REPRESENTATION TYPE OF SCHUR ALGEBRAS

STEPHEN R. DOTY, KARIN ERDMANN, STUART MARTIN, AND DANIEL K. NAKANO

Abstract. We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics.

1 Introduction

1.1 Let A be a finite-dimensional algebra over a field k and let mod(A) be the category of finite-dimensional A-modules. An algebra A is said to have finite representation type if mod(A) contains finitely many indecomposable modules up to isomorphism. Otherwise, the algebra A is of infinite representation type.

The classical Schur algebras, which we shall denote by S(n, d), are finite-dimensional algebras over k which arise in connection with the polynomial representation theory of the general linear group GL_{n} = GL_{n}(k) for k an infinite field. These algebras can be realized as the endomorphism algebra of the symmetric group Σ_{d} acting on d-fold tensor product of the standard representation of GL_{n}. The classification of Schur algebras of finite representation type is known by [Erd2]. Let p equal the characteristic of the field k.

Theorem (A). [Erd2] The Schur algebra S(n, d) has finite representation type if and only if one of the following occurs.

(i) p ≥ 2, n ≥ 3, d < 2p; (ii) p ≥ 2, n = 2, d < p^{2}; (iii) p = 2, n = 2, d = 5, 7.

Now regard G = GL_{n} as an affine algebraic group scheme over k, and assume k is an algebraically closed field of characteristic p > 0. Moreover, let F : G → G be the Frobenius morphism and G_{r}T be the inverse image under the rth iteration of F of the diagonal torus T in G. It is well-known that the group scheme G_{r}T and its rational representations give useful information on the rational representations of the corresponding reductive algebraic group (see [Jan]). The study of the polynomial representations of G_{r}T was initiated in [DNP1]. There certain new finite-dimensional algebras, dubbed the infinitesimal Schur algebras S(n, d)_{r}, were introduced and it was shown that the representation theory for these algebras determines the polynomial representations of G_{r}T. The classification of infinitesimal Schur algebras of finite representation type was achieved in [DNP3].

Date: February 1998.

1991 Mathematics Subject Classification. Primary: 16G10, 20G05; Secondary: 16S99, 17B10.

Research of the first author partially supported by NSF grant DMS-9401576.

Research of the fourth author partially supported by NSF grant DMS-9500715 and Utah St. Univ. grant 845986.

Theorem (B). [DNP3] The infinitesimal Schur algebra S(n, d)_{r} has finite representation type if and only if one of the following occurs.

(i) p ≥ 2, n ≥ 3, d < 2p, r ≥ 2; (ii) p ≥ 2, n ≥ 3, d < p, r = 1; (iii) p = 3, n = 3, d = 4, 5, r = 1; (iv) p = 2, n = 3, d = 2, 3, r = 1;

(v) p ≥ 2, n = 2, d < p^{2}, r ≥ 2; (vi) p = 2, n = 2, d = 5, 7, r ≥ 3; (vii) p = 2, n = 2, d odd, r = 2; (viii) p ≥ 2, n = 2, r = 1.

We remark that the semisimplicity of Schur and infinitesimal Schur algebras was recently settled completely. Here is the result.

Theorem (C). [DN] Let p be the characteristic of k. The Schur algebra S(n, d) is semisimple if and only if

(i) p = 0; (ii) p > d; (iii) p = 2, n = 2, d = 3.

Theorem (D). [DN] Let k be a field of characteristic p > 0. The infinitesimal Schur algebra S(n, d)_{r} is semisimple if and only if

(i) p ≥ 2, n ≥ 2, d < p, r ≥ 1; (ii) p = 2, n = 2, d = 3, r ≥ 1; (iii) p = 2, n = 2, d odd, r = 1.

1.2 Finite-dimensional algebras of infinite representation type can be split into two mutually exclusive categories: tame or wild. This is a deep theorem which was first proved by Drozd. An algebra A has tame representation type if for each dimension there exists finitely many one-parameter families of indecomposable objects in mod(A). For tame algebras the indecomposable modules are in some sense classifiable. On the other hand, the algebras of wild representation type are those whose module theory is as difficult to study as the module theory of the free associative algebra k⟨x, y⟩ on two variables. It is very much an open question whether a satisfying classification of the finite-dimensional k⟨x, y⟩-modules exists. The theory of finite-dimensional k⟨x, y⟩-modules is undecidable [Ben, p. 107]. The formal definitions of tame and wild representation type will be given in Section 2.

The purpose of this paper is to complete the classification of the representation type of the classical Schur algebras and the infinitesimal Schur algebras. Our results are presented in the following two theorems.

Theorem (A). The Schur algebras S(n, d) have tame representation type if and only if

(i) p = 3, n = 3, d = 7, 8; (ii) p = 3, n = 2, d = 9, 10, 11; (iii) p = 2, n = 2, d = 4 or d = 9.

Theorem (B). The infinitesimal Schur algebras S(n, d)_{r} have tame representation type if and only if one of the following occurs.

(i) p ≥ 5, n = 3, p ≤ d ≤ 2p − 1, r = 1; (ii) p = 3, n = 3 d = 3, r = 1; (iii) p = 3, n = 4, d = 3, 4, 5, r = 1; (iv) p = 3, n = 3, d = 7, 8, r ≥ 2;

(v) p = 3, n = 2, d = 9, 10, 11, r ≥ 3; (vi) p = 2, n = 4, d = 2, 3, r = 1; (vii) p = 2, n = 2, d = 4, r ≥ 2; (viii) p = 2, n = 2, d = 9, r ≥ 3.

1.3 In this subsection we fix some notation and conventions used throughout the paper. Let M = M_{n} be the monoid scheme which associates to each commutative k-algebra R the monoid M_{n}(R) of n × n matrices over R. For this k is algebraically closed of characteristic p > 0. We identify the coordinate algebra k[M] with k[c_{ij} : 1 ≤ i, j ≤ n]. The algebra k[M] has a bialgebra structure given by Δ : k[M] → k[M] ⊗ k[M] and ϵ : k[M] → k where

Δ(c_{ij}) = ∑_{ℓ=1}^{n} c_{iℓ} ⊗ c_{ℓj}; ϵ(c_{ij}) = \{ \begin{matrix}1 & i = j \\ 0 & i ≠ j \end{matrix} .  (1.3.1)

Let A(n, d) be the set of homogeneous elements of k[c_{ij}] of total degree d in the generators {c_{ij}}. Since A(n, d) is a subcoalgebra of k[M], its linear dual S(n, d) = A(n, d)^{∗} is a finitedimensional algebra. These algebras are the (ordinary) Schur algebras. For a fixed n, the category of rational M-modules (i.e. k[M]-comodules) is equivalent with the category of polynomial G-modules; moreover, every object in either category splits into a direct sum of homogeneous subobjects of degree d and the subcategory of those subobjects is equivalent with the category of S(n, d)-modules. In this way the family {S(n, d) | d ≥ 0} controls the polynomial representation theory of G = GL_{n}.

Now let F : M → M be the Frobenius morphism and D the subscheme of M determined by the diagonal matrices in each M_{n}(R). Let M_{r}D be the inverse image of D under F^{r}, the rth iterate of F. Its coordinate algebra k[M_{r}D] may be identified with the quotient algebra k[c_{ij}]/J_{r} where J_{r} is the ideal generated by {c_{ij}^{p^{r}} : i ≠ j}. Since M_{r}D is a monoid scheme it is automatic that J_{r} is a coideal in k[M] and that k[M_{r}D] inherits a coalgebra structure from k[M]. Let A(n, d)_{r} be the image of A(n, d) in k[M_{r}D] under the above quotient mapping; we regard elements of A(n, d)_{r} as homogeneous polynomials of degree d on M_{r}D. Clearly A(n, d)_{r} is a subcoalgebra of k[M_{r}D]. So its linear dual S(n, d)_{r} = A(n, d)_{r}^{∗} is a finite-dimensional algebra. These are the so-called infinitesimal Schur algebras introduced in [DNP1].

From the above construction, for each n and d with 0 ≤ d < p^{r} we have S(n, d) = S(n, d)_{r} and S(n, d) = lim_{r^{⟶}} S(n, d)_{r}. The category of rational M_{r}D-modules (i.e. k[M_{r}D]-comodules)

is equivalent with a certain subcategory of the category of rational G_{r}T -modules and one calls objects in this subcategory polynomial G_{r}T -modules. As in the classical case, for fixed n and r the family {S(n, d)_{r} | d ≥ 0} controls the polynomial representation theory of G_{r}T.

Throughout this paper we will adhere to the notation of [DNP1] and [DNP3]. Let X(T) be the set of rational weights and P (D) the set of polynomial weights. We make the natural identifications X(T) ≅ ℤ^{n} and P (D) ≅ ℕ^{n}. Let

X(T)_{+} = {λ = (λ_{1}, λ_{2}, …, λ_{n}) : λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n}},

the set of dominant weights, and

X_{r}(T) = {λ = (λ_{1}, λ_{2}, …, λ_{n}) : 0 ≤ λ_{i} − λ_{i+1} ≤ p^{r} − 1, for 1 ≤ i ≤ n − 1},

the set of p^{r}-restricted weights.

Furthermore, let B be the Borel subgroup determined by lower triangular matrices in G and let H^{0}(λ) be the induced module ind_{B}^{G} λ = ind_{B}^{G} k_{λ}, where k_{λ} is the one-dimensional B-module of character λ. The set of isomorphism classes of the simple G-modules is in oneto-one correspondence with X(T)_{+}; this correspondence is given by λ → L(λ) = soc_{G}H^{0}(λ). Let P (D)_{+} = X(T)_{+} ∩ P (D), the set of dominant polynomial weights. The set of isomorphism classes of the simple M-modules is in one-to-one correspondence with P (D)_{+} and this correspondence is obtained from the above correspondence by restriction. It follows that the set of isomorphism classes of simple S(n, d)-modules is given by

{L(λ) : λ ∈ P (D)_{+}, λ_{1} + λ_{2} + … λ_{n} = d},

so this set is in one-to-one correspondence with the set of partitions of d of not more than n parts.

On the other hand, in the infinitesimal case, the set of isomorphism classes of simple G_{r}T -modules is in one-to-one correspondence with the set X(T). The correspondence is given by λ → L̂_{r}(λ) = soc_{G_{r}T} (ind_{B_{r}T}^{G_{r}T} λ). Let

P_{r}(D) = {λ ∈ P (D) : 0 ≤ λ_{i} − λ_{i+1} ≤ p^{r} − 1, for 1 ≤ i ≤ n}

with the understanding that λ_{n+1} = 0. Moreover, let Γ_{r}(D) = P_{r}(D) + p^{r}P (D) and Γ_{r}^{d}(D) = {λ ∈ Γ_{r}(D) : λ_{1} + λ_{2} + … + λ_{n} = d}. The set of isomorphism classes of simple M_{r}D-modules is obtained by taking the simple G_{r}T -modules L̂_{r}(λ) such that λ ∈ Γ_{r}(D). Finally, the set of isomorphism classes of simple S(n, d)_{r}-modules is obtained by taking the simple G_{r}T -modules with weights in Γ_{r}^{d}(D).

1.4 Acknowledgements: The authors would like to thank Burt Totaro for several useful comments.

2 General results

In this section we will prove general results which allow us to conclude that an algebra has wild representation type by showing that certain subalgebras have wild representation type.

2.1 We begin with the formal definitions of tame and wild representation type [Erd1]. Let x, y be indeterminates. A finite-dimensional algebra B has tame representation type if it has infinite representation type and for each integer d > 0 there exist finitely many B − k[x] bimodules {M_{i} : i = 1, 2, … t} which are free as k[x]-modules, and all but a finite number of indecomposable B-modules of dimension equal to d are isomorphic to M_{i} ⊗_{k[x]} N for some i and some simple k[x]-module N.

An finite-dimensional k-algebra B has wild representation type if and only if there exists a finitely generated B − k⟨x, y⟩-bimodule Q which is free as right k⟨x, y⟩ -module such that the functor F defined by F (N) = Q ⊗_{k⟨x,y⟩} N from mod(k⟨x, y⟩) to mod(B) preserves indecomposability and isomorphism classes.

2.2 Throughout the rest of this section assume that k is an algebraically closed field. The simple k[x]-modules are one-dimensional and have the form k_{a} = k[x]/(x − a) for a ∈ k. Given a subalgebra B of an algebra A let coind_{B}^{A}N = A ⊗_{B} N. The results below relate the representation type of the algebras B and A given suitable conditions on the coinduction functor.

Lemma . Let M be a finitely generated indecomposable B − k[x]-bimodule which is free as k[x]-module. Then M ⊗_{k[x]} k_{a} is an indecomposable B-module.

