as well. For every g ∈ G_{0} and v̄ ∈ V̄, we have α(gv − v) = αgv − αv = αv − αv = 0. Therefore, gv − v ∈ V^{⊥}, and gv̄ − v̄ = 0. Thus G_{0} acts trivially on V̄.

Recall that the isomorphism ρ : G_{1}/G_{0} → H_{1}/H_{0} is defined in the following way (see the proof of Lemma 3.3 in [15]): For a given g ∈ G_{1}, we can find some h ∈ H_{1} such that αg = hα. Then ρ sends ḡ to h̄. Now the reader can check that this isomorphism gives an isomorphism between V̄ and W^{⊤} as k(G_{1}/G_{0}) ≅ k(H_{1}/H_{0})- modules. This proves the only if part.

Conversely, if φ satisfies the given conditions, then by defining R(α)(v) = φ̄(v̄) we indeed obtain a representation R of C. To prove this, it suffices to show that if h_{1}αg_{1} = h_{2}αg_{2} for g_{1}, g_{2} ∈ G and h_{1}, h_{2} ∈ H, then h_{1}φ̄(\overline{g_{1}v}) = h_{2}φ̄(\overline{g_{2}v}) for every v ∈ V. That is, h_{2}^{−1}h_{1}φ̄(\overline{g_{1}v}) = φ̄(\overline{g_{2}v}), where \overline{g_{1}v}, \overline{g_{2}v} are the images of g_{1}v and g_{2}v in V̄ respectively.

Note that h_{1}αg_{1} = h_{2}αg_{2} implies h_{2}^{−1}h_{1}α = αg_{2}g_{1}^{−1}, so h_{2}^{−1}h_{1} and g_{2}g_{1}^{−1} are contained in H_{1} and G_{1} respectively. Passing to the quotient groups, the isomorphism ρ sends \overline{g_{2}g_{1}^{−1}} ∈ G_{1}/G_{0} to \overline{h_{2}^{−1}h_{1}} ∈ H_{1}/H_{0}. Since V̄ is a kG_{1}-module on which G_{0} acts trivially and W^{⊤} is a kH_{1}-module on which H_{0} acts trivially, and φ̄ is a k(G_{1}/G_{0}) ≅ k(H_{1}/H_{0})-module isomorphism, we get

h_{2}^{−1}h_{1}φ̄(\overline{g_{1}v}) = \overline{h_{2}^{−1}h_{1}}φ̄(\overline{g_{1}v}) = φ̄(\overline{g_{2}g_{1}^{−1}} · \overline{g_{1}v}).

Note that in the above identity \overline{g_{1}v} cannot be expresses as g_{1}\overline{v} since V̄ is only a kG_{1}-module instead of a kG-module, and the action of g_{1} ∈ G on v̄ in general is not defined. However, we have

\overline{g_{1}v} = \overline{g_{1}g_{2}^{−1}g_{2}v} = (g_{1}g_{2}^{−1}) · \overline{g_{2}v} = \overline{g_{1}g_{2}^{−1}} · \overline{g_{2}v}

since V̄ is a kG_{1}-module, g_{1}g_{2}^{−1} ∈ G_{1}, and G_{0} acts trivially on V̄. Combining these two identities together, we get

h_{2}^{−1}h_{1}φ̄(\overline{g_{1}v}) = φ̄(\overline{g_{2}g_{1}^{−1}} · \overline{g_{1}v}) = φ̄(\overline{g_{2}g_{1}^{−1}g_{1}g_{2}^{−1}} · \overline{g_{2}v}) = φ̄(\overline{g_{2}v})

as required. This finishes the proof. □

Example 3.2. Let C be a finite EI category with objects x and y; H = C(y, y) is a copy of the symmetric group S_{3} on 3 letters; G = C(x, x) is cyclic of order 2; C(x, y) = S_{3} regarded as an (H, G)-biset where H acts from the left by multiplication, G acts from the right by multiplication after identifying G with a subgroup G^{†} of S_{3}; C(y, x) = ∅. We have G_{0} = 1, G_{1} = G, H_{0} = 1, H_{1} ≅ C_{2}. We set char(k) = 3 instead of 0 as before.

Let V = kG and W be the projective kH-module with W/ rad W ≅ k. Then

V = V ↓_{G_{1}}^{G}= k ↑_{G_{0}}^{G_{1}} ≅ k ⊕ ϵ,  W ↓_{H_{1}}^{H} ≅ k ⊕ k ⊕ ϵ,  k ↑_{H_{0}}^{H_{1}} ≅ k ⊕ ϵ.

Therefore, any linear map φ with the following matrix representation can define a

representation of C: [\begin{matrix}λ & 0 \\ μ & 0 \\ 0 & ν \end{matrix}], λ, μ, ν ∈ k.

3.2. Biset decomposition, induction and restriction. In Section 2 we have defined the restriction functor and induction functor. Now we use these functors to study some special subcategories of C. The main technique we use here is biset decomposition (see [7]).

The first special case we consider is the following category D: Ob D = Ob C, D(x, x) = G_{1}, D(y, y) = H, and D(x, y) = HαG_{1} = Hα. We picture the structure of D below:

[figure: D :; x; G_{1}; HαG_{1}; ...; y; H]

Lemma 3.3. Let D be defined as above. Then kD | (_{kD}kC_{kD}). In particular, kC is of infinite representation type if so is kD.

Proof. Let g_{1} = 1, …, g_{m} be a chosen set of representatives of the right cosets G_{1}\G. Clearly,

Mor C = G ⊔ H ⊔ HαG = (⨆_{i=1}^{m} G_{1}g_{i}) ⊔ H ⊔ (⋃_{i=1}^{m} Hαg_{i}).

We claim that ∪_{i=1}^{m}Hαg_{i} is a disjoint union. If this is not true, we can find some 1 ⩽ i ≠ j ⩽ m such that Hαg_{i} ∩ Hαg_{j} ≠ ∅. In particular, there exist h_{1}, h_{2} ∈ H such that h_{1}αg_{i} = h_{2}αg_{j}. Thus h_{2}^{−1}h_{1}α = αg_{j}g_{i}^{−1}, and g_{j}g_{i}^{−1} ∈ G_{1}. Consequently, g_{i} and g_{j} are in the same coset. This is a contradiction.

Define R_{1} = k(G_{1} ∪ Hα), R_{2} = k(G_{1}g_{2} ∪ Hαg_{2}), …, R_{m} = k(G_{1}g_{m} ∪ Hαg_{m}). They are all left kD-modules. By the above biset decomposition, we get:

kDkC = kH ⊕ R_{1} ⊕ R_{2} ⊕ … ⊕ R_{m} = kD ⊕ R

since R_{1} ⊕ kH = kD.

The above decomposition _{kD}kC = kD ⊕ R is also a decomposition of kC_{kD}. Indeed, R = (⨁_{i=2}^{m} kG_{1}g_{i})⊕(⨁_{i=2}^{m} kHαg_{i}) as a k-vector space. But both ⨁_{i=2}^{m} kG_{1}g_{i} and ⨁_{i=2}^{m} kHαg_{i} are right kD-modules. So R is a right kD-module as well. This proves the first statement. The second one follows from Proposition 2.3. □

Now consider the second special case. We introduce some notation here. Let K ⩽ G_{1} be a subgroup containing G_{0}. Then αK ⊆ αG_{1} ⊆ Hα. Define L = Stab_{H}(αK). That is, L = {h ∈ H | hαK = αK} = {h ∈ H | ∃g ∈ K such that hα = αg}.

Lemma 3.4. Notation as above. Then L is a subgroup of H_{1}, and L/H_{0} ≅ K/G_{0}.

Proof. We can check that L is a subgroup of H_{1} as we did in the proof of Lemma 3.3 in [15]. Since H_{0}◁H_{1} by that lemma, we know H_{0}◁L. The fact that L/H_{0} ≅ K/G_{0} can be shown as in Lemma 3.3 of [15], too. □

Thus H_{0}◁L ⩽ H_{1} ⩽ H and G_{0}◁K ⩽ G_{1} ⩽ G. We then construct a subcategory E as follows:

[figure: E :; x; K; LαK; ...; y; L]

The reader can check that both L and K act transitively on E(x, y) since LαK = Lα = αK by the definitions of K and L.

Lemma 3.5. Let E be constructed as above and suppose that both G and H act transitively on C(x, y). Then kE | (_{kE}kC_{kE}). In particular, kC is of infinite representation type if so is kE.

Proof. Choose a representative from each coset in K\G: g_{1} = 1, …, g_{m}. Since both G and H acts transitively on C(x, y), we know that G = G_{1} and H = H_{1}. Moreover, G_{1}/G_{0} ≅ H_{1}/H_{0}. For each g_{i}, 1 ⩽ i ⩽ m, there is some h_{i} such that αg_{i} = h_{i}α. We claim that Lh_{i}α ∩ Lh_{j}α = ∅, and hence Lh_{i} ∩ Lh_{j} = ∅, 1 ⩽ i ≠ j ⩽ m. Indeed, since Lh_{i}α = Lαg_{i} = αKg_{i} and Lh_{j}α = Lαg_{j} = αKg_{j}. If the intersection is not empty, there exist u, v ∈ K such that αug_{i} = αvg_{j}, i.e, α = αvg_{j}g_{i}^{−1}u^{−1}.

Therefore, vg_{j}g_{i}^{−1}u^{−1} ∈ G_{0} ⩽ K, and hence g_{j}g_{i}^{−1} ∈ K. This is impossible since g_{i} and g_{j} are in different cosets.

Note that {h_{i}}_{i=1}^{m} is a set of representatives of the cosets L\H since G/G_{0} ≅ H/H_{0} and K/G_{0} ≅ L/H_{0}. Clearly,

Mor C = G ⊔ H ⊔ Hα = (⨆_{i=1}^{m} Kg_{i}) ⊔ ( ⨆_{j=1}^{m} Lh_{j}) ⊔ ( ⋃_{j=1}^{m} Lh_{j}α).

The vector space spanned by K ⊔ Lα ⊔ L is precisely kE. Let R be the vector space spanned by (⊔_{i=2}^{m}Kg_{i}) ⊔ (⊔_{j=2}^{m}Lh_{j}) ⊔ (⊔_{j=2}^{m}Lh_{j}α). Then R is a left kE-module, and hence _{kE}kC = kE ⊕ R. We also have kC_{kE} = kE ⊕ R. Indeed, since

⊔_{j=2}^{m}Lh_{j}αK = ⊔_{j=2}^{m}Lh_{j}Lα = ⊔_{j=2}^{m}Lh_{j}α,

we check that R is a right kE-module as well. This proves the first statement. The second one follows from Proposition 2.3. □

Given a subcategory D of C as pictured below and N ∈ kD-mod, where G′ = D(x, x) and H′ = D(y, y).

[figure: D :; x; G′; H′αG′; ...; y; H′].

We want to describe the structure of the induced module N ↑_{D}^{C}. As described in Proposition 3.1, N is determined by a linear map φ : V → W where V = N(x) and W = N(y) are a kG′-module and a kH′-module respectively. In general, N ↑_{D}^{C} (x) ≇ V ↑_{G′}^{G} and N ↑_{D}^{C} (y) ≇ W ↑_{H′}^{H}. Indeed, we have already seen in Example 2.2 that the dimension of the induced module can be even smaller than that of the original module. In the following example we see that the induced module is isomorphic to the original one.

Example 3.6. Let C be a finite EI category such that: Ob C = {x, y}, G = 1, H = ⟨h⟩ is a cyclic group of order 2 = char(k), and C(x, y) = {α}. Let D be the subcategory formed by removing h ∈ H. Let φ : V → W be a representation R of D, where V = ⟨v⟩ ≅ k, W = ⟨w⟩ ≅ k and φ(v) = w. By direct computation we get R ↑_{D}^{C} (x) ≅ V and R ↑_{D}^{C} (y) ≅ W ≇ W ↑_{1}^{H}:

h ⊗_{kD} w = h ⊗_{kD} α · v = hα ⊗_{kD} v = α ⊗_{kD} v = 1_{y} ⊗_{kD} α · v = 1_{y} ⊗ w.

If the pair (C, D) satisfies some extra property, we can explicitly describe the structure of the induced module R ↑_{D}^{C} by a linear map between two induced group representations.

Lemma 3.7. Suppose that H acts transitively on C(x, y). Let D be a subcategory of C pictured as above and R be a kD-module of the form R(α) = φ : V → W. If Stab_{H}(αG′) ⩽ H′, then R̃ = R ↑_{D}^{C} has the form

R̃(α) = φ̃ : kG ⊗_{kG′} V → kH ⊗_{kH′} W, g ⊗ v ↦ h ⊗ φ(v),

where h ∈ H such that αg = hα.

Proof. We first check that φ̃ indeed determines a kC-module. Since kG ⊗_{kG′} V and kH⊗_{kH′} W have a natural kG-module structure and a natural kH-module structure respectively, it is enough to prove that h_{1}φ̃g_{1} and h_{2}φ̃g_{2} coincide as linear maps from kG ⊗_{kG′} V to kH ⊗_{kH′} W if h_{1}αg_{1} = h_{2}αg_{2}, g_{1}, g_{2} ∈ G, h_{1}, h_{2} ∈ H.

Take g ⊗_{kG′} v ∈ kG ⊗_{kG′} V. Since H acts transitively, we can find some h ∈ H such that hα = αg. Moreover, h_{1}αg_{1} = h_{2}αg_{2} implies αg_{1} = h_{1}^{−1}h_{2}αg_{2}, so αg_{1}g = h_{1}^{−1}h_{2}αg_{2}g. Therefore,

h_{1}φ̃g_{1}(g ⊗_{kG′} v) = h_{1}φ̃(g_{1}g ⊗_{kG′} v) = h_{1}h ⊗_{kH′} φ(v),

where hα = αg_{1}g = h_{1}^{−1}h_{2}αg_{2}g. On the other hand, we also have

h_{2}φ̃g_{2}(g ⊗_{kG′} v) = h_{2}φ̃(g_{2}g ⊗_{kG′} v) = h_{2}h′ ⊗_{kH′} φ(v),

where h′α = αg_{2}g. Thus hα = h_{1}^{−1}h_{2}αg_{2}g = h_{1}^{−1}h_{2}h′α, and α = h^{−1}h_{1}^{−1}h_{2}h′α. Consequently, h^{−1}h_{1}^{−1}h_{2}h′ ∈ H_{0} ◁ H′ ⩽ H_{1}. By Proposition 3.1, H_{0} acts trivially on φ(v), so we have

h_{1}h ⊗_{kH′} φ(v) = h_{1}h ⊗_{kH′} h^{−1}h_{1}^{−1}h_{2}h′φ(v) = h_{2}h′ ⊗_{kH′} φ(v)

as required. Therefore, φ̃ is a well defined representation of C.

Now Let {v_{i}}_{i=1}^{m} and {w_{j}}_{i=1}^{n} be bases of V and W respectively. Note that every morphism in C(x, y) can be written as hα for some h ∈ H. We define a k-linear map π : kC × R → R̃ by:

(g, v_{i}) ↦ g ⊗_{kG′} v_{i}, (hα, v_{i}) ↦ h ⊗_{kH′} φ(v_{i}), (h, w_{j}) ↦ h ⊗_{kH′} w_{j}
(h, v_{i}) ↦ 0, (α, w_{j}) ↦ 0, (g, w_{j}) ↦ 0

for g ∈ G, h ∈ H, 1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n. The reader can check that π is kDbalanced using either the condition Stab_{H}(αG′) ⩽ H′, so it induces the following kD-homomorphism π̃ : R ↑_{D}^{C}→ R̃:

g ⊗_{kD} v_{i} ↦ g ⊗_{kG′} v_{i}, h ⊗_{kD} w_{j} ↦ h ⊗_{kH′} w_{j}

for g ∈ G, h ∈ H, 1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n, as shown by the following commutative diagram:

[figure: g ⊗_{kD} v_{i}; π̃_{x}; αg ⊗_{kD} v_{i} = hα ⊗_{kD} v_{i} = h ⊗_{kD} φ(v_{i}); π̃_{y}; g ⊗_{kG′} v_{i}; φ̃; φ̃(g ⊗_{kG′} v_{i}) = h ⊗_{kH′} φ(v_{i})]

Note that π̃ is not only a kD-module homomorphism, but also a kC-module homomorphism. Furthermore, from our definition of π we find that all basis elements of R̃ are images under π. By the universal property of tensor product, they are actually images under π̃. Therefore, π̃ is a surjective kC-module homomorphism.

On the other hand, R ↑_{D}^{C} (x) is spanned by basic tensors g ⊗_{kD} v_{i}, 1 ⩽ i ⩽ m. Since kG′ ⊂ kD, it is actually spanned by the set {g_{t} ⊗_{kD} v_{i} | 1 ⩽ t ⩽ l, 1 ⩽ i ⩽ m}, where {g_{t}}_{t=1}^{l} is a set of representatives of the left cosets G/G′. It turns out that

dim_{k}(R ↑_{D}^{C} (x)) ⩽ |G : G′|dim_{k}V = dim_{k}R̃(x).

Similarly, since H acts transitively on C(x, y), any morphism β in C(x, y) can be written as hα for some h ∈ H. Thus β(g ⊗_{kD} v_{i}) = hαg ⊗_{kD} v_{i} = hh′ ⊗_{kD} φ(v_{i}) where αg = h′α. Therefore, R ↑_{D}^{C} (y) is spanned by basic tensors h ⊗_{kD} w_{j}, 1 ⩽ j ⩽ n, h ∈ H. Since kH′ ⊂ kD, it is actually spanned by the set {h_{s} ⊗_{kD} w_{j} | 1 ⩽ s ⩽ r, 1 ⩽ j ⩽ n}, where {h_{s}}_{s=1}^{r} is a set of representatives of the cosets H/H′. Therefore,

dim_{k}(R ↑_{D}^{C} (y)) ⩽ |H : H′|dim_{k}W = dim_{k}R̃(y).

By the above two inequalities, we have dim_{k}(R ↑_{D}^{C}) ⩽ dim_{k} R̃, so π̃ is actually an isomorphism. □

We will use the following result to get a family of finite EI categories whose category algebras are of finite representation type.

Proposition 3.8. Let D be as in the previous lemma. Suppose that both |G : G′| and |H : H′| are invertible in k. If the pair (C, D) satisfies one of the following conditions:

(1) G acts transitively on C(x, y);

(2) H′ = H,

then M is isomorphic to a direct summand of M ↓_{D}^{C}↑_{D}^{C} for M ∈ kC-mod. In particular, kC has finite representation type if so does kD.

Proof. The second statement follows from Proposition 2.3, so we only need to show the first statement. Firstly we consider a special case. That is, G acts transitively on C(x, y) and G′ = Stab_{G}(H′α). Observe that by this given condition G_{0} ◁ G_{1} = G, H_{1} ◁ H_{1} = H, and H/H_{0} ≅ G/G_{0}. Moreover, by Lemma 3.4, G′/G_{0} ≅ H′/H_{0}. Take {g_{i}}_{i=1}^{n} as a set of representatives of the cosets G/G′ and choose h_{i} ∈ H satisfying αg_{i} = h_{i}α. Then {h_{i}}_{i=1}^{n} is a set of representatives of the cosets H/H′, and αg_{i}G′ = h_{i}H′α. Note that n is invertible.

Suppose that M has the form φ : V → W where: V = M(x) is a kG-module, W = M(y) is a kH-module, and φ = M(α). By the previous lemma, M ↓_{D}^{C}↑_{D}^{C} has the form

φ̃ : kG ⊗_{kG′} V → kH ⊗_{kH′} W, g ⊗ v ↦ h ⊗ φ(v),

where hα = αg.

Define the following maps:

θ_{V} : kG ⊗_{kG′} V → V, g ⊗ v ↦ gv;

δ_{V} : V → kG ⊗_{kG′} V, v ↦ \frac{1}{n} ∑_{i=1}^{n} g_{i} ⊗ g_{i}^{−1}v;

θ_{W} : kH ⊗_{kH′} W → W, h ⊗ w ↦ hw;

δ_{W} : W → kH ⊗_{kH′} W, w ↦ \frac{1}{n} ∑_{i=1}^{n} h_{i} ⊗ h_{i}^{−1}w.

and consider the diagram

\xymatrix{kG ⊗_{kG′} V \ar[r]_{φ̃} \ar[d]^{θ_{V}} & kH ⊗_{kH′} W \ar[d]^{θ_{W}} \\ V \ar[r]_{φ} \ar[u]^{δ_{V}} & W. \ar[u]^{δ_{W}}}

We can show that the diagram commutes. Indeed, for g ∈ G and v ∈ V,

φθ_{V} (g ⊗ v) = φ(gv) = (αg) · v = (hα) · v
= hφ(v) = θ_{W} (h ⊗ φ(v)) = θ_{W} φ̃(g ⊗ v)

where αg = hα; and

φ̃δ_{V} (v) = \frac{1}{n} ∑_{i=1}^{n} φ̃(g_{i} ⊗ g_{i}^{−1}v) = \frac{1}{n} ∑_{i=1}^{n} h_{i} ⊗ φ(g_{i}^{−1}v) = \frac{1}{n} ∑_{i=1}^{n} h_{i} ⊗ (αg_{i}^{−1}) · v

= \frac{1}{n} ∑_{i=1}^{n} h_{i} ⊗ h_{i}^{−1}α · v = \frac{1}{n} ∑_{i=1}^{n} h_{i} ⊗ h_{i}^{−1}φ(v) = δ_{W} φ(v).

But θ_{V} δ_{V} = 1_{V} and θ_{W} δ_{W} = 1_{W}. Therefore, M | M ↓_{D}^{C}↑_{D}^{C}.

Now suppose that D satisfies the second condition, i.e., H′ = H. Thus D(x, y) = Hα = C(x, y). By the previous lemma, M ↓_{D}^{C}↑_{D}^{C} has the form

φ̃ : kG ⊗_{kG′} V → kH ⊗_{kH} W ≅ W, g ⊗ v ↦ h ⊗ φ(v) = 1 ⊗ hφ(v),

where αg = hα.