Proof. Let M be an indecomposable B −k[x]-bimodule and let M = m_{1}k[x]⊕m_{2}k[x]⊕…⊕ m_{t}k[x]. Suppose that M ⊗_{k[x]} k_{a} ≅ Q_{1} ⊕ Q_{2} as B-modules. A basis for M ⊗_{k[x]} k_{a} is given by ⟨m_{i} ⊗ 1 : i = 1, 2, …, t⟩. Let ⟨q_{1}, q_{2}, …, q_{l}⟩ be a basis for Q_{1} and ⟨q_{l+1}, q_{l+2}, …, q_{t}⟩ be a basis for Q_{2}. Furthermore, let

q_{j} = ∑_{i=1}^{t} c_{ij}(m_{i} ⊗ 1) = (∑_{i=1}^{t} c_{ij}m_{i}) ⊗ 1 = z_{j} ⊗ 1

for j = 1, 2, …, t. If X_{1} = ⟨z_{1}, z_{2}, …, z_{l}⟩k[x] and X_{2} = ⟨z_{l+1}, z_{l+2}, …, z_{t}⟩k[x] then X_{1} and X_{2} are sub B − k[x] bimodules of M. The invertibility of the matrix C = (c_{ij}) shows immediately that M = X_{1} + X_{2}. Moreover, if z_{1}r_{1} + z_{2}r_{2} + … + z_{s}r_{s} = 0 where r_{j} ∈ k[x] for j = 1, 2, …, s then C · (r_{1}, r_{2}, …, r_{s})^{t} = 0 since z_{j} = ∑_{i=1}^{s} c_{ij}m_{i}. The invertibility of C implies that r_{j} = 0 for all j. Hence, M = X_{1} ⊕ X_{2} = z_{1}k[x] ⊕ z_{2}k[x] ⊕ … ⊕ z_{s}k[x] which contradicts the indecomposability of M as B − k[x]-bimodule. _{□}

Theorem . Let B be a subalgebra of a finite-dimensional k-algebra A. Assume that every indecomposable B-module N is a direct summand of coind_{B}^{A} N upon restriction to B. If B has wild representation type then so does A.

Proof. Observe that by our assumption every indecomposable B-module appears as a summand of some indecomposable A-module (upon restriction to B). It is clear that if A has finite representation type then so does B. Therefore, it suffices to show that B has tame representation type whenever A does. So assume that A has tame representation type. Fix an integer d > 0 and let

d′ = {max(dim_{k} coind_{B}^{A}N) : N is indecomposable and dim_{k} N = d}.

We remark that d′ < ∞ because dim_{k} coind_{B}^{A}N < (dim_{k} A) · d. For each σ such that 0 < σ ≤ d′ there exists a finite set of A − k[x]-bimodules B_{σ} = {M_{i}^{σ} : i = 1, 2, …, t_{σ}} such that all but a finitely many indecomposable A-modules of dimension σ are isomorphic to a module in C_{σ} = {M_{i}^{σ} ⊗_{k[x]} k_{a} : i = 1, 2, …, t_{σ}, a ∈ k}. Let

E = {M ∈ mod(A) : M is indecomposable, M ∉ ∪_{0<σ≤d′}C_{σ} and dim_{k} M ≤ d′}.

The set E is finite since A has tame representation type.

For each σ, i such that 0 < σ ≤ d′ and 0 ≤ i ≤ t_{σ} let M_{i}^{σ} = X_{iσ}^{1} ⊕ X_{iσ}^{2} ⊕ … ⊕ X_{iσ}^{s_{iσ}} where X_{iσ}^{j} are indecomposable B − k[x]-bimodules for each j = 1, 2, …, s_{iσ}. According to the preceding lemma any indecomposable B-summand of M_{i}^{σ} ⊗_{k[x]} k_{a} must have the form X_{iσ}^{j} ⊗_{k[x]} k_{a} for some j. Let B̂_{d} = {X_{iσ}^{j} : dim_{k}(X_{iσ}^{j} ⊗_{k[x]} k_{a}) = d} and Ĉ_{d} = {X_{iσ}^{j} ⊗_{k[x]} k_{a} : dim_{k}(X_{iσ}^{j} ⊗_{k[x]} k_{a}) = d}. Furthermore, let

Ê_{d} = {N : N is an indecomposable B-module, N is a summand of M ∈ E, and dim_{k} N = d}

.

The set B̂_{d} is a finite set of B − k[x]-bimodules all of which are free as k[x]-modules. Let N be an indecomposable B-module and N′ be an indecomposable A-module of dimension less than or equal to d′ which has N appearing as a direct summand. If N′ ∈ ∪_{0<σ≤d′}C_{σ} then N ∈ Ĉ_{d}. On the other hand if N′ ∉ ∪_{0<σ≤d′}C_{σ} for any such choice of N′ then N is contained in the finite set Ê_{d}. Hence, B has tame representation type. _{□}

These general results can now be applied to relate the representation type of different (infinitesimal) Schur algebras.

Corollary . If S(n, d)_{r} has wild representation type then so does S(n′, d)_{r} for n′ ≥ n.

Proof. According to [DNP3, Prop. 3.2] for n′ ≥ n, there exists a subalgebra B of A = S(n′, d)_{r} such that every indecomposable B-module N is a direct summand of coind_{B}^{S(n′,d)_{r}}N considered as B-module. The subalgebra B contains a subalgebra isomorphic to S(n, d)_{r}. In [DNP3, 3.3] it was shown that any indecomposable S(n, d)_{r}-module can be lifted to a B-module. Since S(n, d)_{r} has wild representation type it follows that B must have wild representation type. Hence, by the preceding theorem S(n′, d)_{r} has wild representation type. _{□}

2.3 Let M be a S(n, d_{1})_{r}-module, N a k⟨x, y⟩-module, and Q an S(n, d_{2})_{r} − k⟨x, y⟩- bimodule which is free as right k⟨x, y⟩-module. Then M ⊗_{k} Q becomes an S(n, d_{1} + d_{2})_{r} − k⟨x, y⟩-bimodule. The left action on the tensor product is obtained from the comultiplication on k[M_{r}D] and the right action is given by (m ⊗ q).a = m ⊗ (q.a) for m ∈ M, q ∈ Q and a ∈ k⟨x, y⟩. Furthermore, there exists an isomorphism of S(n, d_{1} + d_{2})_{r}-modules:

M ⊗_{k} [Q ⊗_{k⟨x,y⟩} N] ≅ [M ⊗_{k} Q] ⊗_{k⟨x,y⟩} N.  (2.3.1)

The isomorphism above also remains valid if M is a G_{r}T -module and Q is a G_{r}T − k⟨x, y⟩- bimodule. Let δ be the one-dimensional determinant representation of G = GL_{n}. This is a polynomial representation of G of degree n, so it may be regarded as a representation of S(n, n). By restriction it is also a representation of S(n, n)_{r}. The existence of this representation allows us to relate the representation type of S(n, d)_{r} with S(n, d + n)_{r}.

Proposition (A). If S(n, d)_{r} has wild representation type then S(n, d + n)_{r} has wild representation type.

Proof. Let S(n, d)_{r} have wild representation type. There exists an S(n, d)_{r} − k⟨x, y⟩- bimodule Q such that the functor F (−) = Q ⊗_{k⟨x,y⟩} − preserves indecomposability and isomorphism classes. Consider the functor F′ from mod(k⟨x, y⟩) to mod(S(n, d + n)_{r}) defined by F′(−) = [δ ⊗_{k} Q] ⊗_{k⟨x,y⟩} − . Suppose N is an indecomposable k⟨x, y⟩-module and F′(N) ≅ X_{1} ⊕ X_{2}. According to (2.3.1) we have

(δ^{−1} ⊗ X_{1}) ⊕ (δ^{−1} ⊗ X_{2}) ≅ δ^{−1} ⊗_{k} F′(N)
≅ δ^{−1} ⊗_{k} [[δ ⊗_{k} Q] ⊗_{k⟨x,y⟩} N]
≅ Q ⊗_{k⟨x,y⟩} N
≅ F (N)

as G_{r}T -modules. By [DNP3, Prop. 2.1a] this is a decomposition of M_{r}D-modules. Therefore, since F preserves indecomposability F′ must also preserve indecomposability. Now suppose that F′(N_{1}) ≅ F′(N_{2}). By tensoring both sides of this isomorphism by δ^{−1} we have Q ⊗_{k⟨x,y⟩} N_{1} ≅ Q ⊗_{k⟨x,y⟩} N_{2} as G_{r}T -modules. From [DNP3, Prop. 2.1c] this must be an isomorphism of M_{r}D-modules. The functor F preserves isomorphisms so N_{1} ≅ N_{2}. Hence, S(n, d + n) has wild representation type. _{□}

Given suitable conditions on the geometry of the weight poset, the preceding result can be improved by using the translation principle (see [DNP3, Sec. 4]). For any G_{r}T -module V and λ, μ ∈ X(T), define pr_{λ}V to be the largest G_{r}T -submodule of V such that μ ∈ W_{p} · λ for every composition factor L̂_{r}(μ) of V. Let R^{+} be a set of positive roots for the root system of GL_{n}, ρ be the half-sum of positive roots and

C̄_{ℤ} = {λ ∈ X(T) | 0 ≤ ⟨λ + δ, α^{∨}⟩ ≤ p for all α ∈ R^{+}}.

Moreover, let λ, μ ∈ C̄_{ℤ} and let ν be the unique weight in X(T)_{+}∩W (μ−λ). The translation functor T_{λ}^{μ} : mod(G_{r}T) → mod(G_{r}T) is defined for any G_{r}T -module V by

T_{λ}^{μ}(V) = pr_{μ}(L̂_{r}(ν) ⊗ pr_{λ}V).

The linkage principle implies that V is a direct sum of pr_{λ}V as λ ranges over a set of representative weights in the linkage classes of X(T). It follows that T_{λ}^{μ} may be viewed as a functor from the subcategory {V ∈ mod(G_{r}T) | pr_{λ}V = V } to the subcategory {V ∈ mod(G_{r}T) | pr_{μ}V = V }.

Let F_{M_{r}D} be the functor from mod(G_{r}T) to mod(M_{r}D) defined by taking the largest M_{r}D-submodule of a given G_{r}T -module. For any λ, ν ∈ C̄_{ℤ}, define a truncated translation functor T̄_{λ}^{μ} to be the composite F_{M_{r}D} ∘ T_{λ}^{μ}. By restriction T̄_{λ}^{μ} induces a functor from mod(M_{r}D) into itself. From this point of view, T̄_{λ}^{μ} maps a M_{r}D-module V with pr_{λ}V = V to an M_{r}D-module V′ with pr_{μ}V′ = V′. Also, if w · μ ∉ Γ_{r}(D) for all w ∈ W_{p}, then T̄_{λ}^{μ}(V) = 0 for any M_{r}D-module V.

Proposition (B). Let B(λ) be a block of S(n, d)_{r} containing the simple module L̂_{r}(λ) and let μ ∈ Γ_{r}^{d′}(D) be a weight in the same facet as λ with μ − λ ∈ P (D). If B(λ) has wild representation type then S(n, d′)_{r} has wild representation type.

Proof. According to [DNP3, Prop. 4.2], under the hypotheses given in the theorem T̄_{μ}^{λ} ∘T̄_{λ}^{μ} is equivalent to the identity functor. Suppose that B(λ) has wild representation type and F is a functor from mod(k⟨x, y⟩) to mod(B(λ)) defined by F (−) = Q ⊗_{k⟨x,y⟩} − where Q is a B(λ) − k⟨x, y⟩-bimodule which is free as right k⟨x, y⟩-module. Let F′ be the functor from mod(k⟨x, y⟩) to mod(B(μ)) defined by F′(−) = T̄_{λ}^{μ}(Q) ⊗_{k⟨x,y⟩} − . From the definition of the translation functor and by the remarks preceding the first proposition of this section T̄_{λ}^{μ}(Q) is a B(μ) − k⟨x, y⟩-bimodule which is also free as k⟨x, y⟩-module. Furthermore, a direct verification shows that F′ = T̄_{λ}^{μ} ∘ F.

Let N be an indecomposable k⟨x, y⟩-module. Suppose F′(N) = T̄_{λ}^{μ} ∘ F (N) ≅ X_{1} ⊕ X_{2} where X_{j} ≠ {0} for j = 1, 2. Since T̄_{μ}^{λ} ∘ T̄_{λ}^{μ} is equivalent to the identity functor it follows that T̄_{μ}^{λ}(X_{1}) ⊕ T̄_{μ}^{λ}(X_{2}) ≅ F (N). By the argument given in [DNP3, Thm. 4.3] we have T̄_{μ}^{λ}(X_{j}) ≠ {0} for j = 1, 2. This contradicts the fact that F preserves indecomposability, thus F′ must preserve indecomposability. Now assume that F′(N_{1}) ≅ F′(N_{2}). It follows that F (N_{1}) ≅ T̄_{μ}^{λ}(T̄_{λ}^{μ}F (N_{1})) ≅ T̄_{μ}^{λ}(T̄_{λ}^{μ}F (N_{2})) ≅ F (N_{2}) and N_{1} ≅ N_{2}. Therefore, F′ preserves isomorphism classes, and B(μ) must have wild representation type. _{□}

We remark that the statements of Corollary 2.2, Propositions 2.3A and 2.3B hold also for the ordinary Schur algebras because for r >> 0 S(n, d)_{r} ≅ S(n, d).

2.4 Several of the methods in the following sections will involve applying knowledge about the existence of extensions between simple M-modules (S(n, d)-modules) to show the existence of extensions between simple M_{r}D-modules (S(n, d)_{r}-modules). The following result will be used extensively for this purpose. For a proof we refer the reader to [DNP3, Prop. 2.3].

Proposition . Let L(λ) and L(μ) be simple M-modules with λ, μ ∈ P_{+}(D).

(a) If λ, μ ∈ X_{r}(T) ∩ P (D) then the restriction map

res_{M,M_{r}D} : Ext_{M}^{1}(L(λ), L(μ)) → Ext_{M_{r}D}^{1}(L(λ), L(μ))

is injective.

(b) Let N be an M-module with two composition factors L(λ) and L(μ), λ ∈ X_{r}(T) and μ ∈ X_{+}(T) with soc_{M} N ≅ L(λ). Moreover, assume that L(μ) = ∑_{j=1}^{s} L̂_{r}(μ_{j}) is the decomposition of L(μ) as M_{r}D-module with λ ≠ μ_{j} in X_{r}(T) for all j. Then soc_{M_{r}D} N ≅ L(λ).

3 Wild representation type for Schur algebras

The purpose of this section is to prove that the ordinary Schur algebras have wild representation type for the algebras not listed in Theorem 1.1A and Theorem 1.2A. This will be accomplished by using some of the calculations given in [Erd2].