Let {g_{i}}_{i=1}^{n} be a set of representatives of the cosets G/G′. Define θ_{V} and δ_{V} as in the first case. We can show the following diagram commutes:

\xymatrix{kG ⊗_{kG′} V \ar[r]_{φ̃} \ar[d]^{θ_{V}} & kH ⊗_{kH} W \ar[d]^{≅} \\ V \ar[r]_{φ} \ar[u]^{δ_{V}} & W. \ar[u]^{≅}}

So the conclusion follows as in the above special case.

In the situation that D satisfies the first condition, i.e., G acts transitively on C(x, y), we define another subcategory E of C as follows:, where Ĝ = Stab_{G}(H′α):

[figure: E :; x; Ĝ; H′αĜ; ...; y; H′].

Note that both Ĝ and H′ act transitively on E(x, y), so Stab_{H}(αĜ) = H′.

The pair (C, E) falls into the special case we consider at the beginning, so we have M | M ↓_{E}^{C}↑_{E}^{C}. The pair (E, D) satisfies the second condition, so we have (M ↓_{E}^{C}) | (M ↓_{E}^{C}) ↓_{D}^{E}↑_{D}^{E}. In conclusion, we get M | M ↓_{E}^{C}↓_{D}^{E}↑_{D}^{E}↑_{E}^{C}, i.e., M | M ↓_{D}^{C}↑_{D}^{C}. This finishes the proof. □

3.3. Transitivity of group actions. Our main task in this subsection is to prove the following proposition, which is condition (2) in Theorem 1.1.

Proposition 3.9. If neither G nor H acts transitively on C(x, y), then kC is of infinite representation type.

This conclusion has been proved in [15] for the case that both |G| and |H| are invertible. We prove the fact for the general case. Define a finite EI category Q as follows, where Stab_{G}(ᾱ) = G_{1} and Stab_{H}(ᾱ) = H_{1}.

[figure: Q :; x; G; HᾱG; ...; y; H].

We claim that Q is a quotient category of C. Indeed, let F : C → Q be the functor such that F is the identity map on objects and automorphisms, and sends hαg to hᾱg for h ∈ H and g ∈ G. If h_{1}αg_{1} = h_{2}αg_{2}, h_{1}, h_{2} ∈ H, g_{1}, g_{2} ∈ G, then h_{2}^{−1}h_{1}α = αg_{2}g_{1}^{−1}, so h_{2}^{−1}h_{1} ∈ H_{1} and g_{2}g_{1}^{−1} ∈ G_{1}. Therefore, h_{2}^{−1}h_{1}ᾱ = ᾱg_{2}g_{1}^{−1} = ᾱ. In other words, h_{1}ᾱg_{1} = h_{2}ᾱg_{2}. Thus F is a well defined quotient functor, and Q is a quotient category of C. Therefore, it suffices to show the infinite representation type of kQ. The strategy is to use permutation modules M = k ↑_{G_{1}}^{G} and N = k ↑_{H_{1}}^{H} to construct infinitely many distinct indecomposable representations (up to isomorphism) for Q. Since neither G nor H acts transitively on C(x, y), we know that G_{1} ≠ G and H_{1} ≠ H, so dim_{k} M > 1 and dim_{k} N > 1.

Note that M has a unique indecomposable summand M_{0} which is characterized by one of the following properties:

(1) Hom_{kG}(M_{0}, k) ≅ k, i.e, Top(M_{0}) has a unique summand isomorphic to k;

(2) Hom_{kG}(k, M_{0}) ≅ k, i.e, Soc(M_{0}) has a unique summand isomorphic to k.

Moreover, M_{0} is isomorphic to its dual, and Top(M_{0}) ≅ Soc(M_{0}). This module is called a Scott module. In one extreme situation that H_{1} contains a Sylow psubgroup of H, M_{0} ≅ k. If |H_{1}| is invertible, then M_{0} ≅ P_{k}, the projective cover of k. For more details, see [8, 14]. In particular, for a finite group with cyclic Sylow p-subgroups (as we study in this paper), the structure of all Scott modules has been described in [22] by considering the Brauer graphs. Similarly, N = k ↑_{H_{1}}^{H} has a Scott module N_{0}.

Proof. We prove the conclusion by considering the permutation modules M and N. There are three cases:

Case I: Both M and N have at least two indecomposable summands. Take an indecomposable summand M_{1} of M which is not the Scott module, and let S be a simple summand of Soc(M_{1}). Clearly, S ≇ k. Moreover, since Hom_{kG}(S, M_{1}) ≠ 0, we deduce that Hom_{kG_{1}}(S ↓_{G_{1}}^{G}, k) ≅ Hom_{kG}(S, M) ≠ 0, so Top(S ↓_{G_{1}}^{G}) has a simple summand k_{1} ≅ k. Dually, we can find a simple kH-module T ≇ k such that Soc(T ↓_{H_{1}}^{H}) has a simple summand k_{2} ≅ k.

As vector spaces, S = k_{1} ⊕ S′ and T = k_{2} ⊕ T′ (S′ and T′ might be 0, but this is fine). Construct a family of representations R_{d} of Q as follows, 0 ≠ d ∈ k. First, R_{d}(x) = k ⊕ S = k ⊕ (k_{1} ⊕ S′), R_{d}(y) = k ⊕ T = k ⊕ (k_{2} ⊕ T′). The linear map R_{d}(ᾱ) is defined by:

[figure: k ⊕ k_{1} ⊕ S′; ^{[\begin{matrix}1 & 1 & 0 \\ 1 & d & 0 \\ 0 & 0 & 0 \end{matrix}]}; k ⊕ k_{2} ⊕ T′]

We check that this linear map R_{d}(ᾱ) indeed gives rise to a representation of Q by Proposition 3.1, and that R_{d} is indecomposable by computing its endomorphism algebra. Moreover, if R_{b} ≅ R_{d}, we have the following matrix identity:

[\begin{matrix}1 & 1 & 0 \\ 1 & d & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix}u & 0 & 0 \\ 0 & v & 0 \\ 0 & 0 & v \end{matrix}] = [\begin{matrix}λ & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t \end{matrix}] [\begin{matrix}1 & 1 & 0 \\ 1 & b & 0 \\ 0 & 0 & 0 \end{matrix}] ,

where b, d, u, t, v, λ ∈ k are nonzero scalars. By computation, we get b = d. In this way we construct infinitely many pairwise non-isomorphic indecomposable representations of Q illustrated as follows, where dotted arrows means that these maps are actually only defined on some subspaces.

\xymatrix{k \ar[r]^{1} \ar@{-->}[dr]^{1} & k \\ S \ar@{-->}[r]^{d} \ar@{-->}[ur]_{1} & T.}

Case II: One of M and N is indecomposable and the other one is not. Without loss of generality we assume that M is indecomposable. Then M ≇ k is the Scott module. Its length is at least 2; and both Top(M) and Soc(M) has a unique composition factor isomorphic to k. Let k_{1} be this composition factor in Top(M).

Observe that dim_{k} End_{kG}(M) ⩾ 2 since there is a nilpotent map φ : M → M such that φ(M) ≅ k is a simple summand of Soc(M). We conclude that

dim_{k} Hom_{kG_{1}}(M ↓_{G_{1}}^{G}, k) = dim_{k} Hom_{kG}(M, k ↑_{G_{1}}^{G}) = dim_{k} Hom_{kG}(M, M) ⩾ 2.

This fact also follows from

dim_{k} Hom_{kG_{1}}(M ↓_{G_{1}}^{G}, k) = dim_{k} Hom_{kG_{1}}(k ↑_{G_{1}}^{G}↓_{G_{1}}^{G}, k)

= dim_{k} Hom_{kG_{1}}( ⨁_{s∈G_{1}\G/G_{1}}(s ⊗ k) ↑_{G_{1}∩sG_{1}s^{−1}}^{G_{1}}, k).

The last dimension equals the number of double cosets G_{1}\G/G_{1}, which is at least 2. Therefore, there are at least two summands isomorphic to k in Top(M ↓_{G_{1}}^{G}). Take a summand k_{2} ≅ k in Top(M ↓_{G_{1}}^{G}) which is different from k_{1}.

Write M = k_{1} ⊕ k_{2} ⊕ M′ as vector spaces. Take a simple kH-module T as we did in Case I. Note that T ↓_{H_{1}}^{H}= k_{3} ⊕ T′ where k_{3} ≅ k. We construct a class of representations R_{d} in the following way, 0 ≠ d ∈ k. First, R_{d}(x) = M, R_{d}(y) = k ⊕ T. The map R_{d}(ᾱ) is defined by

[figure: k_{1} ⊕ k_{2} ⊕ M′; ^{[\begin{matrix}1 & 1 & 0 \\ 1 & d & 0 \\ 0 & 0 & 0 \end{matrix}]}; k ⊕ k_{3} ⊕ T′] .

Again, by Proposition 3.1, R_{d}(ᾱ) gives rise to a representation of Q. It is indecomposable since R(x) = M is an indecomposable kG-module. Moreover, we check that R_{d} ≅ R_{b} if and only if d = b. In this way we construct infinitely many pairwise non-isomorphic indecomposable representations of Q.

Case III: Both M and N are indecomposable. As in Case II, there exist at least two summands k_{1} ≅ k_{2} ≅ k in Top(M ↓_{G_{1}}^{G}), where k_{1} is the restriction of the simple summand k in Top(M) to G_{1}. Dually, there exist at least two summands k_{3} ≅ k_{4} ≅ k in Soc(N ↓_{H_{1}}^{H}), where k_{3} is the restriction of the simple summand k in Soc(N) to H_{1}.

Write M = k_{1} ⊕ k_{2} ⊕ M_{1} and N = k_{3} ⊕ k_{4} ⊕ N_{1} as vector spaces. Then the following map φ

[figure: k_{1} ⊕ k_{2} ⊕ M_{1}; ^{[\begin{matrix}1 & 1 & 0 \\ 1 & d & 0 \\ 0 & 0 & 0 \end{matrix}]}; k_{3} ⊕ k_{4} ⊕ N_{1}]

gives rise to a representation R_{d} of Q by Proposition 3.1. It is obviously indecomposable. Moreover, letting d ≠ 0 vary in k, we check similarly that R_{d} ≅ R_{b} if and only if d = b. □

This proposition implies Theorem 3.35 in [11], which asserts that if both H and G are nontrivial and C(x, y) ≅ H × G as a (H, G)-biset, then kC is of infinite representation type.

3.4. Actions of p-subgroups. In this subsection we set char(k) = p ⩾ 5. It is well known that the representation type of a finite group is completely determined by its Sylow p-subgroups. For the finite EI category C, the Sylow p-subgroups of G and H still play an important role in determining the representation type of kC as illustrated by condition (3) in Theorem 1.1, which will be proved in this subsection.

By the previous proposition, throughout this subsection without loss of generality we assume that H acts transitively on C(x, y). Observe that the condition that H has a p-subgroup acting nontrivially on C(x, y) is equivalent to one of the following conditions:

(1) H has a Sylow p-subgroup acting nontrivially on C(x, y);

(2) all Sylow p-subgroups of H act nontrivially on C(x, y);

(3) O^{p′}H, the normal subgroup generated by all Sylow p-subgroups of H, is not contained in H_{0};

(4) there is some h ∈ H with order a power of p such that hα ≠ α. Moreover, if G has a p-subgroup acting nontrivially on C(x, y), so does H. Indeed, since H acts transitively, G_{0} ◁G_{1} = G. Consequently, G_{0} does not contain a Sylow p-subgroup of G since otherwise O^{p′}G ⩽ G_{0}, contradicting the given condition. Thus p divides |G_{1}/G_{0} = |H_{1}/H_{0}|, so H_{0} does not contain a Sylow p-subgroup of H_{1} ⩽ H. In conclusion, H_{1} and H have a p-subgroup acting nontrivially on C(x, y).

Proposition 3.10. Suppose that char(k) = p ⩾ 5. If both G and H have psubgroups acting nontrivially on C(x, y), then kC is of infinite representation type.

Proof. We have assumed that H acts transitively on C(x, y). Consequently, G = G_{1}. Define a subcategory D of C as follows:

[figure: D :; x; G; H_{1}αG; ...; y; H_{1}]

By Lemma 3.3, it suffices to show the infinite representation type of kD.

We claim that both G and H_{1} act transitively on D(x, y) = H_{1}αG. Indeed, by the definition of H_{1}, H_{1}α ⊆ αG, so H_{1}αG = αG, i.e., G acts transitively on D(x, y). On the other hand, since H acts transitively on C(x, y), for every g ∈ G, there exists some h ∈ H with hα = αg. But again by the definition of H_{1}, h ∈ H_{1}. Therefore, αG ⊆ H_{1}α. This proves the claim.

Let Ḡ = G/G_{0} and H̄ = H_{1}/H_{0}. Since Ḡ ≅ H̄, we identify these two groups. Moreover, p divides |Ḡ| = |H̄|. Indeed, since G has a p-subgroup acting nontrivially on C(x, y), we can find some g ∈ G with order a power of p such that αg ≠ α, i.e., g ∉ G_{0}. Therefore, its image ḡ ∈ Ḡ has order a power of p, so p divides |Ḡ|.

Define another finite EI category E as follows:

[figure: E :; x; Ḡ; ^{H̄ᾱḠ}; ...; y; H̄]

where Ḡ acts regularly on E(x, y) from the right side and H̄ acts regularly on it from the left side. The reader can check that E is a quotient category of D by factoring out G_{0} and H_{0}. Consequently, it is enough to show the infinite representation type of kE.

Since p divides |Ḡ|, we can take a nontrivial p-subgroup P ⩽ Ḡ. Correspondingly, Q = StabH̄(ᾱP) ⩽ H̄, and P ≅ Q. Let F be the following subcategory of E:

[figure: F :; x; P; QᾱP; ...; y; Q].

By Theorem 5.3 in [12], kF is of infinite representation type. By Lemma 3.5, kE is of infinite representation type as well. This finishes the proof. □

By the remark before this proposition, if G has a p-subgroup acting nontrivially on C(x, y), so does H since it acts transitively. Consequently, kC has infinite representation type. Thus in the rest of this subsection we assume that O^{p′}G acts trivially on C(x, y), i.e., O^{p′}G ⩽ G_{0}. By this assumption, we know that p does not divide |G_{1} : G_{0}| = |H_{1} : H_{0}|. Therefore, for every Sylow p-subgroup P ⩽ H, P ∩ H_{0} = P ∩ H_{1}. In particular, p divides |H_{1}| if and only if it divides |H_{0}|, and O^{p′}H ⩽ H_{0} if and only if O^{p′}H ⩽ H_{1}.

Lemma 3.11. Suppose that G = 1 and p divides |H : H_{0}|. If dim_{k} End_{kH}(P_{k}) ⩾ 6 where P_{k} is the projective cover of the simple kH-module k, then kC is of infinite representation type.

The condition that p divides |H : H_{0}| implies that there is a Sylow p-subgroup P ⩽ H acting nontrivially on C(x, y). But the converse statement is not true. Indeed, p does not divide |H : H_{0}| if and only if H_{0} contains a Sylow p-subgroup of H, weaker than the condition O^{p′} ⩽ H_{0}, which by the above remark is equivalent to saying that a Sylow p-subgroup of H acting trivially on C(x, y). We also remind the reader that P_{k} can also be viewed as a projective kC-module and End_{kC}(P_{k}) ≅ End_{kH}(P_{k}).

Proof. Consider the projective kC-module P_{k} ⊕kC1_{x} and Λ = End_{kC}(P_{k} ⊕kC1_{x})^{op}. By Proposition 2.5 on page 36 in [3], Λ-mod is equivalent to a subcategory of kC-mod. Thus it suffices to show the infinite representation type of Λ. We have

End_{kC}(P_{k}) ≅ k[t]/(t^{d}), End_{kC}(kC1_{x}) ≅ 1_{x}kC1_{x} ≅ k, Hom_{kC}(kC1_{x}, P_{k}) = 0,

where d = dim_{k} End_{kH}(P_{k}), and

dim_{k} Hom_{kC}(P_{k}, kC1_{x}) = dim_{k} Hom_{kH}(P_{k}, kHα) = dim_{k} Hom_{kH}(P_{k}, k ↑_{H_{0}}^{H}).

Let M be the Scott module of k ↑_{H_{0}}^{H}. Since p divides |H : H_{0}|, M ≇ k. Therefore, M has at least two composition factors isomorphic to k. Consequently,

t = dim_{k} Hom_{kC}(P_{k}, kC1_{x}) ⩾ dim_{k} Hom_{kH}(P_{k}, M) ⩾ 2.

Thus Λ is isomorphic to the path algebra of the following quiver with relations δ^{d} = 0 and δ^{t}β = 0 for some t ⩾ 2:

[figure: •; β; •; δ].

From Bongartz’s list in [4, 6] we conclude that Λ and hence kC are of infinite representation type if d ⩾ 6. □

Before stating and proving the next two propositions, let us do some reduction. Define a finite EI category G as follows:

[figure: G :; x; 1; Hᾱ; ...; y; H].

where Stab_{H}(ᾱ) = H_{1}. Let F : C → G be the functor defined by: F (g) = 1, F (h) = h, F (hαg) = hᾱ for g ∈ G and h ∈ H. The reader can check that F is well defined. Therefore, G is a quotient category of C. Moreover, if a Sylow p-subgroup P ⩽ H acts nontrivially on C(x, y), then it acts nontrivially on G(x, y) as well. This conclusion comes from the fact that O^{p′}H ⩽ H_{0} if and only if O^{p′}H ⩽ H_{1}, see the remark before this lemma and the equivalent conditions introduced at the beginning of this subsection.

Proposition 3.12. Suppose that char(k) = p ⩾ 5. If G (resp., H) has a p-subgroup P (resp., Q) such that 1 ≠ P ∩ G_{0} ≠ G_{0} (resp., 1 ≠ Q ∩ H_{0} ≠ H_{0}), then kC is of infinite representation type.

Proof. By the reduction before the previous lemma, we know O^{p′}G ⩽ G_{0}. Therefore, we only consider the case that a p-subgroup Q ⩽ H such that Q ∩ H_{0} ≠ 1 and Q ⩽̸ H_{0}, i.e., Qα ≠ {α} and Stab_{Q}(α) ≠ 1. Moreover, since p does not divide |H_{1}/H_{0}| = |G_{1}/G_{0}|, we conclude that 1 ≠ Q ∩ H_{0} = Q ∩ H_{1} and Q ⩽̸ H_{1}. Consequently, the quotient category G defined above satisfies the given condition as well since Stab_{H}(ᾱ) = H_{1}. Therefore, it suffices to show the infinite representation type of kG. Observe that p divides both |H_{1}| and |H : H_{1}|, so the Scott module N of k ↑_{H_{1}}^{H} is neither isomorphic to k nor projective. We have two cases:

Case I: Soc(N) contains a simple summand S ≇ k. Since Soc(N) ≅ Top(N) (see [22]), Top(N) contains a summand isomorphic to S. Thus

0 ≠ Hom_{kH}(N, S) ⊆ Hom_{kH}(k ↑_{H_{1}}^{H}, S) ≅ Hom_{kH_{1}}(k, S ↓_{H_{1}}^{H}),

we conclude that Soc(S ↓_{H_{1}}^{H}) contains a summand k_{1} isomorphic to k.

Write N = k ⊕ S ⊕ N′ = k ⊕ (k_{1} ⊕ S′) ⊕ N′ as vector spaces, where both k and S are simple summands in Soc(N). Note that N is indecomposable, k ≇ S, and they both are contained in Soc(N). Therefore, every non-invertible endomorphism on N maps k ⊕ S to 0.

Construct a class of representations R_{d} of G in the following way, 0 ≠ d ∈ k. First, R_{d}(x) = k, R_{d}(y) = N. The map R_{d}(α) is defined by

[figure: k; ^{[\begin{matrix}1 \\ d \\ 0 \\ 0 \end{matrix}]}; k ⊕ (k_{1} ⊕ S′) ⊕ M′] .

By Proposition 3.1, this determines a representation of G. It is indecomposable. Moreover, we check that R_{b} ≅ R_{d} if and only if d = b. In this way we construct infinitely many pairwise non-isomorphic indecomposable representations of G.

Case II: Soc(N) ≅ Top(N) ≅ k. In this situation N is a proper quotient module of the projective kH-module P_{k}. In particular, the multiplicity [P_{k}, k] > [N : k] ⩾ 2. Therefore, in the Brauer graph Γ of the principal block B_{0}(kH), there is an exceptional vertex to which the edge k is adjacent.

Let m be the multiplicity of this exceptional vertex and e be the number of edges in Γ. Then e | (p − 1) and em = |D| − 1, where the defect group D of B_{0}(kH) is a Sylow p-subgroup of kH (see [1, 5]). By the given condition, |D| ⩾ p^{2}. Therefore, m ⩾ p + 1, so P_{k} has at least p + 2 composition factors isomorphic to k. Consequently, dim_{k} End_{kH}(P_{k}) ⩾ p + 2 ⩾ 7. The conclusion follows from Lemma 3.11 (note that Stab_{H}(ᾱ) is H_{1} instead of H_{0} for G). □

Actually, we have shown the following conclusion in the above proof:

Corollary 3.13. Suppose that char(k) = p ≠ 2, 3. If kC is of finite representation type, then the Scott module of k ↑_{H_{0}}^{H} (or k ↑_{H_{1}}^{H}) is either k or projective.

Proof. This is clearly true for p = 0. Thus we assume p ⩾ 5. Since p does not divide |H_{1} : H_{0}| by the assumption we made before, the Scott module of k ↑_{H_{0}}^{H} is isomorphic to that of k ↑_{H_{1}}^{H}, see [8, 14, 22]. By the above proposition, H_{0} either contains a Sylow p-subgroup of H or only has a trivial p-subgroup. In the first case the Scott module is isomorphic to k, and in the second case it is isomorphic to the the uniserial projective module P_{k}. □

3.5. Permutation modules. In this subsection we study the structure of permutation modules to develop a few more criteria. By previous results, we can assume that H acts transitively on C(x, y). Therefore, G_{1} = G. We identify k(G_{1}/G_{0}) and k(H_{1}/H_{0}). We do not suppose that O^{p′}G acts trivially on this biset, so |H_{1} : H_{0}| = |G_{1} : G_{0}| might not be invertible in k, and hence k ↑_{H_{0}}^{H_{1}} ≅ k(H_{1}/H_{0}) might not be semisimple. We will find in many situations the representation type of kC is determined by the structure of permutation module k ↑_{H_{1}}^{H}.