3.1 The algebra S(3, 2p) contains a block Morita equivalent to the algebra L_{6} for p ≥ 5 [Erd2, Prop. 5.3.1]. The algebra L_{6} is given by the quiver in Figure 1 along with the relations given in [Erd2, 3.9].

The two-nilpotent algebra L_{6}/J^{2} has the same quiver as L_{6}. Moreover, the separated quiver for the quiver given in Figure 1 is not a union of extended Dynkin diagrams [DR], thus L_{6}/J^{2} has wild representation type. Therefore, S(3, 2p) has wild representation type for p ≥ 5.

Proposition . The algebra S(n, d) has wild representation type if p ≥ 5, n ≥ 3 and d ≥ 2p.

[figure: •; 1; •; 2; •; 4; •; 5; •; •; 3; 6] Figure 1

Proof. The block B(λ) of S(3, 2p) which is Morita equivalent to L_{6} is the one containing the simple module L(λ) with λ = (2p − 2, 1, 1). Consider the weights μ_{1} = (2p − 2, 2, 1) and μ_{2} = (2p − 2, 2, 2). The weights λ, μ_{1} and μ_{2} belong to the same facet and λ − μ_{j} ∈ P (D) for j = 1, 2. Therefore, by the remark at the end of 2.3, and Proposition 2.3B, it follows that S(3, 2p + j) has wild representation type for j = 1, 2. For d ≥ 2p, the algebras S(3, d) have wild representation type by Proposition 2.3A. One can now apply Corollary 2.2 to show that S(n, d) has wild representation type for n ≥ 3 and d ≥ 2p. _{□}

3.2 Let p = 3. The algebra S(3, 6) is Morita equivalent to L_{6}, thus has wild representation type. Now consider the algebra S(3, 10). The induced modules H^{0}(λ) for λ = (10, 0, 0), (7, 2, 1), (5, 5, 0) and (4, 3, 3) have the following structures:

H^{0}(10, 0, 0) :  L(8, 2, 0)  H^{0}(7, 2, 1) :  L(4, 3, 3)
L(7, 3, 0)  L(7, 2, 1)
L(10, 0, 0)
H^{0}(5, 5, 0) :  L(4, 3, 3)  H^{0}(4, 3, 3) :  L(4, 3, 3)
L(5, 5, 0)

where the vertical display of simple modules determines the socle and radical series of the given module (the socle is the bottom layer). Next we claim that Ext_{M}^{1}(L(7, 3, 0), L(4, 3, 3)) ≠ 0. First observe that

Ext_{M}^{1}(L(7, 3, 0), L(4, 3, 3)) ≅ Ext_{G}^{1}(L(7, 3, 0), L(4, 3, 3))

and

L(7, 3, 0) ≅ L(2, 1, 0)^{(1)} ⊗ L(1, 0, 0)
L(4, 3, 3) ≅ L(1, 1, 1)^{(1)} ⊗ L(1, 0, 0).

From the five term exact sequence arising from the Lyndon-Hochschild-Serre spectral sequence for G/G_{1} and the fact that Ext_{G_{1}}^{1}(L(1, 0, 0), L(1, 0, 0)) = 0, it follows that

Ext_{G}^{1}(L(7, 3, 0), L(4, 3, 3)) ≅ Ext_{G/G_{1}}^{1}(L(2, 1, 0)^{(1)}, L(1, 1, 1)^{(1)})
≅ k.

Therefore, L(4, 3, 3) appears in the second socle layer of H^{0}(7, 3, 0). Its projective cover P (4, 3, 3) (in the category of S(3, 10)-modules) has a filtration the sections of which are the transpose duals of the induced modules containing L(4, 3, 3) as a composition factor. The decomposition above along with the fact that H^{0}(7, 3, 0) has L(4, 3, 3) in the second socle layer shows that P (4, 3, 3) must have at least three copies of L(4, 3, 3) in the third radical layer. Let B = B(10, 0, 0) be the block containing L(10, 0, 0). Consider B/J^{3} where J = Rad B. Then End_{B/J^{3}}(P (4, 3, 3)/J^{3}) is a local algebra isomorphic to k < x_{1}, x_{2}, …, x_{s} > /J^{2} where s ≥ 3. Hence, B/J^{3} is wild, thus S(3, 10) is wild. Since B(10, 0, 0) is of wild representation type and μ = (10, 1, 0) satisfies the conditions of Proposition 2.3B, it follows that S(3, 11) is of wild representation type.

Proposition . Let p = 3. The algebra S(n, d) has wild representation type for n ≥ 3 and d = 6, d ≥ 9.

Proof. Let p = 3. The algebras S(3, d) for d = 6, 10, 11 have wild representation type. Therefore, by Proposition 2.3A, S(3, d) has wild representation type for d = 6 and d ≥ 9. One can now apply Corollary 2.2 to obtain the desired result. _{□}

3.3 Once again, let p = 3 and consider the algebra S(4, 7). The composition factors and structure diagrams for certain induced modules are shown below.

H^{0}(7, 0, 0, 0) : L(2, 2, 2, 1)  H^{0}(4, 1, 1, 1) : L(3, 2, 1, 1)
L(5, 2, 0, 0)  L(4, 1, 1, 1)
L(7, 0, 0, 0)
H^{0}(3, 2, 1, 1) : L(2, 2, 2, 1)  H^{0}(2, 2, 2, 1) : L(2, 2, 2, 1)
L(3, 2, 1, 1)

Over S(4, 4), there exists a non-split extension

0 → L(2, 2, 0, 0) → N → L(1, 1, 1, 1) → 0

where N ≅ H^{0}(2, 2, 0, 0). By Proposition 2.4A this is a non-split sequence upon restriction to M_{1}D. Tensoring this sequence by L(1, 0, 0, 0)^{(1)}, we get a sequence

0 → L(5, 2, 0, 0) → N ⊗ L(1, 0, 0, 0)^{(1)} → L(4, 1, 1, 1) → 0.

Suppose that this sequence splits as M-modules. Then

N ⊗ L(1, 0, 0, 0)^{(1)} ≅ L(4, 1, 1, 1) ⊕ L(5, 2, 0, 0).

As M_{1}D-modules, it would then follow that N ≅ L(1, 1, 1, 1) ⊕ L(2, 2, 0, 0), thus reaching a contradiction. Referring to the correspondence given in the table below, one can use a similar argument to show that there is a non-split extension between L(5, 1, 1, 1) and L(5, 2, 1, 0).

simple module S(4, 7)  S(4, 8)
X_{1}  L(4, 1, 1, 1) L(5, 1, 1, 1)
X_{2}  L(3, 2, 1, 1) L(3, 3, 1, 1)
X_{3}  L(7, 0, 0, 0) L(8, 0, 0, 0)
X_{4}  L(5, 2, 0, 0) L(5, 2, 1, 0)
X_{5}  L(2, 2, 2, 1) L(2, 2, 2, 2)

Therefore, the Ext^{1}-quiver of either algebra has a subquiver of the form given in Figure 2. This does not separate as a union of extended Dynkin diagrams. Consequently, S(4, 7) and S(4, 8) have wild representation type.

[figure: •; X_{3}; •; X_{4}; •; X_{5}; •; X_{1}; X_{2}; •] Figure 2

Proposition . Let p = 3. Then S(n, d) has wild representation type for n ≥ 4 and d ≥ 6.

Proof. In Section 3.2, we have seen that S(3, 6) and S(3, 9) have wild representation type. Therefore, S(4, 6) and S(4, 9) have wild representation type. From the argument given above S(4, 7) and S(4, 8) have wild representation type. The result now follows by Proposition 2.3A and Corollary 2.2. _{□}

3.4 Let p ≥ 3. The algebras S(2, p^{2}) and S(2, p^{2} + 1) have a block which is Morita equivalent to a basic algebra D_{p+1} [Erd2, Prop. 5.2] with quiver given in Figure 3 and relations α_{1}β_{1} = 0 = α_{2}β_{2}, α_{1}β_{2}α_{2} = 0 = β_{2}α_{2}β_{1}, δ_{3}β_{2} = 0 = δ_{3}β_{1}, α_{2}γ_{3} = 0 = α_{1}γ_{3}, γ_{3}δ_{3} = β_{2}α_{2}, γ_{i}γ_{i+1} = 0 = δ_{i+1}δ_{i} and δ_{i}γ_{i} = γ_{i+1}δ_{i+1} for 3 ≤ i ≤ p − 1.

We will show that the algebras D_{m} for m ≥ 5 are wild. Let Q be the quiver given in Figure 4.

There is a functor F : mod(kQ) → mod(D_{m}) which preserves indecomposables and reflects isomorphism (with a few obvious exceptions). This functor is constructed as follows. Given a representation V of Q, define F V = M where

Me_{3} = V e_{3} ⊕ V e_{3′}, Me_{2} = V e_{2} ⊕ V e_{2′}, Me_{4} = V e_{4} ⊕ V e_{4′}, Me_{1} = V e_{1}, Me_{5} = V e_{5}

[figure: 1; 2; •; •; •; 3; •; 4; •; 5; •; p; •; p + 1; γ_{3}; δ_{3}; γ_{4}; δ_{4}; ⋯; γ_{p}; δ_{p}; α_{1}; α_{2}; β_{1}; β_{2}] Figure 3

[figure: 2; 3; 1; 3′; 2′; β̄_{2}; β̄_{1}; ᾱ_{1}; ᾱ_{2}; 4′; δ̄_{3}; γ̄_{3}; 4; 5; δ̄_{4}] Figure 4

and Me_{i} = 0 for i ≥ 6. On morphisms the definition of F is the obvious one. To show that F preserves indecomposability, one has to use the fact that β̄_{2} is onto (unless the module is the simple module at vertex 2), and that V e_{4} = Im(γ̄_{3}) + Im(δ̄_{4}) etc. The exceptions come from the fact that the simples at vertex 2 and 2′ of Q are mapped to the same (similarly for 3, 3′ and for 4, 4′). By [Rin], the algebra kQ is of wild type; for example, by comparison

with ˜ D̃_{4}. It follows that D_{m} is wild, m ≥ 5.

The structure of the induced modules for S(2, 12) when p = 3 are as follows:

H^{0}(12, 0) :  L(8, 4)  H^{0}(11, 1) :  L(9, 3)
L(6, 6) L(11, 1)  L(8, 4)
L(12, 0)  L(11, 1)
H^{0}(9, 3) :  L(8, 4)  H^{0}(8, 4) :  L(6, 6)  H^{0}(6, 6) :  L(6, 6)
L(9, 3)  L(8, 4)

It follows that the Ext^{1}-quiver is given by Figure 2. The correspondence between vertices and simple modules is given by the following table.

simple module S(2, 12)
X_{1}  L̂(11, 1)
X_{2}  L̂(12, 0)
X_{3}  L̂(9, 3)
X_{4}  L̂(8, 4)
X_{5}  L̂(6, 6)

The separated quiver is not a union of extended Dynkin diagrams, hence S(2, 12) is of wild representation type.

Proposition . The algebra S(2, d) has wild representation type if

(a) p ≥ 5, d ≥ p^{2};

(b) p = 3, d ≥ 12.

Proof. The algebras S(2, p^{2}) and S(2, p^{2} + 1), p ≥ 5, have wild representation type. Therefore, by Proposition 2.3A, S(2, d) has wild representation type for d ≥ p^{2}. For p = 3, S(2, 12) has wild representation type. By using Proposition 2.3B, one can show that S(2, 13) also has wild representation type. Consequently, S(2, d), d ≥ 12 has wild representation type by Proposition 2.3A. _{□}

3.5 The non-semisimple blocks for S(2, 6) and S(2, 11) with p = 2 are Morita equivalent. Consider the algebra S(2, 6) with p = 2. The modules H^{0}(λ), λ ∈ P (D)_{+} are uniserial with composition factors given below:

H^{0}(6, 0) :  L(3, 3)  H^{0}(5, 1) :  L(4, 2)
L(5, 1)  L(3, 3)
L(6, 0)  L(5, 1)
H^{0}(4, 2) :  L(3, 3)  H^{0}(3, 3) :  L(3, 3)
L(4, 2)

For an M-module N, let N^{τ} denote the transpose dual of the module N. Consider the algebra S(2, 3) for p = 2. By the Steinberg tensor product theorem,

L(3, 0) ≅ L(1, 0)^{(1)} ⊗ L(1, 0).

Therefore, H^{0}(3, 0) = L(3, 0) because both modules have dimension four and L(3, 0) is the socle of H^{0}(3, 0). Moreover, H^{0}(2, 1) = L(2, 1) also by dimension considerations. This implies that S(2, 3) for p = 2 is semisimple.

Let P (3, 0) be the projective cover of L(3, 0) so P (3, 0) ≅ L(3, 0). We claim that P (3, 0)⊗ P (3, 0)^{τ} is an indecomposable M-module with simple head L(3, 3). We have seen that L(3, 0) ≅ V (3, 0) where V (3, 0) is the Weyl module corresponding to the weight (3, 0). Let λ = (6, 0), (5, 1), (4, 2), (3, 3). A character computation shows that

Hom_{G}(P (3, 0) ⊗ P (3, 0)^{τ} , L(λ)) ≅ Hom_{G}(V (3, 0) ⊗ L(3, 0), L(λ))
≅ Hom_{B}((3, 0) ⊗ L(3, 0)^{τ} , L(λ))
= \begin{cases}k & λ = (3, 3) \\ 0 & λ = (6, 0), (5, 1), (4, 2) \end{cases}

It follows that P (3, 0) ⊗ P (3, 0)^{τ} is a homomorphic image of P (3, 3). The decompositions of the induced modules above, along with Brauer-Humphreys reciprocity, imply that dim_{k} P (3, 3) = 16. Therefore, P (3, 3) ≅ P (3, 0) ⊗ P (3, 0)^{τ} and the module P (3, 3) is selfdual. This fact along with the structures of the projective modules for S(2, 4) given in [DNP3, 5.2], can be used to determine the structure of the projective indecomposable S(2, 6)-modules. Set X_{1} = L(6, 0), X_{2} = L(5, 1), X_{3} = L(3, 3), and X_{4} = L(4, 2). The structure of the projective covers for these simple modules is depicted in Figure 5. The lines in the figure indicate extensions between simple modules and one can read off the submodules structure directly.

[figure: P (X_{1}); X_{1}; X_{2}; X_{3}; P (X_{2}); X_{2}; X_{1}; X_{3}; X_{2}; X_{4}; X_{3}; P (X_{3}); X_{3}; X_{4}; X_{2}; X_{3}; X_{1}; X_{3}; X_{2}; X_{4}; X_{3}; P (X_{4}); X_{4}; X_{3}; X_{2}; X_{3}; X_{4}] Figure 5

The quiver for the basic algebra B associated to S(2, 6) is given in Figure 6, with the relations α_{1}β_{1} = α_{2}β_{2} = β_{3}α_{3} = 0, α_{1}α_{2}α_{3} = 0 = β_{3}β_{2}β_{1}, β_{1}α_{1}α_{2} = α_{2}α_{3}β_{3}, and β_{2}β_{1}α_{1} = α_{3}β_{3}β_{2}.

[figure: •; X_{1}; •; X_{2}; •; X_{3}; •; X_{4}; α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 6

Let 1_{X_{i}}, i = 1, 2, 3, 4 be the identity maps on the vertices X_{i}. Consider the idempotent e = 1_{X_{1}} + 1_{X_{3}}. There is a map from X_{3} to X_{1} given by β_{2}β_{1}. Furthermore, there are two distinct self maps of X_{3} given by α_{3}β_{3} and β_{2}α_{2}. This gives rise to a subquiver of the quiver of eBe as shown in Figure 7. Therefore, by [Erd1, I.10.8], eBe is of wild representation type, and B is of wild representation type.

Proposition . Let p = 2. The algebra S(2, d) has wild representation type for d = 6, 8 and d ≥ 10.

[figure: •; •] Figure 7

Proof. The algebra S(2, 6) has wild representation type, so by Proposition 2.3A, S(2, 6+2j) has wild representation type for j ∈ ℕ. Similarly, S(2, 11) has wild representation type, so S(2, 11 + 2j) has wild representation type for j ∈ ℕ. The result follows by combining these two statements. _{□}

3.6 Let p = 2. Consider first the algebra S(3, 4) with Y_{1} = L(4, 0, 0), Y_{2} = L(2, 1, 1), Y_{3} = L(3, 1, 0), and X = L(2, 2, 0). The structure for the projective covers for these simple modules is given in Figure 8.

[figure: P (Y_{1}); Y_{1}; X; Y_{3}; P (Y_{2}); Y_{2}; X; Y_{3}; Y_{2}; X; Y_{2}; P (Y_{3}); Y_{3}; X; Y_{1}; Y_{2}; X; Y_{3}; P (X); X; Y_{3}; Y_{1}; Y_{2}; X; X; Y_{2}; Y_{3}] Figure 8

Let B be the basic algebra corresponding to S(3, 4). The algebra B can be described by the quiver in Figure 9, along with the relations α_{1}β_{1} = 0 = α_{3}β_{3}, α_{1}β_{2} = 0 = α_{2}β_{1}, α_{1}β_{3}α_{3} = 0 = β_{3}α_{3}β_{1}, and β_{1}α_{1} = β_{2}α_{2}. We will show that the algebra S(3, 4) for p = 2 has wild representation type. Consider the quiver Q given in Figure 10.

There is a functor F : mod(kQ) → mod(B) which preserves indecomposability and reflects isomorphisms, except for finitely many simple modules. Now, Q is one of the wild quivers in the list of [Rin], thus B is wild.

Now consider the algebra S(3, 5), p = 2. According to [Erd2, Prop. 5.8], there exists a block of S(3, 5) which is Morita equivalent to the basic algebra U_{4} [Erd2, 3.7]. This algebra is given by the quiver in Figure 11, with relations α_{1}β_{1} = 0 = α_{2}β_{2}, α_{1}α_{2}α_{3} = 0 = β_{3}β_{2}β_{1}, and α_{3}β_{3} = β_{2}α_{2}.

There is a functor from mod(kQ/I) → mod(U_{4}) which preserves indecomposability and reflects isomorphisms (with finitely many exceptions) where Q is the quiver given in Figure

[figure: •; Y_{1}; •; X; •; Y_{3}; •; Y_{2}; α_{1}; β_{1}; α_{2}; β_{2}; β_{3}; α_{3}] Figure 9

[figure: x; 3; x; 2; 1; 2; α_{3}; β_{3}; α_{2}; β_{1}; β_{2}] Figure 10

[figure: •; 1; •; 2; •; 3; •; 4; α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 11

12, and I is generated by the two commutativity conditions: the squares in Figure 12 commute.

[figure: 3; 3; 4; 2; 4; α_{3}; β_{3}; β_{2}; α_{2}; β_{2}; α_{2}; α_{3}; β_{3}; β_{1}; α_{1}; α_{3}; 1; 3; 3; 4; 2] Figure 12

We form the push-outs of the commutative squares, as described in [Rin, p. 192]. This

gives a quiver (Figure 13) which contains ˜ D̃_{7} as a proper subquiver, hence it is wild. Another way to see this is to consider the representations of this quiver where the arrows between the branch points act as isomorphisms. This is ˜ D̃_{4}-double tilde which is in Ringel’s list as a wild quiver.

[figure: •; •; •; 1; •; •; •; •; 4; •; α_{3}; β_{1}; α_{1}] Figure 13

Proposition . Let p = 2. The algebra S(n, d) has wild representation type for n ≥ 3 and d ≥ 4.

Proof. From the result above, S(3, 4) and S(3, 5) have wild representation type. Moreover, Proposition 3.5 and Corollary 2.2 imply that S(3, 6) has wild representation type. Therefore, S(3, d) has wild representation type by Proposition 2.3A for d ≥ 4. The result now follows by Corollary 2.2. _{□}

4 Wild representation type for the infinitesimal Schur algebras

4.1 We will show that S(n, d)_{r} has wild representation type for the cases not listed in Theorem 1.1B and Theorem 1.2B. The first result handles the general cases.

Proposition . The infinitesimal Schur algebra S(n, d)_{r} has wild representation type if

(a) p ≥ 5, n ≥ 3, d ≥ 2p, r ≥ 2;

(b) p = 3, n ≥ 3, d = 6, r ≥ 2;

(c) p = 3, n ≥ 3, d ≥ 9, r ≥ 3;

(d) p = 3, n ≥ 4, d ≥ 6, r ≥ 2;

(e) p = 2, n ≥ 3, d ≥ 2p, r ≥ 3;

(f) p ≥ 5, n = 2, d ≥ p^{2}, r ≥ 3;

(g) p = 3, n = 2, d ≥ 12, r ≥ 3;

(h) p = 2, n = 2, d = 6, 8, d ≥ 10, r ≥ 4;

(i) p = 2, n = 2, d = 6 + 2j, j ≥ 0, r ≥ 3;

Proof. For part (a) let p ≥ 5, r ≥ 2. Since S(3, d) = S(3, d)_{r} for d = 2p, 2p + 1, 2p + 2, the infinitesimal Schur algebra has wild representation type in these cases by Proposition 3.1. Therefore, S(3, d)_{r} has wild representation type for d ≥ 2p by Proposition 2.3A, and so does S(n, d)_{r} for n ≥ 3, d ≥ 2p by Corollary 2.2. The second assertion holds because S(n, 6) = S(n, 6)_{r} for n ≥ 3, p = 3, and r ≥ 2. The ordinary Schur algebra is of wild representation type in this case. For part (c), S(3, d) = S(3, d)_{r} for r ≥ 3 and d = 9, 10, 11. By results in Section 3.2, the Schur algebras is of wild representation type. The assertion follows by Proposition 2.3A and Corollary 2.2. Part (d) follows by a similar argument as in part (c) using the results in Section 3.3.

The assertion (e) holds because under the hypotheses, S(3, d) = S(3, d)_{r} for d = 2p, 2p + 1, 2p + 2 for p = 2 and r ≥ 3. In these cases the ordinary Schur algebra has wild representation type by Proposition 3.6. Now let p ≥ 5, n = 2, d ≥ p^{2} and r ≥ 3. We have S(2, d) = S(2, d)_{r} for d = p^{2}, p^{2} + 1. Also for p = 3 and r ≥ 3, S(2, d) = S(2, d)_{r} for d = 12, 13. These algebras have wild representation type by Proposition 3.4. Parts (f) and (g) now follows by Proposition 2.3A and Corollary 2.2.

Similarly, (h) holds because for p = 2, S(2, d) = S(2, d)_{r} for r ≥ 4 and d = 6, 11. Finally, S(2, 6) = S(2, 6)_{r} when r ≥ 3 and p = 2 so (i) follows by Proposition 2.3A. _{□}

4.2 Let p = 3 and consider the algebra S(3, 10)_{2}. The following table provides labels for a subset of simple modules.

simple module S(3, 10)_{2}
γ  L̂_{2}(4, 3, 3)
η  L̂_{2}(8, 2, 0)
μ  L̂_{2}(7, 3, 0)
λ  L̂_{2}(10, 0, 0)
δ  L̂_{2}(1, 9, 0)
β  L̂_{2}(1, 0, 9)

The information given in Section 3.2 and Proposition 2.4A shows that for σ = γ, η,

Ext_{M_{2}D}^{1}(L̂_{2}(μ), L̂_{2}(γ)) ≠ 0.

Furthermore, as M_{2}D-modules,

L(10, 0, 0) ≅ L̂_{2}(10, 0, 0) ⊕ L̂_{2}(1, 9, 0) ⊕ L̂_{2}(1, 0, 9)

By Proposition 2.4B, Ext_{M_{2}D}^{1}(L̂_{2}(μ), L̂_{2}(σ)) ≠ 0, for σ = λ, δ, β. The Ext^{1}-quiver has a subquiver of the form given by Figure 14, thus S(3, 10)_{2} has wild representation type. Once again Proposition 2.3B with μ = (10, 1, 0) shows that S(3, 11)_{2} has wild representation type.

Theorem . Let p = 3. The infinitesimal Schur algebra S(n, d)_{r} has wild representation type if n ≥ 3, d = 6, d ≥ 9, and r ≥ 2.

Proof. From Proposition 4.1, S(3, 6)_{r} has wild representation type for r ≥ 2. Therefore, S(3, 9)_{r} has wild representation type for r ≥ 2. By Proposition 4.1, S(3, 10)_{r} and S(3, 11)_{r} have wild representation type for r ≥ 3. Furthermore, S(3, 10)_{2} and S(3, 11)_{2} have wild representation type from the argument given above. The theorem now follows by Proposition 2.3A and Corollary 2.2. _{□}

4.3 Let p ≥ 3 and consider modules over the ordinary Schur algebra S(3, 2p). The induced module H^{0}(2p, 0, 0) has the following radical structure:

H^{0}(2p, 0, 0) : L(p − 1, p − 1, 2)

[figure: •; λ; •; η; •; γ; •; δ; •; β; •; μ] Figure 14

L(2p − 1, 1, 0) L(2p, 0, 0).

Upon restriction to M_{1}D, the module L(2p − 1, 1, 0) decomposes as follows:

L(2p − 1, 1, 0) ≅ L(1, 0, 0)^{(1)} ⊗ L(p − 1, 1, 0)
≅ L̂_{1}(2p − 1, 1, 0) ⊕ L̂_{1}(p − 1, p + 1, 0) ⊕ L̂_{1}(p − 1, 1, p).

Let σ = (p − 1, p − 1, 2), α = (2p − 1, 1, 0), β = (p − 1, p + 1, 0), and δ = (p − 1, 1, p). Since Ext_{M}^{1}(L(σ), L(α)) ≠ 0, it follows by Proposition 2.4B that

Ext_{M_{1}D}^{1}(L̂_{1}(σ), L̂_{1}(ν)) = Ext_{S(3,2p)_{1}}^{1}(L̂_{1}(σ), L̂_{1}(ν)) ≠ 0

where ν = α, β, δ. For the algebra S(3, p) one can use the decomposition of H^{0}(p, 0, 0) to show that Ext_{M}^{1}(L(p−1, 1, 0), L(p, 0, 0)) ≠ 0. Furthermore, we have L(p, 0, 0) ≅ L(1, 0, 0)^{(1)} ≅ L̂_{1}(p, 0, 0) ⊕ L̂_{1}(0, p, 0) ⊕ L̂_{1}(0, 0, p) as M_{1}D-modules. Let ζ_{1} = (p, 0, 0), ζ_{2} = (0, p, 0) and ζ_{3} = (0, 0, p). By Proposition 2.4B

Ext_{M_{1}D}^{1}(L̂_{1}(p − 1, 1, 0), L̂_{1}(ζ_{i})) ≠ 0

for i = 1, 2, 3. These extensions can be tensored by the one-dimensional M_{1}D-modules (p, 0, 0) and (0, p, 0) to show that Ext_{M_{1}D}^{1}(L̂_{1}(2p−1, 1, 0), L̂_{1}(p, p, 0)) ≠ 0 and Ext_{M_{1}D}^{1}(L̂_{1}(p− 1, p + 1, 0), L̂_{1}(p, p, 0)) ≠ 0. Set γ = (p, p, 0). The algebra S(3, 2p)_{1} has the subquiver given in Figure 15 appearing as a subquiver of its Ext^{1}-quiver. Therefore, S(3, 2p)_{1} has wild representation type for p ≥ 3.