The following lemma will be used frequently.

Lemma 3.14. Let M be a kH-module. Then Hom_{kH}(M, k ↑_{H_{1}}^{H}) ≠ 0 if and only if k is a summand of Top(M ↓_{H_{1}}^{H}). Dually, Hom_{kH}(k ↑_{H_{1}}^{H}, M) ≠ 0 if and only if k is a summand of Soc(M ↓_{H_{1}}^{H}).

Proof. Use Frobenius reciprocity. □

The following proposition generalizes Corollary 6.5 in [15], where kH is supposed to be semisimple.

Proposition 3.15. If kC is of finite representation type, then the following conditions must be true:

(1) Top(k ↑_{H_{1}}^{H}) has no repeated summands;

(2) every indecomposable summand M of k ↑_{H_{1}}^{H} is uniserial or biserial, and if M is biserial, it is projective;

(3) if M ≇ N are indecomposable summands of k ↑_{H_{1}}^{H}, then Hom_{kH}(M, N) = 0.

Proof. Suppose that a simple kH-module U appears at least twice in Top(k ↑_{H_{1}}^{H}). Therefore, by Frobenius reciprocity k appears at least twice in Soc(U ↓_{H_{1}}^{H}). Take two summands k_{1} ≅ k ≅ k_{2} in Top(U ↓_{H_{1}}^{H}) and write U = k_{1} ⊕ k_{2} ⊕ U′ as vector spaces. Define a family of representations R_{d} for C as follows, 0 ≠ d ∈ k:

[figure: φ_{d} : k; ^{[\begin{matrix}1 \\ d \\ 0 \end{matrix}]}; k_{1} ⊕ k_{2} ⊕ U′] .

As we did in the proof of Proposition 3.12, we can check that R_{d} is indecomposable, and that R_{d} ≅ R_{b} if and only if b = d. Thus (1) must be true.

We show that every indecomposable summand M of k ↑_{H_{1}}^{H} has a simple top. This implies (2). Indeed, since M has a simple top, it is a quotient module of an indecomposable projective kH-module, and hence must be uniserial or biserial (see [1]). Because k ↑_{H_{1}}^{H} is isomorphic to its dual module, M must has a simple socle. Therefore, if M is biserial, it must be isomorphic to its projective cover.

Suppose that Top(M) contains more than one simple summands. Take U and V from Top(M). By (1), U is not isomorphic to V. By the previous lemma and (1), U ↓_{H_{1}}^{H} contains exactly a simple summand k_{1} ≅ k. Similarly, V ↓_{H_{1}}^{H} contains exactly a simple summand k_{2} ≅ k as well.

Let ΩM be the first syzygy of M. Then ΩM ≠ 0 since M is indecomposable and its top is not simple. Moreover, U and V appears as summands of Soc(ΩM). Therefore, as we did in the proof of (1), using ΩM we can construct a family of infinitely many non-isomorphic indecomposable representations of C. This contradiction shows that M must have a simple top, and (2) is proved.

Now we turn to (3). Suppose that Hom_{kH}(M, N) ≠ 0. By (2) we can suppose that M (resp., N) is a quotient module of an indecomposable projective module P_{U} (resp., P_{V}). Therefore, if Hom_{kH}(M, N) ≠ 0, the multiplicity [N : U] ≠ 0, so [P_{V} , U] ≠ 0. This happens if and only if U and V lie in the same block B of kH, and U and V are connected to the same vertex in the Brauer graph Γ of B. Therefore, Γ has a piece as follows:

•_{...}  _{U}_{V}  _{...}

So we can define a uniserial kH-module L such that Top(L) ≅ Top(N) ≅ V, Soc(L) ≅ Soc(M) ≅ U, and [L : U] = [L : V] = 1. Therefore, L is a quotient module of P_{V}, and it is actually a quotient module of N.

Clearly, Hom_{kH}(N, L) ≠ 0 ≠ Hom_{kH}(M, L). Therefore,

dim_{k} Hom_{kH_{1}}(S, L ↓_{H_{1}}^{H}) = dim_{k} Hom_{kH}(S ↑_{H_{1}}^{H}, L) ⩾ 2.

So we can find two simple summands S_{1} ≅ S ≅ S_{2} in Soc(L ↓_{H_{1}}^{H}). Note that End_{kH}(N) ≅ k. Using this module L, as we did in the proof of (1) we can construct infinitely many indecomposable non-isomorphic representations of C. The conclusion of (3) follows from this contradiction. □

In the following lemma we give some algebras of infinite representation types. They will be used in the proof of the next proposition.

Lemma 3.16. The path algebras of the following quivers with relations are of infinite representation type:

[figure: •; t; •; γ; β; •; t^{3} = 0,]
[figure: •; t; •; γ; β; •; s; t^{2} = s^{2} = 0,]
[figure: •; •; θ; β; γ; •; t; t^{2} = 0.; •]

Proof. These results can be proved by the covering theory. As example, we show the first algebra is of infinite representation type, and leave the other two to the reader. A universal covering of the first quiver is described as follows with relation t^{3} = 0:

\xymatrix{… & • & • & • & • & • & … \\ … & • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} & … \\ … \ar[r]^{t} & • \ar[r]^{t} & • \ar[r]^{t} & • \ar[r]^{t} & • \ar[r]^{t} & • \ar[r]^{t} & …}

This covering is not locally representation-finite since it contains a subquiver of infinite representation type as below:

\xymatrix{• & • & • \\ • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} & • \ar[u]^{β} \ar[d]_{γ} \\ • \ar[r]^{t} & • \ar[r]^{t} & •}

Therefore, the first algebra is of infinite representation type. □

Proposition 3.17. If kC is of finite representation type, then dim_{k} End_{kH}(k ↑_{H_{1}}^{H} ) ⩽ 3. That is, H_{1}\H/H_{1} has at most three double cosets.

The second statement comes from

Hom_{kH}(k ↑_{H_{1}}^{H}, k ↑_{H_{1}}^{H}) ≅ Hom_{kH_{1}}(k ↑_{H_{1}}^{H}↓_{H_{1}}^{H}, k)

≅ ⨁_{s∈H_{1}\H/H_{1}} Hom_{kH_{1}}((s ⊗ k) ↑_{H_{1}∩sH_{1}s^{−1}}^{H_{1}}, k)

≅ ⨁_{s∈H_{1}\H/H_{1}} Hom_{k(H_{1}∩sH_{1}s^{−1})}(s ⊗ k, k).

So dim_{k} End_{kH}(k ↑_{H_{1}}^{H}) coincides with the number of double cosets of H_{1}\H/H_{1}.

Proof. Note that C has a quotient category G (defined before Proposition 3.12) such that G(x, x) = 1, G(y, y) = H, and C(x, y) = Hᾱ where Stab_{H}(ᾱ) = H_{1}. Therefore, it suffices to show that kG is of infinite representation type if dim_{k} End_{kH}(k ↑_{H_{1}}^{H} ) ⩾ 4.

Denote all indecomposable summands of k ↑_{H_{1}}^{H} by M_{1}, M_{2}, …, M_{n}. By Proposition 3.15, each of them is a quotient module of an indecomposable projective module of kH. Let e_{1}, e_{2}, …, e_{n} be primitive idempotent elements in kH such that P_{i} = kHe_{i} is a projective cover of M_{i}, 1 ⩽ i ⩽ n. By Proposition 6.4, we can assume the following properties: n ⩽ 3; Hom_{kH}(P_{i}, M_{j}) = 0, 1 ⩽ i ≠ j ⩽ n. Consider the algebra

Λ = End_{kG}(kG(1_{x} + e_{1} … + e_{n}))^{op} ≅ (e_{1} + … + e_{n} + 1_{x})kG(e_{1} + … + e_{n} + 1_{x}).

We only need to show that Λ is of infinite representation type if dim_{k} End_{kH}(k ↑_{H_{1}}^{H} ) ⩾ 4 since by Proposition 2.5 on page 36 in [3] Λ-mod is equivalent to a subcategory of kG-mod.

Observe that 1_{x}kG(e_{1} + … e_{n}) = 0, 1_{x}kG1_{x} ≅ k, and

(e_{1} + … + e_{n})kG1_{x} = (e_{1} + … + e_{n})kHᾱ

≅ Hom_{kH}(kH(e_{1} + … + e_{n}), kHᾱ)

≅ Hom_{kH}(kH(e_{1} + … + e_{n}), k ↑_{H_{1}}^{H})

≅ Hom_{kH}(kH(e_{1} + … + e_{n}), M_{1} ⊕ … ⊕ M_{n})

≅ Hom_{kH}(P_{1} + … + P_{n}, M_{1} ⊕ … ⊕ M_{n})

≅ ⨁_{i=1}^{n} Hom_{kH}(P_{i}, M_{i})

as left (e_{1} +…+e_{n})kG(e_{1} +…+e_{n})-modules since Hom_{kG}(P_{i}, M_{j}) = 0 for 1 ⩽ i ≠ j ⩽ n. By the same reasoning, the product of an element in the two-sided Λ-ideal generated by ⊕_{1⩽i≠j⩽n} Hom_{kH}(P_{i}, P_{j}) and an element in (e_{1} + … + e_{n})kG1_{x} is 0. We also have:

dim_{k} Hom_{kH}(kHe_{i}, M_{i}) = dim_{k} End_{kH}(M_{i}).

Therefore, modulo the two-sided Λ-ideal generated by ⊕_{1⩽i≠j⩽n} Hom_{kH}(P_{i}, P_{j}), Λ has a quotient algebra Λ̄ which is isomorphic to the path algebra of the following bounded quiver

[figure: •; t_{1}; •; t_{n}; •; β_{1}; β_{...}; β_{n}; •; t_{...}]

with relations t_{i}^{a_{i}} = 0 = t_{i}^{d_{i}}β_{i}, where a_{i} ⩾ d_{i} = dim_{k} End_{kH}(M_{i}), 1 ⩽ i ⩽ n.

Suppose that d = dim_{k} End_{kH}(k ↑_{H_{1}}^{H}) ⩾ 4. Note that d = ∑_{i=1}^{n} d_{i}. We have several cases:

Case I: n = 1. Therefore, k ↑_{H_{1}}^{H} is an indecomposable kH-module, and Λ̄ is isomorphic to the path algebra of the following quiver with relation t^{a} = 0 = t^{d}β, a ⩾ d. we deduce that Λ̄ is of infinite representation type from Bongartz’s list in [4, 6] if d ⩾ 4.

[figure: •; β; •; t]

Case II: n = 2, so d_{1} + d_{2} = d ⩾ 4. If one of them is 1, then Λ̄ has the first algebra in the previous lemma as a quotient algebra; if both numbers are at least 2, then Λ̄ has the second algebra in the previous lemma as a quotient algebra.

Case III: n = 3, so d_{1} + d_{2} + d_{3} ⩾ 4. Therefore, at least one of them must be bigger than 1, and Λ̄ has the third algebra in the previous lemma as a quotient algebra.