Let A be one of the algebras S(3, 7)_{1}, S(3, 8)_{1} for p = 3 or S(3, 4)_{1}, S(3, 5)_{1} for p = 2. Moreover, let γ, α, β, σ, δ correspond with the following A-modules.

simple module S(3, 7)_{1} p = 3 S(3, 8)_{1} p = 3 S(3, 4)_{1} p = 2 S(3, 5)_{1} p = 2
γ  L̂_{1}(2, 5, 0)  L̂_{1}(3, 5, 0)  L̂_{1}(1, 3, 0)  L̂_{1}(3, 0, 2)
α  L̂_{1}(4, 3, 0)  L̂_{1}(4, 4, 0)  L̂_{1}(2, 2, 0)  L̂_{1}(3, 1, 1)
β  L̂_{1}(1, 3, 3)  L̂_{1}(1, 4, 3)  L̂_{1}(0, 2, 2)  L̂_{1}(1, 1, 3)
σ  L̂_{1}(4, 2, 1)  L̂_{1}(4, 2, 2)  L̂_{1}(2, 1, 1)  L̂_{1}(2, 2, 1)
δ  L̂_{1}(4, 0, 3)  L̂_{1}(4, 1, 3)  L̂_{1}(2, 0, 2)  L̂_{1}(1, 3, 1)

[figure: •; γ; •; σ; •; β; •; α; •; δ] Figure 15

There exists a non-trivial extension between the M-modules L(α) and L(σ). Observe that as M_{1}D-module L(α) decomposes as

L(4, 3, 0) ≅ L̂_{1}(4, 3, 0) ⊕ L̂_{1}(1, 3, 3) ⊕ L̂_{1}(4, 0, 3)
L(4, 4, 0) ≅ L̂_{1}(4, 4, 0) ⊕ L̂_{1}(1, 4, 3) ⊕ L̂_{1}(4, 1, 3)
L(2, 2, 0) ≅ L̂_{1}(2, 2, 0) ⊕ L̂_{1}(0, 0, 2) ⊕ L̂_{1}(2, 0, 2)
L(3, 1, 1) ≅ L̂_{1}(3, 1, 1) ⊕ L̂_{1}(1, 1, 3) ⊕ L̂_{1}(1, 3, 1)

Therefore, by Proposition 2.4B, Ext_{M_{1}D}^{1}(L̂_{1}(σ), L̂_{1}(δ)) ≅ Ext_{A}^{1}(L̂_{1}(σ), L̂_{1}(δ)) ≠ 0. By [DNP3, 5.5] the Ext^{1}-quiver for A has a subquiver of the form given in Figure 15. Hence, A has wild representation type.

For the algebra S(3, 6)_{1} when p = 2, one can tensor the simple S(3, 4)_{1}-modules L̂_{1}(λ) for λ = γ, α, β, δ, σ by the one-dimensional representation L̂_{1}(2, 0, 0). This shows that the Ext^{1}-quiver for S(3, 6)_{1} also has a subquiver of the form given in Figure 15, thus S(3, 6)_{1} has wild representation type.

Proposition . The algebra S(n, d)_{1} has wild representation type for p ≥ 2, n ≥ 3 and d ≥ 2p.

Proof. For p ≥ 5 the block B(2p, 0, 0) of S(3, 2p)_{1} has wild representation type. Consider the weights μ_{1} = (2p + 1, 0, 0) ∈ Γ_{1}^{2p+1}(D) and μ_{2} = (2p + 2, 0, 0) ∈ Γ_{1}^{2p+2}(D). Observe that μ_{i} and λ belong to the same facet and μ_{i} − λ ∈ P (D) for i = 1, 2. Therefore, by Proposition 2.3B the algebras S(3, 2p + 1)_{1} and S(3, 2p + 2)_{1} have wild representation type. We have also seen that for p = 2, 3 the algebras S(3, d)_{1} have wild representation type for d = 2p, 2p + 1, 2p + 2. The result now follows by Proposition 2.3A and Corollary 2.2. _{□}

4.4 Consider the algebra S(4, p) for p ≥ 5. The radical series of the S(4, p)-module H^{0}(p, 0, 0, 0) is given by

H^{0}(p, 0, 0, 0) : L(p − 1, 1, 0, 0)

L(p, 0, 0, 0).

Therefore, Ext_{M}^{1}(L(p − 1, 1, 0, 0), L(p, 0, 0, 0)) ≠ 0. As M_{1}D-module L(p, 0, 0, 0) decomposes as follows:

L(p, 0, 0, 0) ≅ L̂_{1}(p, 0, 0, 0) ⊕ L̂_{1}(0, p, 0, 0) ⊕ L̂_{1}(0, 0, p, 0) ⊕ L̂_{1}(0, 0, 0, p).

Let μ = (p − 1, 1, 0, 0), λ = (p, 0, 0, 0), γ = (0, p, 0, 0), η = (0, 0, p, 0) and β = (0, 0, 0, p). Then by Proposition 2.4B,

Ext_{M_{1}D}^{1}(L̂_{1}(μ), L̂_{1}(σ)) = Ext_{S(4,p)_{1}}^{1}(L̂_{1}(μ), L̂_{1}(σ)) ≠ 0

for σ = λ, γ, η, β.

Let {α_{1}, α_{2}, α_{3}} be a set of simple roots for the root system of GL_{4}. Furthermore, let s_{α_{1}+α_{2}} be the reflection in W associated to the root α_{1} + α_{2}. There exists a reflection w ∈ W_{p} given by

w · μ = s_{α_{1}+α_{2}} · μ + p(α_{1} + α_{2}) = (p − 2, 1, 1, 0)  (4.4.1)

Let δ = (p − 2, 1, 1, 0). The equation (4.4.1) shows that μ and δ are mirror reflections across some hyperplane defined by the action of the affine Weyl group on X(T). The validity of the Lusztig conjecture for GL_{4} insures that Ext_{M}^{1}(L(μ), L(δ)) ≠ 0. By Proposition 2.4A we can conclude that Ext_{S(4,p)_{1}}^{1}(L̂_{1}(μ), L̂_{1}(δ)) ≠ 0. The Ext^{1}-quiver for S(4, p)_{1} contains a five-subspace quiver (see Figure 14) as a subquiver. Hence, S(4, p)_{1} has wild representation type.

Proposition . Let p ≥ 5. The algebra S(n, d)_{1} has wild representation type for n ≥ 4, d ≥ p.

Proof. It was shown above that the block B(p, 0, 0, 0) of S(4, p)_{1} has wild representation type for p ≥ 5. Let μ_{1} = (p + 1, 0, 0, 0) ∈ Γ_{1}^{2p+1}(D), μ_{2} = (p + 2, 0, 0, 0) ∈ Γ_{1}^{2p+2}(D) and μ_{3} = (p+2, 1, 0, 0) ∈ Γ_{1}^{2p+3}(D). Observe that μ_{j} and λ = (p, 0, 0, 0) belong to the same facet and μ_{j} − λ ∈ P (D) for j = 1, 2, 3. Therefore, by Theorem 2.3B the algebra S(4, p + j)_{1} has wild representation type for j = 1, 2, 3. The proposition now follows by Proposition 2.3A and Corollary 2.2. _{□}

4.5 In this subsection assume that p = 2 or p = 3. For p = 2 let A = S(5, 2) or S(5, 3) and for p = 3 let A = S(5, 3), S(5, 4) or S(5, 5). Morover, let A_{1} denote the corresponding infinitesimal Schur algebra. By [Dot] the A-module H^{0}(d, 0, 0, 0, 0) has the following decomposition:

Schur algebra H^{0}(d, 0, 0, 0) composition factors of H^{0}(d, 0, 0, 0)
S(5, 2) p = 2 H^{0}(2, 0, 0, 0, 0) L(1, 1, 0, 0, 0) L(2, 0, 0, 0, 0)
S(5, 3) p = 2 H^{0}(3, 0, 0, 0, 0) L(1, 1, 1, 0, 0) L(3, 0, 0, 0, 0)
S(5, 3) p = 3 H^{0}(3, 0, 0, 0, 0) L(2, 1, 0, 0, 0) L(3, 0, 0, 0, 0)
S(5, 4) p = 3 H^{0}(4, 0, 0, 0, 0) L(2, 2, 0, 0, 0) L(4, 0, 0, 0, 0)
S(5, 5) p = 3 H^{0}(5, 0, 0, 0, 0) L(2, 2, 1, 0, 0) L(5, 0, 0, 0, 0)

The decomposition of the simple M-module L(d, 0, 0, 0, 0) as M_{1}D-module is given by

L(d, 0, 0, 0, 0) ≅ L̂_{1}(γ) ⊕ L̂_{1}(λ) ⊕ L̂_{1}(η) ⊕ L̂_{1}(β) ⊕ L̂_{1}(δ)

where γ, λ, η, β, and δ are defined for each algebra in the following table:

S(5, 2)_{1} S(5, 3)_{1} S(5, 3)_{1} S(5, 4)_{1} S(5, 5)_{1}
γ = (2,0,0,0,0) (3, 0, 0, 0, 0) (3, 0, 0, 0, 0) (4, 0, 0, 0, 0) (5, 0, 0, 0, 0)
λ = (0,2,0,0,0) (1, 2, 0, 0, 0) (0, 3, 0, 0, 0) (1, 3, 0, 0, 0) (2, 3, 0, 0, 0)
η = (0,0,2,0,0) (1, 0, 2, 0, 0) (0, 0, 3, 0, 0) (1, 0, 3, 0, 0) (2, 0, 3, 0, 0)
β = (0,0,0,2,0) (1, 0, 0, 2, 0) (0, 0, 0, 3, 0) (1, 0, 0, 3, 0) (2, 0, 0, 3, 0)
δ = (0,0,0,0,2) (1, 0, 0, 0, 2) (0, 0, 0, 0, 3) (1, 0, 0, 0, 3) (2, 0, 0, 0, 3)
μ = (1,1,0,0,0) (1, 1, 1, 0, 0) (2, 1, 0, 0, 0) (2, 2, 0, 0, 0) (2, 2, 1, 0, 0)

By Propositions 2.4A and 2.4B, we can identify a five-subspace quiver as in Figure 14 as a subquiver of the Ext^{1}-quiver for A_{1}. Hence, A_{1} has wild representation type.

Proposition . For n ≥ 5, the algebra S(n, d)_{1} has wild representation type if p = 3 and d ≥ 3 or p = 2 and d ≥ 2.

Proof. The algebras S(5, 2)_{1} and S(5, 3)_{1} for p = 2 have wild representation type by the argument preceding this proposition and, by Proposition 4.3, the algebras S(5, j)_{1}, j = 4, 5, 6, have wild representation type. Again by the preceding argument, the algebras S(5, 3)_{1}, S(5, 4)_{1} and S(5, 5)_{1} have wild representation type for p = 3. Furthermore, by Proposition 4.3, S(5, 6)_{1} and S(5, 7)_{1} for p = 3 have wild representation type. The statement now follows by Proposition 2.3A and Corollary 2.2. _{□}

4.6 Let p ≥ 5 and consider the algebra S(2, p^{2})_{2}. Let λ = (p^{2} − p, p), β = (p^{2} − 1, 1), η = (p^{2} − p − 1, p + 1), γ = (p^{2}, 0) and τ = (0, p^{2}). From [DNP3, 5.3], we have Ext_{S(2,p^{2})_{2}}^{1}(L̂_{2}(λ), L̂_{2}(σ)) ≠ 0 for σ = β, η, γ, τ. By computing characters one can show that the radical series of the M-module H^{0}(p^{2} − p − 1, p + 1) is given by

H^{0}(p^{2} − p − 1, p + 1) : L(p^{2} − 2p, 2p)
L(p^{2} − p − 1, p + 1).

Set δ = (p^{2} − 2p, 2p). Then Ext_{M}^{1}(L(η), L(δ)) ≠ 0. Since δ, η ∈ P_{2}(D) it follows by Proposition 2.4A that Ext_{M_{2}D}^{1}(L̂_{2}(η), L̂_{2}(δ)) ≅ Ext_{S(2,p^{2})_{2}}^{1}(L̂_{2}(η), L̂_{2}(δ)) ≠ 0. Therefore, the Ext^{1}-quiver of S(2, p^{2})_{2} has a subquiver of the form given in Figure 16 and must therefore have wild representation type.

Let p = 3 and consider S(2, p^{2})_{2} = S(2, 9)_{2}. There is one non-semisimple block for this algebra with simple modules given by Y_{1} = L̂_{2}(9, 0), Y_{2} = L̂_{2}(0, 9), Y_{3} = L̂_{2}(8, 1), Y_{4} = L̂_{2}(5, 4), and X = L̂_{2}(6, 3). The structures for the projective covers of these simple modules is given in Figure 17. These were obtained from the quiver in Section 3.4 and Proposition 2.4B. The basic algebra B for this block of S(2, 9)_{2} is given by the quiver in Figure 18 with relations α_{i}β_{j} = 0 for i = 1, 2, 4 and j = 1, 2, 4, α_{3}β_{3} = 0 = α_{3}β_{4}, β_{4}α_{4} = 0, β_{1}α_{1} = β_{2}α_{2}, α_{3}β_{1}α_{1} = β_{1}α_{1}β_{3}, β_{3}α_{3}β_{1} = 0 = β_{3}α_{3}β_{1}, and α_{1}β_{3}α_{3} = 0 = α_{2}β_{3}α_{3}. Consider the idempotent e = 1_{X} + 1_{Y_{4}}. There are self maps of the vertex X given by β_{2}α_{2} and β_{3}α_{3}, and a map from X to Y_{4} given by α_{4}. It follows that eBe has a subquiver of the form given in Figure 7, thus has wild representation type. Consequently, S(2, 9)_{2} has wild representation type.

Proposition . Let p ≥ 3. The algebra S(2, d)_{2} has wild representation type for d ≥ p^{2}.

[figure: •; β; •; η; •; τ; •; γ; λ; •; •; δ] Figure 16

[figure: P (Y_{1}); Y_{1}; X; Y_{3}; P (Y_{2}); Y_{2}; X; Y_{3}; P (Y_{3}); Y_{3}; X; Y_{1}; Y_{2}; X; Y_{3}; P (Y_{4}); Y_{4}; X; Y_{4}; P (X); X; Y_{1}; Y_{2}; X; Y_{3}; Y_{3}; X; Y_{4}] Figure 17

Proof. By the remarks preceding the statement of the proposition, the block B(p^{2}, 0) of S(2, p^{2})_{2} containing L̂_{2}(p^{2}, 0) has wild representation type for p ≥ 3. Let λ = (p^{2}, 0) and μ = (p^{2} + 1, 0) ∈ Γ_{2}^{p^{2}+1}(D). Clearly, λ and μ are in the same facet. Furthermore, μ − λ ∈ P (D) so by Proposition 2.3B the algebra S(2, p^{2} + 1)_{2} has wild representation type. Now apply Proposition 2.3A and Corollary 2.2. _{□}

4.7 The non-semisimple blocks of S(2, 6)_{2} and S(2, 11)_{3} are Morita equivalent for p = 2. Let B be the basic algebra corresponding to one of these blocks. The basic algebra has six simple modules X_{j}, j = 1, 2, 3, 4, 5, 6. The structure of the projective covers for these simple modules is given in Figures 19 and 20. This information can be obtained from the structure of the projectives for S(2, 6) given in Section 3.5. The algebra B is presented by the quiver in Figure 21, with relations α_{1}β_{1} = 0 = α_{3}β_{3}, β_{4}β_{3} = 0 = β_{4}α_{4}, α_{3}α_{4} = 0, β_{3}α_{3} = α_{4}β_{4}, α_{1}α_{3}α_{2} = 0 = β_{2}β_{3}β_{1}, α_{2}β_{2}α_{4} = 0 = β_{2}α_{4}α_{5}, β_{5}β_{4}α_{2} = 0, β_{1}α_{1}α_{3} = α_{3}α_{2}β_{2}, β_{3}α_{3}α_{2} = α_{4}β_{4}α_{2}, β_{3}β_{1}α_{1} = α_{2}β_{2}β_{3}, β_{4}α_{2}β_{2} = α_{5}β_{5}β_{4}, α_{4}α_{5}β_{5} = α_{2}β_{2}α_{4}. Consider the

[figure: •; Y_{1}; •; X; •; Y_{3}; •; Y_{2}; •; Y_{4}; α_{1}; β_{1}; α_{3}; β_{3}; α_{2}; β_{2}; α_{4}; β_{4}] Figure 18

idempotent e = 1_{X_{1}} + 1_{X_{4}}. There are self maps of X_{4} by taking α_{4}β_{4} and α_{2}β_{2}, and a map from the vertex X_{4} to X_{1} given by β_{3}β_{1}. Therefore, the quiver for eBe has a subquiver of the form given by Figure 7, thus eBe and B have wild representation type.

[figure: P (X_{1}); X_{1}; X_{3}; X_{4}; P (X_{2}); X_{2}; X_{4}; X_{3}; X_{5}; X_{4}; X_{2}; P (X_{3}); X_{3}; X_{1}; X_{4}; X_{3}; X_{2}; X_{4}] Figure 19

Proposition . For p = 2, S(2, d)_{r} has wild representation type if

(a) r = 2, d = 6 + 2j, j = 0, 1, 2, …;

(b) r = 3, d = 6, 8, d ≥ 10.

Proof. Let r = 2. From the discussion given above, S(2, 6)_{2} has wild representation type. Therefore, S(2, 6+2j)_{2} for j = 0, 1, 2, … has wild representation type by Proposition 2.3A. Now let r = 3, observe that S(2, 6) = S(2, 6)_{3} so by Proposition 3.5, S(2, 6)_{3} has wild representation type. Hence, S(2, 6 + 2j)_{3} has wild representation type for j = 0, 1, … by using Proposition 2.3A. The statement of part (b) follows from Proposition 2.3A again and the fact that S(2, 11)_{3} has wild representation type. _{□} [figure: P (X_{4}); X_{4}; X_{3}; X_{2}; X_{5}; X_{4}; X_{1}; X_{6}; X_{4}; X_{5}; X_{3}; X_{2}; X_{4}; P (X_{5}); X_{5}; X_{6}; X_{4}; X_{5}; X_{2}; X_{4}; P (X_{6}); X_{6}; X_{5}; X_{4}] Figure 20

[figure: •; X_{1}; •; X_{3}; •; X_{4}; •; X_{5}; •; X_{6}; α_{1}; β_{1}; α_{3}; β_{3}; α_{4}; β_{4}; α_{5}; β_{5}; •; α_{2}; β_{2}; X_{2}] Figure 21

4.8 Consider the algebras S(3, 4)_{2} and S(3, 5)_{2} for p = 2. For S(3, 4)_{2}, let the following table provide a correspondence between the simple modules and their labels.

simple module S(3, 4)_{2} p = 2
γ  L̂_{2}(2, 1, 1)
η  L̂_{2}(3, 1, 0)
μ  L̂_{2}(2, 2, 0)
λ  L̂_{2}(4, 0, 0)
δ  L̂_{2}(0, 4, 0)
β  L̂_{2}(0, 0, 4)

As a S(3, 4)_{2}-module we have L(4, 0, 0) ≅ L(1, 0, 0)^{(2)} ≅ L̂_{2}(4, 0, 0) ⊕ L̂_{2}(0, 4, 0) ⊕ L̂_{2}(0, 0, 4).

For the ordinary Schur algebra S(3, 4) from Section 3.6 we have Ext_{S(3,4)}^{1}(L(2, 2, 0), L(σ)) ≠ 0 for σ = (2, 1, 1), (3, 1, 0) and (4, 0, 0). Therefore, by Theorem 2.4(a) and (b) the Ext^{1}- quiver for S(3, 4)_{2} will have the five subspace quiver appearing as a subquiver as in Figure 14. Hence, S(3, 4) has wild representation type.

For the algebra S(3, 5)_{2} let the following table provide a correspondence between the simple modules and their labels.

simple module S(3, 5)_{2} p = 2
γ  L̂_{2}(5, 0, 0)
τ  L̂_{2}(1, 4, 0)
β  L̂_{2}(1, 0, 4)
δ  L̂_{2}(2, 2, 1)
η  L̂_{2}(3, 1, 1)
λ  L̂_{2}(3, 2, 0)

From our analysis for S(3, 5) when p = 2 we have Ext_{S(3,5)}^{1}(L(σ), L(λ)) ≠ 0 for σ = γ, η. Furthermore, Ext_{S(3,5)}^{1}(L(η), L(δ)) ≠ 0. As a S(3, 5)-module we have

L(5, 0, 0) ≅ L̂_{2}(5, 0, 0) ⊕ L̂_{2}(1, 4, 0) ⊕ L̂_{2}(1, 0, 4).

Again by Theorem 2.4(a) and (b) we can conclude that the Ext^{1}-quiver will have a subquiver of the type given in Figure 16, thus S(3, 5)_{2} has wild representation type.

Proposition . Let p = 2. The algebra S(n, d)_{r} has wild representation type when n ≥ 3, d ≥ 4 and r ≥ 2.

Proof. For r ≥ 3 this follows by Proposition 4.1(e). Let r = 2. By the above analysis S(3, 4)_{2} and S(3, 5)_{2} have wild representation type when p = 2. Furthermore, S(3, 6)_{2} has wild representation type because S(2, 6)_{2} has wild representation type. The result for r = 2 now follows by using Proposition 2.3A and Corollary 2.2. _{□}

4.9 This subsection is devoted to organizing all the relevent results about infinite and wild representation type of infinitesimal Schur algebras. According to [DNP3], the infinitesimal Schur algebras of infinite representation type are given by the following list.

Theorem . The infinitesimal Schur algebra S(n, d)_{r} has infinite representation type if and only if

(a) p ≥ 2, n ≥ 3, d ≥ 2p, r ≥ 2;

(b) p ≥ 5, n ≥ 3, d ≥ p, r = 1;

(c) p = 2, 3, n ≥ 4, d ≥ p, r = 1;

(d) p = 3, n = 3, d = 3 or d ≥ 6, r = 1;

(e) p = 2, n = 3, d ≥ 4, r = 1;

(f) p ≥ 3, n = 2, d ≥ p^{2}, r ≥ 2;

(g) p = 2, n = 2, d = 4, 6 or d ≥ 8, r ≥ 3;

(h) p = 2, n = 2, d ≥ 4 with d even, r = 2.

In this section we have prove that the infinitesimal Schur algebras have wild representation type under the following conditions. Next to each condition is a reference number.

(i) p ≥ 5, n ≥ 3, d ≥ 2p, r ≥ 2; (4.1a) (ii) p = 3, n ≥ 4, d ≥ 6, r ≥ 2; (4.1d) (iii) p = 3, n = 3, d = 6 or d ≥ 9, r ≥ 2; (4.1b) (4.2) (iv) p = 2, n ≥ 3, d ≥ 2p, r ≥ 2; (4.8)

(v) p ≥ 5, n ≥ 4, d ≥ p, r = 1; (4.4) (vi) p ≥ 5, n = 3, d ≥ 2p, r = 1; (4.3) (vii) p = 3, n = 4, d ≥ 6, r = 1; (4.3) (viii) p = 3, n ≥ 5, d ≥ 3, r = 1; (4.4) (ix) p = 3, n = 3, d ≥ 6, r = 1; (4.3)

(x) p = 2, n ≥ 5, d ≥ 2, r = 1; (4.4) (xi) p = 2, n = 4, d ≥ 4, r = 1; (4.3) (xii) p = 2, n = 3, d ≥ 4, r ≥ 1; (4.3) (xiii) p ≥ 5, n = 2, d ≥ p^{2}, r ≥ 2; (4.1f) (4.6) (xiv) p = 3, n = 2, d ≥ 12, r ≥ 3; (4.1g) (xv) p = 3, n = 2, d ≥ 9, r = 2; (4.6) (xvi) p = 2, n = 2, d = 6, 8 or d ≥ 10, r ≥ 3; (4.1h) (4.7b) (xvii) p = 2, n = 2, d ≥ 6 with d even, r = 2. (4.7a)

In the following section we will show that the remaining cases not covered under these conditions which also fall under aforementioned theorem are of tame representation type.

5 Tame representation type

5.1 We begin by proving (iii) of Theorem 1.2A and a good portion of (vii) and (viii) of Theorem 1.2B.

Proposition . Let p = 2. Then

(a) the Schur algebras S(2, 4) and S(2, 9) have tame representation type;

(b) the infinitesimal Schur algebras S(2, d)_{r} have tame representation type for d = 4, r ≥ 3, and d = 9, r ≥ 4.

Proof. The non-semisimple blocks of S(2, 4) and S(2, 9) are Morita equivalent to the algebra D_{3} [Erd2, Prop. 5.2]. The quiver and relations are provided in Section 3.4. This algebra is a special biserial algebra [Erd1, II.1]. Any special biserial algebra is either of finite or tame representation type. Since D_{3} is not finite by Theorem 1.1A, it follows that D_{3} must be tame. For the second assertion, observe that S(2, 4)_{r} = S(2, 4) for r ≥ 3 and S(2, 9)_{r} = S(2, 9) for r ≥ 4. _{□}

5.2 The infinitesimal Schur algebra S(2, 4)_{2} and S(2, 9)_{3} have non-semisimple blocks which are Morita equivalent to a basic algebra D [DNP3, 5.2]. There are four simple modules X_{1}, X_{2}, X_{3}, Y. The projective cover of Y has the following structure.

[figure: P (Y) :; X_{3}; Y; Y; X_{1}; X_{2}; X_{3}; Y]

The structure of the projective covers for the other simple modules are given below.

P (X_{1}) :  X_{1}  P (X_{2}) :  X_{2}  P (X_{3}) :  X_{3}
Y  Y  Y
X_{3}  X_{3}  X_{1} X_{2}
Y
X_{3}

The Ext^{1}-quiver for D is given by Figure 22 with relations α_{1}β_{1} = α_{1}β_{2} = 0 = α_{2}β_{2} = α_{2}β_{1}, α_{3}β_{3} = 0, β_{1}α_{1} = β_{2}α_{2}, α_{1}β_{3}α_{3} = 0 = α_{2}β_{3}α_{3}, β_{3}α_{3}β_{2} = 0 = β_{3}α_{3}β_{1}.

[figure: •; X_{1}; •; Y; •; X_{3}; •; X_{2}; α_{1}; β_{1}; α_{2}; β_{2}; β_{3}; α_{3}] Figure 22

Proposition . The algebras S(2, 4)_{2} and S(2, 9)_{3} for p = 2 have tame representation type.

Proof. In order to show that D is tame, we first factor out the socle of the projective-injective module P (X_{3}). This does not change the representation type and gives the additional relation α_{3}β_{1}α_{1}β_{3} = 0. Denote this algebra by A.