Consequently, in all cases we get infinite representation type for Λ̄. This contradiction tells us that d ⩽ 3. □

4. Classifications of representation types.

As in the previous section C is a connected skeletal finite EI category with two objects x and y such that C(y, x) = ∅. Let α, G_{0} ◁ G_{1} ⩽ G and H_{0} ◁ H_{1} ⩽ H as defined before. In this section we use the criteria developed in the previous section to classify the representation type of C. We point out that this classification is not complete. However, it covers most cases.

When both |G| and |H| are invertible, we can use the algorithm described in [15] to construct the ordinary quiver Q of kC. Moreover, we proved in that paper that kC is Morita equivalent to kQ. Therefore, the representation type of kC can be determined by Gabriel’s theorem.

When both G and H are p-groups, this classification has been described in [11]. We record his result here.

Proposition 4.1. (Theorem 5.3 in [12]) Let C be a finite EI category with two objects for which the automorphism groups are p-groups. Suppose that C(y, x) = ∅. Let G = C(x, x) and H = C(y, y). Then kC is of finite representation type if and only if the following conditions are satisfied

(1) both G and H are cyclic;

(2) either G or H acts transitively on C(x, y);

(3) one of the following is true:

(a) |C(x, y)| ⩽ 1;

(b) |G||H| ⩽ 3;

(c) p = 2 = C(x, y), either G or H is trivial;

(d) p = 2 = C(x, y), either G or H has order 2 and acts transitively;

(e) p = 3 = |C(x, y)| = |G| = |H|, and both G and H act transitively.

An immediate corollary is:

Corollary 4.2. Suppose that p ⩾ 5. Let C be a skeletal finite EI category for which the automorphism groups of all objects are p-groups. If kC is of finite representation type, then |C(x, y)| ⩽ 1 for every pair of distinct objects x and y.

Proof. Consider the full subcategory with objects x and y and use the previous proposition. □

Now suppose that both G and H are arbitrary. By the previous section, in many cases the representation type of kC is determined by two pieces of information: the transitivity of the actions by G and H on C(x, y), and the triviality of the actions by Sylow p-subgroups in G and H. Indeed, the combination of conditions (a-c)

(a) Both G and H act transitively.

(b) One of G and H acts transitively, and the other one does not.

(c) Neither G nor H acts transitively. and conditions (1-3)

(1) Both O^{p′}G and O^{p′}H act trivially.

(2) One of O^{p′}G and O^{p′}H acts trivially, and the other one does not.

(3) Neither O^{p′}G nor O^{p′}H acts trivially.

give us 8 situations (the combination (a)+(2) cannot happen). By Theorem 1.1, if char(k) = p ≠ 2, 3, we get infinite representation type for five cases: (c)+(1), (c)+(2), (c)+(3), (a)+(3), (b)+(3). In the first subsection we will prove the finite representation type for the case (a)+(1).

4.1. Type (a) + (1). We first prove the following lemma.

Lemma 4.3. If C(x, y) has only one morphism, then kC is of finite representation type.

Proof. Let p = char(k). Take a Sylow p-subgroup P of G and a Sylow p-subgroup Q of H. This can always be done since if G or H has order invertible in k, we then take the trivial group as the unique Sylow p-subgroup by our convention. Let S be the following subcategory:

[figure: S :; x; P; α; y; Q],

and define another subcategory J as below:

[figure: J :; x; G; α; y; Q].

By Proposition 4.1, kS is of finite representation type. Applying Proposition 3.8 to the pair (J , S), we conclude that kJ is of finite representation type, and so is the category algebra of the opposite category J^{op}. Applying Proposition 3.8 again to the pair (C^{op}, J^{op}), we obtain the finite representation type of kC^{op}. Therefore, kC is of finite representation type. □

Now we prove Theorem 1.2.

Theorem 4.4. Suppose that both G and H act transitively on C(x, y). If char(k) = p ≠ 2, 3, then kC is of finite representation type if and only if all p-subgroups of G and H act trivially on C(x, y).

Proof. Since both G and H act transitively on C(x, y), we have G_{0} ◁ G_{1} = G, H_{0} ◁ H_{1} = H. If G or H has a p-subgroup acting nontrivially on C(x, y), so does the other one by the remark before Proposition 3.10. Therefore, kC has infinite representation type by Proposition 3.10.

On the other hand, if all p-groups of G and H act trivially on C(x, y), then |G/G_{0}| = |H/H_{0}| = n is invertible in k. Let K be the following subcategory:

[figure: K :; x; G_{0}; α; y; H_{0}],

whose category algebra is of finite representation type by the previous lemma. Applying Proposition 3.8 to the pair (C, K), we conclude that kC is of finite representation type as well. □

4.2. Both G and H are abelian. Throughout this subsection we assume that G and H are abelian, and char(k) = p ≠ 2, 3. As before, we suppose that H acts transitively on C(x, y). Note that when p = 0, by our convention 1 is the unique p-subgroup of a finite group.

We first describe some easy observation.

Lemma 4.5. Suppose that kC is of finite representation type and char(k) = p ≠ 2, 3. Then both O^{p′}G and O^{p′}H act trivially on C(x, y), and |H : H_{1}| ⩽ 3.

Proof. Since char(p) = p ≠ 2, 3, we know that O^{p′}G must act trivially on C(x, y). Let P be the unique Sylow p-subgroup of H. Without loss of generality we assume that P ≠ 1 (so p ⩾ 5). By Proposition 3.12, either P ⩽ H_{0} ⩽ H_{1} or P ∩ H_{0} = 1. We show the second possibility cannot happen, so P ⩽ H_{0} acts trivially on C(x, y), as asserted by the first statement

Otherwise, if P ∩H_{0} = 1, then P ∩H_{1} = 1 as well (see the remark before Lemma 3.11). Write H = P × H′. By our assumption, we can choose H_{1} ⩽ H′. Factoring out H′, we get a quotient category Q with Q(x, x) = 1, Q(y, y) = P, and Q(y, y) acts regularly on Q(x, y). Thus kQ is of infinite representation type, so does kC, contradicting the given condition.

The second statement comes from Proposition 3.17. □

Therefore, if kC is of finite representation type, H/H_{1} ≅ C_{n} where C_{n} is a cyclic group of order n, 1 ⩽ n ⩽ 3.

From now on we suppose that O^{p′}G ⩽ G_{0} and O^{p′}H ⩽ H_{0}. Note that k(G/G_{0}) = k(G_{1}/G_{0}) is a semisimple kG-module with pairwise non-isomorphic simple summands. Let S_{1}, …, S_{r} be all simple summands. Correspondingly, let e_{1}, …, e_{r} be primitive idempotents in kG such that kGe_{i} is a projective cover of S_{i}, 1 ⩽ i ⩽ r. Denote 1 − e_{1} − … − e_{r} by f. These simple modules S_{i} can be viewed as kH_{1}- modules as well since G/G_{0} ≅ H_{1}/H_{0}, and we let E_{i} be idempotents in kH such that kHE_{i} is the projective cover of kH ⊗_{kH_{1}} S_{i}, 1 ⩽ i ⩽ r. By the previous lemma, H/H_{1} ≅ C_{n} with n ⩽ 3, so kH ⊗_{kH_{1}} S_{i} has n simple summands, and each E_{i} is a sum of n primitive idempotents in kH. Let F = 1 − E_{1} − … − E_{r}. Note that

kHα ≅ k ↑_{H_{0}}^{H} ≅ kH ⊗_{kH_{1}} (k ↑_{H_{0}}^{H_{1}}) ≅ ⊕_{i=1}^{r}kH ⊗_{kH_{1}} S_{i}.

Lemma 4.6. Notation as above. Then kC = ⊕_{i=1}^{r}Λ_{i} ⊕ kHF ⊕ kGf as algebras, where Λ_{i} = kHE_{i} ⊕ kGe_{i} ⊕ kHE_{i}α as vector spaces.

Proof. Since C(x, y) = Hα, 1_{y} = F + E_{1} + … + E_{r}, and

KHF α ≅ ⊕_{i=1}^{r}F (kH ⊗_{kH_{1}} S_{i}) = 0

since kH ⊗_{kH_{1}} S_{i} has a projective cover kHE_{i}. Therefore, the first decomposition is true as vector spaces, and we want to show that all summands are kC-ideals. Particularly, it suffices to show the following identities: kHE_{i}αG ⊆ kHE_{i}α and kHαGe_{i} ⊆ kHE_{i}α for 1 ⩽ i ⩽ r; kHF αG = 0; and kHαGf = 0.

First, we have

kHαGf = kHαf = ⊕_{i=1}^{r}kH ⊗_{kH_{1}} S_{i}f = 0

since f = 1 − ∑_{i=1}^{r} e_{i} and kGe_{i} is the projective cover of S_{i}.

The identity KHF αG = 0 is clear as we have shown kHF α = 0. The identity kHE_{i}αG ⊆ kHE_{i}α is also clear since αG ⊆ Hα and H is abelian. We claim kHαe_{i} = kHE_{i}α = kHE_{i}αe_{i}, implying kHαGe_{i} = kHαe_{i} ⊆ kHE_{i}α. Indeed,

E_{i}kHα(1 − e_{i}) ≅ ⊕_{j=1}^{r}E_{i}(kH ⊗_{kH_{1}} S_{j})(1 − e_{i})
≅ E_{i}(kH ⊗_{kH_{1}} (S_{1} ⊕ … ⊕ S_{i−1} ⊕ S_{i+1} ⊕ ⋯ ⊕ S_{n}))
≅ E_{i}(S_{1} ↑_{H_{1}}^{H} ⊕ … ⊕ S_{i−1} ↑_{H_{1}}^{H} ⊕S_{i+1} ↑_{H_{1}}^{H} ⊕ … ⊕ S_{n} ↑_{H_{1}}^{H})

is 0 since kHE_{i} is the projective cover of S_{i} ↑_{H_{1}}^{H}, so kHE_{i}α = kHE_{i}αe_{i}. Similarly, we can show (1 − E_{i})kHαe_{i} = 0. Therefore, kHαe_{i} = kHE_{i}αe_{i}. □

We see at once that kC is of finite representation type if and only if all Λ_{i} are of finite representation type, 1 ⩽ i ⩽ r, for which the structure is described by the following lemma.

Lemma 4.7. Let Λ_{i} be as in the previous lemma, 1 ⩽ i ⩽ r. Then it is isomorphic to the path algebra of the following bounded quiver with relations: δ_{i}^{t} = δ_{i}α_{i} = α_{i}θ = θ^{s} = 0, 1 ⩽ i ⩽ n = |H : H_{1}|, where t = |O^{p′}H| and s = |O^{p′}G|.

[figure: •; δ_{...}; •; δ_{1}; •; θ; α_{...}; α_{n}; α_{1}; •; δ_{n}]

Proof. Note that H/H_{1} ≅ C_{n}. Therefore, S_{i} ↑_{H_{1}}^{H} is semisimple and has n nonisomorphic simple summands, and kHE_{i} = E_{i}kHE_{i} is a direct sum of endomorphism algebras of n non-isomorphism projective kH-modules. Consequently, kHE_{i} ≅ ⊕_{i=1}^{n}k[δ_{i}]/(δ_{i}^{t}) where t = |O^{p′}H|. Similarly, kGe_{i} ≅ e_{i}kGe_{i} is the endomorphism algebra of an indecomposable projective kG-module. Therefore, kGe_{i} ≅ k[θ]/(θ^{s}) where s = |O^{p′}G|.

As a left E_{i}kHE_{i}-module, we have

E_{i}kHα ≅ Hom_{kH}(kHE_{i}, kHα) ≅ Hom_{kH}(kHE_{i}, ⊕_{j=1}^{r}(S_{j} ↑_{H_{1}}^{H}))
≅ Hom_{kH}(kHE_{i}, S_{i} ↑_{H_{1}}^{H})

which is the space of all homomorphisms from kHE_{i} to its top S_{i} ↑_{H_{1}}^{H}. Since kHE_{i}α = kHE_{i}αe_{i} from the proof of the previous lemma, as an e_{i}kGe_{i}-module,

E_{i}kHαe_{i} ≅ ⊕_{j=1}^{r}E_{i}(kH ⊗_{kH_{1}} S_{j})e_{i} ≅= (kH ⊗_{kH_{1}} S_{i})e_{i}
≅ (S_{i}e_{i})^{d} ≅ Hom_{e_{i}kGe_{i}}(e_{i}kG, S_{i})^{d}.

which is the direct sum of d copies of projections from e_{i}kG to its top S_{i}, where d = |H : H_{1}|. The conclusion follows from these observations. □

Now we are ready to classify the representation types of finite EI categories with two objects whose automorphism groups are abelian for p ≠ 2, 3.

Theorem 4.8. Suppose that both G and H are abelian, char(k) = p ≠ 2, 3. Without loss of generality assume that H acts transitively on C(x, y). Let s = |O^{p′}G|, t = |O^{p′}H|, and n = |H : H_{1}|. Then kC is of finite representation type if and only if both O^{p′}G and O^{p′}H act trivially on C(x, y), and one of the following conditions hold:

(1) n = 1 for s, t ⩾ 5;

(2) n ⩽ 2 for s = 1, t ⩾ p or t = 1, s ⩾ p;
(3) n ⩽ 3 for t = s = 1.

Proof. By Lemma 4.5, we know that n cannot be bigger than 3 for all cases. Lemmas 4.6 and 4.7 tell us that it suffices to consider the representation types of path algebras described in Lemma 4.8. Now we can use covering theory, Auslander-Reiten quivers, and Proposition 4.1 to get the conclusion.

As an example, let us show the case t = 1, s ⩾ p ⩾ 5, and n = 3 is of infinite representation type. In that case the corresponding path algebra has the following covering:

\xymatrix{ & • & • & • & • & • & • \\ … \ar[rr]^{θ} &  & • \ar[rr]^{θ} \ar[ul]^{ρ} \ar[u]^{β} \ar[d]^{γ} &  & • \ar[rr]^{θ} \ar[ul]^{ρ} \ar[u]^{β} \ar[d]^{γ} &  & • \ar[r]^{θ} \ar[ul]^{ρ} \ar[u]^{β} \ar[d]^{γ} & … \\  &  & • &  & • &  & •}

with relations θ^{s} = ρθ = βθ = γθ = 0. This covering is not locally representationfinite since it has a subquiver as

\xymatrix{ & • \\ • & • \ar[r]^{ρ} \ar[l]^{β} \ar[u]^{γ} \ar[d]^{θ} & • \\ & •}.

Therefore, kC has infinite representation type for t = 1, s ⩾ 5 and n = 3. □

5. Finite EI categories with three objects

The reader can see from our previous analysis that the problem to classify the representation types of category algebras of arbitrary finite EI categories is very complicated. Indeed, the answer for finite EI categories with two objects is still not completely obtained. However, for an arbitrary finite EI category with more than two objects, by considering its full subcategories with two objects, we can still deduce a lot of useful information on the representation type of its category algebra. In this section we consider two special collections.

5.1. Finite EI categories with three objects whose automorphism groups of objects are p-groups. Again, we assume that p ⩾ 5. The main techniques we use are covering theory (see [6]) and representations of string algebras (see [9]). By Proposition 4.1 and Corollary 4.2, we can make the following assumptions: all automorphism groups are cyclic, and there is at most one morphism between every pair of distinct objects.

Consider two families of finite EI categories C as below:

[figure: x; G; α; y; H; β; z; L; x; G; α; y; H; z; L; β]

Let |G| = p^{r}, |H| = p^{s}, and |L| = p^{t}. Note that the category algebra kC is isomorphic to the path algebra of one of the following bounded quivers with relations g^{p^{r}} = h^{p^{s}} = l^{r^{t}} = 0, hα = αg = 0, and lβ = βh = 0 (or βl = hβ = 0 for the second class).

[figure: •; g; α; •; h; β; •; l; •; g; α; •; h; •; l; β]

Proposition 5.1. Let C be a connected skeletal finite EI category with three objects such that the automorphism groups of all objects are p-groups and suppose that char(k) = p ⩾ 5. Then kC is of finite representation type if and only if C or C^{op} satisfies one of the following conditions:

(1) it lies in the first family;

(2) it lies in the second family and s = 0;

(3) it lies in the second family and r = t = 0.

Proof. If kC is of finite representation type, then either C or C^{op} lies in one of these two families. Without loss of generality we suppose that C are in one of these two families. There are four cases:

Case I: If C is in the first family, then kC is a string algebra. For a definition of string algebras, see [9]. In that paper an algorithm to construct all indecomposable representations (up to isomorphism) of a string algebra is described. Thus we can describe all indecomposable representations R of C as follows: for w ∈ {x, y, z}, R(w) is an indecomposable representation of the automorphism group of the corresponding object; the maps R(α) and R(β) send the top (if nonzero) of a group representation isomorphically onto the socle (if nonzero) of another group representation. Obviously, kC is of finite representation type.

Case II: C lies in the second family, and s = 0. In this case kC is still a string algebra, and we can describe all indecomposable representations R of C as above. Similarly, the value of R on each object is an indecomposable representation of the automorphism group of this object, so kC is of finite representation type as well.

Case III: C lies in the second family, s ≠ 0, and either r ≠ 0 or t ≠ 0. Without loss of generality we assume that r ≠ 0. Then kC has a covering (see [6]) as shown below with relations g^{p^{r}} = h^{p^{s}} = l^{g^{t}} = 0, αg = hα = 0, and hβ = βl = 0. Here the arrow marked by l does not exist if t = 0.

\xymatrix{… \ar[r]^{g} & • \ar[r]^{g} \ar[d]^{α} & • \ar[r]^{g} \ar[d]^{α} & • \ar[r]^{g} \ar[d]^{α} & … \\ … \ar[r]^{h} & • \ar[r]^{h} & • \ar[r]^{h} & • \ar[r]^{h} & … \\ … \ar@{-->}[r]^{l} & • \ar@{-->}[r]^{l} \ar[u]^{β} & • \ar@{-->}[r]^{l} \ar[u]^{β} & • \ar@{-->}[r]^{l} \ar[u]^{β} & …}

This quiver is not locally representation-finite since it has the following subquiver with 2p + 1 ⩾ 11 vertices, which is of infinite representation type:

\xymatrix{1 \ar[r]^{h} & 2 \ar[r]^{h} & … \ar[r]^{h} & p & p + 1 \ar[l]^{α} \ar[r]^{g} & p + 2 \ar[r] \ar[r]^{g} & … \ar[r]^{g} & 2p \\  &  &  & 0 \ar[u]^{β}}

Therefore, kC is of infinite representation type.

Case IV: C lies in the second family, and r = t = 0. We claim that the category algebra of this category is of finite representation type. To prove this, let R be an indecomposable representation of C. It suffices to show that R(y) is an indecomposable kH-module. Otherwise, R(y) = M ⊕ N where M, N ∈ kH-mod are nonzero and M is indecomposable. Note that R(α) and R(β) are injective, and their images are contained in Soc(R(y)). Take 0 ≠ v ∈ Soc(M), and let u and w be the preimages of v in R(x) and R(z) respectively (if a preimage does not exist, we take 0 alternately). Then the reader can check that the kC-submodule R′ of R with R′(x) = ⟨u⟩, R′(y) = M and R′(z) = ⟨w⟩ is a nonzero proper summand of R. This contradicts the assumption that R is indecomposable. Therefore, R(y) is indecomposable, so kC is of finite representation type. □

5.2. Finite free EI categories with three objects. The concepts of unfactorizable morphisms and finite free EI categories are introduced in [15].

Definition 5.2. A morphism α : x → z in a finite EI category C is unfactorizable if α is not an isomorphism and whenever it has a factorization as a composite

\xymatrix{x \ar[r]^{β} & y \ar[r]^{γ} & z}, then either β or γ is an isomorphism.

For finite EI categories, unfactorizable morphisms are precisely irreducible morphisms which are widely used in [3, 4, 26]. But in a more general context, they are different, see Example 2.4 in [15].

Definition 5.3. A finite EI category C is called a finite free EI category if it satisfies the following Unique Factorization Property (UFP): whenever a non-isomorphism α has two decompositions into unfactorizable morphisms,

\xymatrix{x = x_{0} \ar[r]^{α_{1}} & x_{1} \ar[r]^{α_{2}} & … \ar[r]^{α_{m}} & x_{m} = y \\ x = x_{0} \ar[r]^{β_{1}} & y_{1} \ar[r]^{β_{2}} & … \ar[r]^{β_{n}} & y_{n} = y}

then m = n, x_{i} = y_{i}, and there are h_{i} ∈ C(x_{i}, x_{i}) such that the following diagram commutes, 1 ⩽ i ⩽ n − 1:

[figure: x_{0}; α_{1}; id; x_{1}; α_{2}; h_{1}; …; α_{...}; h_{...}; x_{n−1}; α_{n}; h_{n−1}; x_{n}; id; x_{0}; β_{1}; x_{1}; β_{2}; …; β_{...}; x_{n−1}; β_{n}; x_{n}] .

Note that every finite free EI category C can be graded as follows: automorphisms lie in degree 0, and unfactorizable morphisms are in degree 1. The reader can easily check the following fact: an arbitrary finite EI category C is a finite free EI category if and only if C can be graded in the above way, and the graded category algebra kC is isomorphic to the tensor algebra generated by the degree 0 and the degree 1 components.

Some properties of finite free EI categories are collected below.

Proposition 5.4. Let C be a finite EI category.

(1) There is a finite free EI category Ĉ and a full functor F̂ : Ĉ → C such that F̂ is the identity map restricted to objects, automorphisms and unfactorizable morphisms in C. This finite free EI category Ĉ is unique up to isomorphism.

(2) C is a finite free EI category if and only if so are all full subcategories.

(3) The algebra kC is hereditary if and only if C is a finite free EI category and the automorphism group of each object has order invertible in k.

Proof. See Propositions 2.9, 2.10 and Theorem 5.3 in [15]. The category Ĉ in (1) is called the free EI cover of C. □

From now on we let C be a finite free EI category with three objects. Then C has one of the following two descriptions.

[figure: C :; x; G; HαG; ...; y; H; LβH; ...; z; L; x; G; HαG; ...; y; H; z; L; HβL; ...].

For the first category, C(x, z) = LβH ×_{H} HαG, where ×_{H} is the biset product defined in [23]. Let H_{1} = Stab_{H}(αG), H_{2} = Stab_{H}(Lβ) (or H_{2} = Stab_{H}(βL) for the second one).