We define now an algebra B as follows. The quiver is given by Figure 23 with relations

ϵ^{2} = ϵ, κη = 0, ηκ = 0, ᾱ_{3}β̄_{3} = 0, β̄_{3}ᾱ_{3}κ = 0, ηβ̄_{3}ᾱ_{3} = 0
ᾱ_{3}κϵηβ̄_{3} = 0

The ideal is not admissible. Next, we define a functor F : mod(A) → mod(B) as follows. For M ∈ mod(A), let F M = V where V e_{y} = Me_{y}, V e_{3} := Me_{3} and moreover V e_{5} := Me_{1} ⊕ Me_{2}. The arrows act as follows. Set (m_{1}, m_{2})ϵ = (m_{1}, 0), and let

(m_{1}, m_{2})η := m_{1}α_{1} + m_{2}α_{2}, m_{y}κ := (m_{y}β_{1}, −m_{y}β_{2})

[figure: •; 3; •; y; 5; ᾱ_{3}; β̄_{3}; •; κ; η; ϵ] Figure 23

and let ᾱ_{3}, β̄_{3} act on V as α_{3}, β_{3} act on M. It is straightforward to check that the relations of B are satisfied.

The definition of F on maps is the obvious one. One checks that F induces an isomorphism Hom_{A}(M, N) → Hom_{B}(F M, F N), and thus preserves indecomposability and reflects isomorphisms. Next, let k̃ = k[t] where t is an indeterminate. Define an algebra B_{t} over k̃ by setting B_{t} := k̃Q/I_{t} where Q is the quiver in Figure 23 and I_{t} is generated by the same relations as B, except that the relation for ϵ^{2} is replaced by ϵ^{2} = tϵ. If one specializes t to a non-zero element in k then one obtains an algebra isomorphic to B. On the other hand, if one specializes t = 0 then one obtains a special biserial algebra, which we call C. It remains to show that B and C have the same dimension. Both algebras have a monomial basis. One finds that e_{3}B has dimension 5, moreover e_{y}B has dimension 7 and e_{5}B has dimension 6. Moreover, one sees that the dimensions of e_{i}C and of e_{i}B are equal. Hence, D has tame representation type. _{□}

This completes the proof of Theorem 1.2B (vii) and (viii).

5.3 Let A be S(3, d) for p ≤ d ≤ 2p − 1 if p ≥ 5 and d = 3 if p = 3. From [Erd2, 6.2] the non-semisimple blocks B of A can be described as follows. The first case occurs when B has two simple modules X_{1} and X_{2}. The projective indecomposable modules have the following structure.

P (X_{1}) :  X_{1}  P (X_{2}) :  X_{2}
X_{2}  X_{1}
X_{1}

Moreover, X_{1} and X_{2} correspond to simple modules whose weights are in the restricted region X_{1}(T). This implies that the block in S(3, d)_{1} containing these simple modules is Morita equivalent to B, thus has finite representation type.

The other case occurs when B has three simple modules X_{1}, X_{2} and X_{3}. The projective covers for these simple module are described below.

P (X_{1}) :  X_{1}  P (X_{2}) :  X_{2}  P (X_{3}) :  X_{3}
X_{2}  X_{1} X_{3}  X_{2}
X_{1}  X_{2}

The simple modules X_{1} and X_{2} correspond to simple S(3, d)-modules with weights in X_{1}(T). However, the simple module X_{3} is of the form L(λ_{1}, λ_{2}, λ_{3}) with p ≤ λ_{1}−λ_{2} ≤ 2p−1 and 0 ≤ λ_{2} − λ_{3} ≤ p − 1. First observe that λ_{1} ≥ p so set λ_{1} = p + r where 0 ≤ r ≤ p − 1. Since p ≤ λ_{1} − λ_{2} it follows that 0 ≤ r − λ_{2}. Hence,

X_{3} ≅ L(λ_{1}, λ_{2}, λ_{3}) ≅ L(1, 0, 0)^{(1)} ⊗ L(r, λ_{2}, λ_{3}).

Upon restriction to M_{1}D we have

X_{3} ≅ L̂_{1}(λ_{1}, λ_{2}, λ_{3}) ⊕ L̂_{1}(r, λ_{2} + p, λ_{3}) ⊕ L̂_{1}(r, λ_{2}, λ_{3} + p).

Set Y_{1} = L̂_{1}(λ_{1}, λ_{2}, λ_{3}), Y_{2} = L̂_{1}(r, λ_{2} + p, λ_{3}), and Y_{3} = L̂_{1}(r, λ_{2}, λ_{3} + p). The block B(λ_{1}, λ_{2}, λ_{3}) of S(3, d)_{1} has five simple modules X_{1}, X_{2}, Y_{1}, Y_{2} and Y_{3}. The projective covers for these simple modules have the following structure.

P (Y_{1}) :  Y_{1} P (Y_{2}) :  Y_{2} P (Y_{3}) :  Y_{3}
X_{2}  X_{2}  X_{2}
P (X_{1}) :  X_{1}  P (X_{2}) :  X_{2}
X_{2}  X_{1} Y_{1} Y_{2} Y_{3}
X_{1}  X_{2}

The quiver for the block B(λ_{1}, λ_{2}, λ_{3}) is given in Figure 24 with relations α_{1}β_{1} = α_{2}β_{2} = α_{3}β_{3} = γδ, β_{j}γ = 0 and α_{i}β_{j} = 0 for i, j = 1, 2, 3.

[figure: •; Y_{1}; •; X_{2}; •; Y_{3}; •; Y_{2}; •; X_{1}; β_{1}; α_{1}; β_{3}; α_{3}; β_{2}; α_{2}; γ; δ] Figure 24

Proposition . The infinitesimal Schur algebras S(3, d)_{1} have tame representation type if one of the following occurs.

(a) p ≥ 5, p ≤ d ≤ 2p − 1;

(b) p = 3, d = 3.

Proof. From Theorem 1.1B these algebras are not of finite representation type. Therefore, these algebras must contain a block B with five simple modules whose description was given above. The only non-projective indecomposable modules have radical length two. The separated quiver for B/J^{2} where J is the Jacobson radical of B is two copies of the four-subspace quiver D̃_{4}. By [DR] B/J^{2} has tame representation type, thus B has tame representation type. _{□}

This completes (i) and (ii) of Theorem 1.2B.

5.4 For p = 2, let A = S(4, 2) or S(4, 3), and for p = 3, let A = S(4, 3), S(4, 4) or S(4, 5). By a similar computation to the one used in [DNP2, 6.2], it follows that the blocks for these algebras are Morita equivalent to a 14-dimensional basic algebra denoted by B. This algebra has five simple modules X, Y_{1}, Y_{2}, Y_{3} and Y_{4}. The following tables provides the correspondence between the simple modules for this basic algebra and A.

simple module S(4, 2)_{1}  S(4, 3)_{1}
Y_{1}  L̂_{1}(2, 0, 0, 0) L̂_{1}(3, 0, 0, 0)
Y_{2}  L̂_{1}(0, 2, 0, 0) L̂_{1}(1, 2, 0, 0)
Y_{3}  L̂_{1}(0, 0, 2, 0) L̂_{1}(1, 0, 2, 0)
Y_{4}  L̂_{1}(0, 0, 0, 2) L̂_{1}(1, 0, 0, 2)
X  L̂_{1}(1, 1, 0, 0) L̂_{1}(1, 1, 1, 0)
simple module S(4, 3)_{1}  S(4, 4)_{1}  S(4, 5)_{1} (block 1) S(4, 5)_{1} (block 2)
Y_{1}  L̂_{1}(3, 0, 0, 0) L̂_{1}(4, 0, 0, 0) L̂_{1}(4, 1, 0, 0)  L̂_{1}(5, 0, 0, 0)
Y_{2}  L̂_{1}(0, 3, 0, 0) L̂_{1}(1, 3, 0, 0) L̂_{1}(1, 4, 0, 0)  L̂_{1}(2, 3, 0, 0)
Y_{3}  L̂_{1}(0, 0, 3, 0) L̂_{1}(1, 0, 3, 0) L̂_{1}(1, 1, 3, 0)  L̂_{1}(2, 0, 3, 0)
Y_{4}  L̂_{1}(0, 0, 0, 3) L̂_{1}(1, 0, 0, 3) L̂_{1}(1, 1, 0, 3)  L̂_{1}(2, 0, 0, 3)
X  L̂_{1}(2, 1, 0, 0) L̂_{1}(2, 1, 1, 0) L̂_{1}(3, 2, 0, 0)  L̂_{1}(2, 2, 1, 0)

The structure of the projective covers of these modules is given by

P (X) :  X  P (Y_{j}) :  Y_{j}
Y_{1} Y_{2} Y_{3} Y_{4}  j = 1, 2, 3, 4  X
X

The quiver for this algebra is given by Figure 25 with relations α_{1}β_{1} = α_{2}β_{2} = α_{3}β_{3} = α_{4}β_{4} and all other products being zero.

[figure: •; Y_{1}; •; X; •; Y_{3}; •; Y_{2}; •; Y_{4}; β_{1}; α_{1}; β_{3}; α_{3}; β_{2}; α_{2}; α_{4}; β_{4}] Figure 25

From this information we can conclude (iii) and (vi) of Theorem 1.2B.

Theorem . The algebras S(4, 2)_{1}, S(4, 3)_{1} for p = 2 and S(4, 3), S(4, 4)_{1}, S(4, 5)_{1} for p = 3, have tame representation type.

Proof. Since P (X) is an injective module and the other projective covers have radical length two, the projective modules are the only indecomposable modules of radical length greater than two. Therefore, in order to show that B has tame representation type it suffices to show that B/J^{2} has tame representation type. The separated quiver for B/J^{2} is two copies of D̃_{4}. By [DR] it follows that B/J^{2} has tame representation type. _{□}

5.5 The non-semisimple blocks for S(2, 9), S(2, 10) and S(2, 11) for p = 3 are Morita equivalent to D_{4}. The quiver and relations for this algebra are given in Section 3.4. One can factor out the socles of the projective modules P_{2} and P_{4} for D_{4} to obtain an algebra A. Note that the representation type of A and D_{4} are the same. Consider the algebra B with the quiver in Figure 26, along with the relations:

ϵ^{2} = ϵ, ᾱ_{1}β̄_{1} = 0, ηκ = 0, κη = 0, ηβ̄_{1}ᾱ_{1}κ = 0, ᾱ_{1}κϵη = 0, κϵηβ̄_{1} = 0.

Note that the ideal is not admissible; but the algebra is finite-dimensional.

[figure: •; 1; •; 3; 5; ᾱ_{1}; β̄_{1}; •; κ; η; ϵ] Figure 26

Now we define the functor F. Given a right A-module M, define a right B-module F M = V as follows. Set V e_{i} = Me_{i} for i = 1, 3; and set V e_{5} = Me_{2} ⊕ Me_{4}. The action of the arrows is defined as follows (we write m_{i} for an element of M_{i}) .

(m_{2}, m_{4})η := m_{2}α_{2} + m_{4}δ, m_{3}κ := (m_{3}β_{2}, −m_{3}γ), (m_{2}, m_{4})ϵ = (m_{2}, 0).

Moreover, we let ᾱ_{1}, β̄_{1} act as α_{1}, β_{1} act on M. It is straightforward to check that these satisfy the relations for B. If f : M → N is a A-homomorphism then F f is defined in the obvious way (note that F f as a linear map is the same as f but written differently). Next we must show that if M is indecomposable then so is F M, and moreover that F M ≅ F N implies M ≅ N. Both will follow if we show that F induces an isomorphism

Hom_{A}(M, N) → Hom_{B}(F M, F N)

It is clear that the map is 1-1. Now suppose g : F M → F N is a B-homomorphism. Write F M = V and F N = W as above. Clearly g maps V e_{i} into W e_{i}. Moreover since g commutes with ϵ and e_{5} − ϵ, it takes Me_{2} into Ne_{2} and Me_{4} into Ne_{4}. Then it follows easily that g commutes with the arrows of A appropriately; and hence g arises from a A-homomorphism from M to N, as required.

We construct now a degeneration of B. Let k̃ = k[t] where t is an indeterminate. Let B_{t} be the algebra over F given by B_{t} := F Q/I_{t} where Q is the above quiver, and I_{t} is the ideal generated by the same relations as the ideal for B except that the relation for ϵ^{2} is replaced by

ϵ^{2} = tϵ

If we specialize t to a non-zero element in k, c say, then we obtain an algebra isomorphic to B; this is seen by re-scaling a generator (replace ϵ by c^{−1}ϵ). On the other hand, if we specialize t = 0 then we obtain an algebra C which is special biserial, hence tame.

It remains to show that C is really a degeneration of B; to do this we must show that dim_{k} C = dim_{k} B. (Since ϵ ∈ B is an idempotent, we must be careful.) We find that e_{1}B has a basis {e_{1}, ᾱ_{1}, ᾱ_{1}κ, ᾱ_{1}κϵ}. Moreover, e_{3}B has a basis

{e_{3}, β̄_{1}, β̄_{1}ᾱ_{1}, β̄_{1}ᾱ_{1}κ, β̄_{1}ᾱ_{1}κϵ, κ, κϵ, κϵη}

Finally, a basis for e_{5}B is given by

{e_{5}, η, ηβ̄_{1}, ηβ̄_{1}ᾱ_{1}, ϵ, ϵη, ϵηβ̄_{1}, ϵηβ̄_{1}ᾱ_{1}}

In particular B has dimension 20. It is straightforward to check that C has a similar basis and is of the same dimension. We can now finish off (ii) of Theorem 1.2A, and (v) of Theorem 1.2B.

Theorem . Let p = 3. The algebras S(2, d) and S(2, d)_{r} for d = 9, 10, 11, and r ≥ 3 have tame representation type.