We need the following technical lemma to prove the main result of this subsection.

Lemma 5.5. Let G be a finite group with cyclic Sylow p-subgroups. Let G_{1} and G_{2} be two proper subgroups. Suppose that k ↑_{G_{1}}^{G} and k ↑_{G_{2}}^{G} has no common composition factors except k. If |G_{1}\G/G_{2}| ⩾ 2, then P_{k} has only composition factors isomorphic to k, and the Scott modules of both permutation modules are quotient modules of P_{k} and are not isomorphic to k.

Proof. Observe that

dim_{k} Hom_{kG}(k ↑_{G_{1}}^{G}, k ↑_{G_{2}}^{G}) = dim_{k} Hom_{kG_{1}}(k, k ↑_{G_{2}}^{G}↓_{G_{1}}^{G})

= dim_{k} Hom_{kG_{1}}(k, ⨁_{s∈G_{1}\G/G_{2}}(s ⊗ k) ↑_{G_{1}∩sG_{2}s^{−1}}^{G_{1}})

which is exactly the number of double cosets in G_{1}\G/G_{2}.

Decompose k ↑_{G_{1}}^{G}= M_{1}⊕…⊕M_{m} and k ↑_{G_{2}}^{G}= N_{1}⊕…⊕N_{m} into indecomposable summands, and suppose that M_{1} and N_{1} are the Scott modules. By the given condition, P_{k} is a uniserial module. Suppose that P_{k} has a composition factor S ≇ k. We want to get a contradiction. Note that by this assumption, Top(P_{k}/k) ≇ k.

Since k ↑_{G_{1}}^{G} and k ↑_{G_{2}}^{G} have no other common composition factors except k, for 2 ⩽ i ⩽ m and 1 ⩽ j ⩽ n, Hom_{kH}(M_{i}, N_{j}) = 0 because the simple summands of Top(M_{i}) are not isomorphic to k, so are not composition factors (up to isomorphism) of N_{j}. Therefore,

Hom_{kG}(k ↑_{G_{1}}^{G}, k ↑_{G_{2}}^{G}) = ⨁_{i=1}^{m} ⨁_{j=1}^{n} Hom_{kG}(M_{i}, N_{j}) ≅ ⨁_{j=1}^{n} Hom_{kG}(M_{1}, N_{j}).

Note that M_{1} is a string module, and Top(P_{k}/k) ≇ k. Therefore, for any 0 ≠ φ ∈ Hom_{kG}(M_{1}, N_{j}), if the image of φ is not contained in Soc(N_{j}), we can get a common composition factor S ≇ k of M_{1} and N_{j}. This is not allowed by our assumption. Moreover, the image of φ must be isomorphic to k by the same reason. Consequently, we get Hom_{kG}(M_{1}, N_{j}) = 0 for 2 ⩽ j ⩽ n, and Hom_{kG}(M_{1}, N_{1}) is spanned by the morphism sending the composition factor k in Top(M_{1}) onto the composition factor k in Soc(N_{1}). So Hom_{kG}(k ↑_{G_{1}}^{G}, k ↑_{G_{2}}^{G}) is one-dimensional, contradicting the given condition. This contradiction tells us that P_{k} only has composition factors isomorphic to k, and the principal block B_{0}(kG) has only one simple module k lying in it. Thus both Scott modules are quotient modules of P_{k}. Moreover, since

dim_{k} Hom_{kG}(M_{1}, N_{1}) = dim_{k} Hom_{kG}(k ↑_{G_{1}}^{G}, k ↑_{G_{2}}^{G}) = |G_{1}\G/G_{2}| ⩾ 2,

we conclude that neither of them is isomorphic to k. □

Now we can prove:

Proposition 5.6. Let C be one of the above two categories, and suppose that its category algebra is of finite representation type. If char(k) = p ≠ 2, 3, then H acts transitively on either C(x, y) or D(y, z) (or C(z, y) for the second one). Moreover, |H_{1}\H/H_{2}| = 1.

Proof. We only prove the first statement for the first category since the proof also works for the second one with a very small modification. Let G_{1} = Stab_{G}(Hα) and L_{1} = Stab_{L}(βH). Suppose that H does not act transitively on either C(x, y) or C(y, z). By considering the full subcategory E with objects x and y we conclude that G acts transitively on C(x, y). Similarly, L acts transitively on C(y, z). Therefore, G_{1} and L_{1} are proper subgroups of G and L respectively. We claim that there is an indecomposable projective kG-module P_{S} such that S ≇ k_{G} and Top(P_{S} ↓_{G_{1}}^{G}) has a simple summand isomorphic to k. Dually, there is an indecomposable projective kLmodule P_{T} such that T ≇ k_{L} and Soc(P_{T} ↓_{L_{1}}^{L}) has a simple summand isomorphic to k.

We prove the first claim since the second one is a dual statement. By Lemma 3.14, it is equivalent to saying that k ↑_{G_{1}}^{G} has a composition factor S ≇ k_{G}, which is obviously true for p = 0. Thus we let p ⩾ 5. If k ↑_{G_{1}}^{G} has only composition factors isomorphic to k, then it is indecomposable. By Proposition 3.12, either |G_{1}| is invertible or G_{1} contains a Sylow p-subgroup of G. The second case cannot happen. Otherwise k ↑_{G_{1}}^{G} ≅ k, which is absurd since G_{1} is a proper subgroup of G. Therefore, |G_{1}| must be invertible in k and k ↑_{G_{1}}^{G} ≅ P_{k}. Moreover, G has a nontrivial Sylow p-subgroup since dim_{k} P_{k} = |G : G_{1}| > 1. Consequently, dim_{k} End_{kG}(P_{k}) = dim_{k} P_{k} ⩾ p ⩾ 5, so kE is of infinite representation type by Proposition 3.17. This is not allowed, and the claim is proved.

For an indecomposable representation R of the following quiver with dimension vector (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}),

[figure: 1; k^{d_{1}}; φ_{1}; 2; 5; 4; R; k^{d_{2}}; φ_{2}; k^{d_{5}}; φ_{4}; φ_{3}; k^{d_{4}}; 3; k^{d_{3}}]

we can construct an indecomposable corresponding representation R̃ of C as follows: R̃(x) ≅ k^{d_{1}} ⊕ P_{S}^{d_{2}}, R̃_{(}y) ≅ k^{d_{5}}, R̃(z) ≅ k^{d_{3}} ⊕ P_{T}^{d_{4}}. Linear maps R̃(α) and R̃(β) can be defined in a way as we did in the proofs of Propositions 3.9 and 3.12. Consequently, kC is of infinite representation type. This contradiction shows that H must act transitively on at least one biset, proving the first statement.

Now we turn to the proof of the second statement. There are three cases:

Case I: C is the first category. We consider the full subcategory E with objects x and z. Note that E(x, z) = LβHαG. Since kE is of finite representation type as well, either L or G should act transitively on E(x, z). Without loss of generality suppose that L acts transitively, i.e., LβHαG = Lβα.

Take an arbitrary h ∈ H and consider βhα. Since βhα ∈ E(x, y) = Lβα, we can find some l ∈ L such that βhα = lβα. Consequently, this non-isomorphism in C(x, z) has two decompositions into unfactorizable morphisms. By the unique factorization property, there should exist h_{1} ∈ H such that the following diagram commutes:

\xymatrix{x \ar[r]^{α} \ar[d]^{1} & y \ar[r]^{lβ} \ar[d]^{h_{1}} & z \ar[d]^{1} \\ x \ar[r]^{hα} & y \ar[r]^{β} & z}.

Consequently, lβ = βh_{1} and h_{1}α = hα. So h_{1} ∈ H_{2} and h^{−1}h_{1} ∈ H_{1}. Therefore, h ∈ H_{2}H_{1}, and |H_{1}\H/H_{2}| = 1.

In the next two cases we consider the second category. Suppose that |H_{1}\H/H_{2}| ⩾ 2 (so H_{1} ≠ H ≠ H_{2}), we want to get a contradiction.

Case II: k ↑_{H_{1}}^{H} and k ↑_{H_{2}}^{H} has a common composition factor S ≇ k. Then the projective kH-module P_{S} ≇ P_{k} satisfies that Top(P_{S} ↓_{H_{1}}^{H}) and Soc(P_{S} ↓_{H_{2}}^{H}) have a summand isomorphic to k_{H_{1}} and k_{H_{2}} respectively. For an indecomposable representation R of the following quiver with dimension vector (d_{1}, d_{2}, d_{3}, d_{4}),

[figure: 2; k^{d_{2}}; 1; 4; R; k^{d_{1}}; φ_{1}; φ_{3}; k^{d_{4}}; φ_{4}; φ_{3}; 3; k^{d_{3}}]

we can construct a corresponding representation R̃ of C as follows: R̃(x) ≅ k^{d_{1}}, R̃_{(}y) ≅ k^{d_{2}} ⊕P_{S}^{d_{3}}, R̃(z) ≅ k^{d_{4}}. Linear maps R̃(α) and R̃(β) can be defined in a way as we did before. We deduce that kC is of infinite representation type, contradicting the given condition.

Case III: k ↑_{H_{1}}^{H} and k ↑_{H_{2}}^{H} has no common composition factors except k. By the previous lemma, the projective kH-module P_{k} and the Scott modules of both k ↑_{H_{1}}^{H} and k ↑_{H_{2}}^{H} only have composition factors isomorphic to k. Moreover, their dimensions are all bigger than 1.

Consider the following category E:

[figure: E :; x; 1; Hᾱ; ...; y; H; z; 1; Hβ̄; ...],

where Stab_{H}(ᾱ) = H_{1} and Stab_{H}(β̄) = H_{2}. The reader can check that E is a quotient category of C. So it suffices to prove the infinite representation type of kE.

Let e be a primitive idempotent in kH such that kHe ≅ P_{k}. Consider the algebra

Λ = End_{kE}(kE1_{x} ⊕ kEe ⊕ kE1_{z})^{op} ≅ (1_{x} + e + 1_{z})kE(1_{x} + e + 1_{z}).

It is easy to check that 1_{x}kE1_{x} ≅ k ≅ 1_{z}kE1_{z}, ekEe ≅ k[t]/(t^{d}) where d = dim_{k} P_{k} ⩾ 2, and

dim_{k} ekE1_{x} = dim_{k} ekHα = dim_{k} Hom_{kH}(P_{k}, k ↑_{H_{1}}^{H}) = s ⩾ 2,

where s is the dimension of the Scott module of k ↑_{H_{1}}^{H}. Similarly, dim_{k} ekE1_{z} = t ⩾ 2, where t is the dimension of the Scott module of k ↑_{H_{2}}^{H}. Therefore, Λ is isomorphic to the path algebra of the following bounded quiver

[figure: •; γ; •; δ; •; μ]

with relations δ^{d} = δ^{s}γ = δ^{t}μ = 0. Its covering is not locally representation-finite since it contains the following subquiver:

\xymatrix{• \ar[r] & • \ar[d] & • \ar[l] \\ • \ar[r] & • & • \ar[l]}

Therefore, Λ and hence kE are of infinite representation type, contradicting the given condition.

□

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School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

E-mail address: lixxx480@math.umn.edu