Proof. Note that for r ≥ 3, S(2, d) = S(2, d)_{r} for d = 9, 10, 11. The result now follows because D_{4} has tame representation type. _{□}

5.6 The algebra S(3, 8), p = 3, is Morita equivalent to H_{4} [Erd2, 3.13]. This basic algebra is given by the quiver in Figure 27 with relations αβ = αρ = 0, ζβ = ζρ = 0, αδ = 0 = γβ, and δγ = βα + ρζ.

[figure: •; 1; •; 3; •; 2; •; 4; α; β; γ; δ; ρ; ζ] Figure 27

Proposition . Let p = 3. The algebras S(3, 8) and S(3, 8)_{r} for r ≥ 2 have tame representation type.

Proof. Observe that S(3, 8) = S(3, 8)_{r} for r ≥ 2. It suffices to show the algebra H_{4} is tame; this is done by almost the same method as in the previous section. First factor out the socle of the indecomposable projective-injective P_{4}; this does not change the representation type. This gives us the additional relation γρζδ = 0. Denote this new algebra by A.

Define an algebra B with the quiver given by Figure 28 with relations

ϵ^{2} = ϵ, ᾱβ̄ = 0, ᾱκ = 0, ηβ̄ = 0, κη = −β̄ᾱ,
ϵηκϵ = 0, ηκϵηκ = 0

[figure: •; 1; •; 3; 5; ᾱ; β̄; •; κ; η; ϵ] Figure 28

We now define a functor F : mod(A) → mod(B). If M is an A-module then set F M = V where V e_{1} = Me_{1}, V e_{3} = Me_{3} and V e_{5} = Me_{2} ⊕ Me_{4}. The action of the arrows on V is defined as follows. Let (m_{2}, m_{4})ϵ := (m_{2}, 0), and define

(m_{2}, m_{4})η := m_{2}ζ + m_{4}γ, m_{3}κ := (m_{3}ρ, −m_{3}δ)

and ᾱ, β̄ act on V as α, β act on M. Then the relations of B hold. As before, this gives a functor which preserves indecomposability and reflects isomorphisms. We claim that B is tame. Let k̃ = k[t] as before, and define an k̃-algebra B_{t} by B_{t} := k̃Q/I_{t} where Q is as in Figure 28, and I_{t} is defined by the set of relations for B with two relations replaced. First, the relation for ϵ^{2} is replaced by ϵ^{2} = tϵ. Next, the commutativity relation is replaced by κη = −tβα. If one specializes t to a non-zero element in k then the algebra is isomorphic to B. In order to see this one rescales generators ϵ and β. On the other hand, if C is the algebra obtained from B_{t} by specializing t = 0 then C is special biserial. One can verify that C is a degeneration of B, as before with the dimensions equaling 17. _{□}

5.7 The algebra S(3, 7), p = 3, is Morita equivalent to R_{4} [Erd2, 3.11]. This algebra is given by the quiver in Figure 29 with relations α_{1}β_{1} = 0 = α_{1}α_{2} = β_{2}β_{1}, β_{1}α_{1} = α_{2}β_{2}, and β_{2}α_{2} = α_{3}β_{3}. Note that S(3, d) = S(3, d)_{r} for d = 7, 8 and r ≥ 2.

[figure: •; 1; •; 2; •; 3; •; 4; α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 29

Proposition . Let p = 3. The algebras S(3, 7) and S(3, 7)_{r} for r ≥ 2 have tame representation type.

Proof. Observe that S(3, 7) = S(3, 7)_{r} for r ≥ 2. We will now show that the algebra R_{4} is tame. The representation type does not change if one factors out socles of projectiveinjectives. So we may factor out the socle of P_{4}. The resulting algebra has again indecomposable projective-injectives, namely P_{3} and P_{4}. After factoring out their socles, P_{3} is again projective-injective, and we factor out the socle again. We denote this algebra by A. It has the same quiver as R_{4} and the ideal is generated by the relations

α_{1}β_{1} = 0 = α_{1}α_{2} = β_{2}β_{1} = β_{2}α_{2} = α_{3}β_{3}, β_{1}α_{1} = α_{2}β_{2}

Define an algebra D := kQ/I where Q is the quiver given in Figure 30 and I is generated by the relation

β̄_{1}ᾱ_{1} = ᾱ_{2}β̄_{2}

[figure: 2; 2^{∗}; 3; 1; 4; 4^{∗}; β̄_{1}; ᾱ_{1}; ᾱ_{2}; β̄_{2}; β̄_{3}; ᾱ_{3}] Figure 30

This algebra is tame, for example by comparison with the algebra labelled as 54 in [Rin, p. 162]. The modules of D may be identified with the representations of this algebra for which the vertical arrows in the commutative figure act as isomorphisms. We will now show that the module categories of A and of D are ‘almost equivalent’, i.e. except for finitely many indecomposables. To be precise, we will construct functors F : mod(A) → mod(D) and G : mod(D) → mod(A) such that GF (M) ≅ M for all A-modules M and F G(X) ≅ X for all but two indecomposable D-modules. This implies that A is tame.

Let F : mod(A) → mod(D) be defined as follows. Given an A-module M, define first subspaces M_{2}∗ and M_{4}∗ of Me_{2} and Me_{4} respectively as follows. Let M_{2}∗ := Im(α_{1}) + Im(β_{2}) and let M_{4}∗ = Im(α_{3}). Let F M = V where V e_{i} = Me_{i} for i = 1, 3; and where we set

V e_{2} := Me_{2}/M_{2}∗, V e_{2∗} = M_{2}∗, V e_{4} = Me_{4}/M_{4}∗, V e_{4∗} = M_{4}∗

We define β̄_{1} to be the map induced on the quotient by β_{1}, and β̄_{3} is the map induced by β_{3}. Note that these are well-defined, by the relations of A. Moreover, we let ᾱ_{1}, β̄_{2} and ᾱ_{3} act as α_{1}, β_{1} and α_{3} on M. This defines a representation of D; the relation follows since β_{1}α_{1} = α_{2}β_{2} on M. The definition of F on maps is the obvious one.

Next define a functor G : mod(D) → mod(A). Suppose X is a module for D, set G(X) = N where Ne_{i} = Xe_{i} for i = 1, 3; and where we set

Ne_{2} := Xe_{2} ⊕ Xe_{2∗}, Ne_{4} := Xe_{4} ⊕ Xe_{4∗}

The actions of the arrows of A are defined as follows. Let n_{i} be an element of Ne_{i} and n_{i∗} an element of Ne_{i∗}. Then set

(n_{1})α_{1} = (0, n_{1}ᾱ_{1}), (n_{2}, n_{2∗})β_{1} := n_{2}β̄_{2}
(n_{2}, n_{2∗})α_{2} = n_{2}ᾱ_{2}, n_{3}β_{2} = (0, n_{3}β̄_{2})
n_{3}α_{3} = (0, n_{3}ᾱ_{3}), (n_{4}, n_{4∗})β_{3} = n_{4}β̄_{3}

Then it is straightforward to check that these satisfy the relations of A, hence N is an A-module. The definition of F on maps is the obvious one.

We consider now F G(X) where X is an indecomposable D-module. We set G(X) = N and F (N) = V. Then V e_{1} = Xe_{1} and V e_{3} = Xe_{3}. Consider Xe_{2∗}. We assume that X is not the simple module at vertex 2∗, then we have Xe_{2∗} = ℑ(ᾱ_{1}) + ℑ(β̄_{1}). Moreover, we assume that X is not the simple module at vertex 4∗, then we have Xe_{4∗} = ℑ(ᾱ_{3}). These imply that V e_{2} = Xe_{2} and V e_{2∗} = Xe_{2∗}; and moreover V e_{4} = Xe_{4} and V e_{4∗} = Xe_{4∗}. We get F G(X) = X if X is indecomposable and not one of the starred simples.

Finally, consider now GF (M) where M is an indecomposable A-module. Set F (M) = X and GX = N. One can now check that N is isomorphic to M. We have Ne_{i} = Me_{i} for i = 1, 3, and Ne_{i} ≅ Me_{i} as vector spaces for i = 2, 4. Define a k-linear map ψ : N → M such that ψ = id on Ne_{1} and on Ne_{3}; and on Ne_{2}, Ne_{4} define ψ on a vector space basis which comes from a basis of Me_{i} containing a basis of M_{i∗}, in the obvious way. Then ψ is clearly a k-isomorphism; and it is clear that it commutes with α_{1}, β_{2}, α_{3} since these are the same on M as on N. Moreover, ψ also commutes with the other arrows.

For example, take a basis element b of Me_{2} which does not lie in M_{2∗}, then

ψ((b + M_{2∗}), 0)β_{1}) = ψ(bβ_{1}) = bβ_{1}

(the last step since ψ = id on Ne_{1}). On the other hand,

[ψ(b + M_{2∗}, 0)]β_{1} = bβ_{1}

(using the definition of ψ). _{□}

The preceding theorem in conjunction with Proposition 5.6 proves (i) of Theorem 1.2A, and (iv) of Theorem 1.2B. This completes the proof of the main result.

6 Appendix: Representation type of infinitesimal Schur algebras

6.1 In this appendix we provide tables for the reader to indicate the various cases of the representation type for the infinitesimal Schur algebras. Each table list the parameters d and r for S(n, d)_{r}. For the algebras of tame and wild representation type references are also provided where the proofs are given.

\underline{p = 2, n = 2}

Finite  Tame  reference wild  reference
d < 4, r ≥ 2 d = 4, r ≥ 2 (5.1), (5.2) d = 6, 8; r ≥ 4  (4.1h)
d = 5, 7; r ≥ 3 d = 9, r ≥ 3 (5.1), (5.2) d ≥ 10, r ≥ 4  (4.1h)
d odd, r = 2  d = 6 + 2j, j ≥ 0, r ≥ 3 (4.1i)
d ≥ 1, r = 1  d = 6 + 2j, j ≥ 0, r = 2 (4.7)
d = 6, 8; r = 3  (4.7)
d ≥ 10, r = 3  (4.7)

\underline{p = 2, n = 3}

Finite  Tame reference Wild  reference
d < 4, r ≥ 2 none none  d ≥ 4, r ≥ 3 (4.1e)
d < 2, r = 1  d ≥ 4, r ≥ 2 (4.8)
d = 2, 3; r = 1  d ≥ 4, r = 1 (4.3)

\underline{p = 2, n = 4}

Finite  Tame  reference Wild  reference
d < 4, r ≥ 2 d = 2, 3; r = 1 (5.4)  d ≥ 4, r ≥ 3 (4.1e)
d < 2, r = 1  d ≥ 4, r = 1 (4.3)
d ≥ 4, r ≥ 2 (4.8)

\underline{p = 2, n ≥ 5}

Finite  Tame reference Wild  reference
d < 4, r ≥ 2 none none  d ≥ 4, r ≥ 3 (4.1e)
d < 2, r = 1  d ≥ 4, r = 1 (4.3)
d ≥ 4, r ≥ 2 (4.8)
d ≥ 2, r = 1 (4.4)

\underline{p = 3, n = 2}

Finite  Tame  reference Wild  reference
d < 9, r ≥ 2 d = 9, 10, 11; r ≥ 3 (5.5)  d ≥ 12, r ≥ 3 (4.1g)
d ≥ 1, r = 1  d ≥ 9, r = 2 (4.6)

\underline{p = 3, n = 3}

Finite  Tame  reference Wild  reference
d < 6, r ≥ 2 d = 3, r = 1 (5.3)  d = 6, r ≥ 2 (4.1b)
d < 3, r = 1 d = 7, 8; r ≥ 2 (5.7) (5.6) d ≥ 9, r ≥ 3 (4.1c)
d = 4, 5; r = 1  d = 6, r ≥ 2 (4.2)
d ≥ 9, r ≥ 2 (4.2)
d ≥ 6, r = 1 (4.3)

\underline{p = 3, n = 4}

Finite  Tame  reference Wild  reference
d < 6, r ≥ 2 d = 3, 4, 5; r = 1 (5.4)  d ≥ 6, r ≥ 2 (4.1d)
d < 3, r = 1  d ≥ 9, r ≥ 3 (4.1c)
d ≥ 6, r = 1 4.3

\underline{p = 3, n ≥ 5}

Finite  Tame reference Wild  reference
d < 6, r ≥ 2 none none  d ≥ 6, r ≥ 2  (4.1d)
d < 3, r = 1  d ≥ 9, r ≥ 3  (4.1c)
d ≥ 6, r = 1  (4.3)
d = 3, 4, 5, r = 1 (4.4)

\underline{p ≥ 5, n = 2}

Finite  Tame reference Wild  reference
d < p^{2}, r ≥ 2 none none  d ≥ p^{2}, r ≥ 3 (4.1f)
d ≥ 1, r = 1  d ≥ p^{2}, r = 2 (4.6)

\underline{p ≥ 5, n = 3}

Finite  Tame  reference Wild  reference
d < 2p, r ≥ 2 p ≤ d ≤ 2p − 1, r = 1 (5.3)  d ≥ 2p, r ≥ 2 (4.1a)
d < p, r = 1  d ≥ 2p, r = 1 (4.3)

\underline{p ≥ 5, n ≥ 4}

Finite  Tame reference Wild  reference
d < 2p, r ≥ 2 none none  d ≥ 2p, r ≥ 2 (4.1a)
d < p, r = 1  d ≥ 2p, r = 1 (4.3)
d ≥ p, r = 1 (4.4)

References

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E-mail address: doty@math.luc.edu

Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford, OX1 3LB England

E-mail address: erdmann@vax.ox.ac.uk

Magdalene College, Cambridge University, Cambridge, CB3 0AG England

E-mail address: s.martin@pmms.cam.ac.uk

Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322 U.S.A.

E-mail address: nakano@sunfs.math.usu.edu