REPRESENTATION TYPE OF SCHUR SUPERALGEBRAS

DAVID J. HEMMER, JONATHAN KUJAWA, AND DANIEL K. NAKANO

Abstract. Let S(m|n, d) be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup GL(m|n) of degree d. In this paper we determine the representation type of these algebras (i.e. classify the ones which are semisimple, have finite, tame and wild representation type). Moreover, we prove that these algebras are in general not quasi-hereditary and have infinite global dimension.

1 Introduction

1.1 A central problem in the study of finite-dimensional algebras is to understand the structure of the indecomposable modules. As a first step one would like to know how many indecomposable modules a given algebra admits. Any finite-dimensional algebra can be classified into one of three categories: finite, tame, or wild representation type. A finitedimensional algebra A has finite representation type if A has finitely many indecomposable modules up to isomorphism. If A is not of finite type then A is of infinite representation type. Algebras of infinite representation type are either of tame representation type or of wild representation type. For algebras of tame representation type one has a chance of classifying all the indecomposable modules up to isomorphism.

There have been much progress in the past ten years in determining the representation type of important classes of finite-dimensional algebras. The Schur algebras and the q-Schur algebra are finite-dimensional algebras which arise in the representation theory of the general linear groups and symmetric groups. A complete classification of the representation type of these algebras was given by Doty, Erdmann, Martin and Nakano [E, DN, DEMN, EN1]. These algebras are fundamental examples of quasi-hereditary algebras (or equivalently highest weight categories) which were introduced by Cline, Parshall and Scott [CPS]. The blocks for the ordinary and parabolic BGG category O for finite-dimensional complex semisimple Lie algebras are other important examples of highest weight categories. Results pertaining to the representation type for these blocks were obtained independently by Futorny, Nakano and Pollack [FNP], and Brüstle, König and Mazorchuk [BKM] for the ordinary category O and more recently for the parabolic category O by Boe and Nakano [BN]. For quasihereditary algebras the projective modules admit filtrations by certain standard modules. This information allows one to determine the structure of the projective modules which in turn can lead to an expression of the algebra via quiver and relations. In many cases this information can be used to determine the representation type of the algebra.

Date: October 2004.

1991 Mathematics Subject Classification. Primary 20C30.

Research of the first author was supported in part by NSF grant DMS-0102019.

Research of the second author was supported in part by NSF grant DMS-0402916.

Research of the third author was supported in part by NSF grant DMS-0400548.

Other important classes of finite-dimensional algebras are quasi-Frobenius algebras where all projective modules are injective. These algebras all have infinite global dimension (except when the algebra is semisimple). For group algebras, restricted enveloping algebras, and Frobenius kernels homological information involving the theory of complexity and support varieties can be used to deduce information about the representation type of these algebras. For Hecke algebras of type A, the representation type of the blocks was given by Erdmann and Nakano in [EN2]. Ariki and Mathas [AM] classified the representation type for Hecke algebras of type B using Fock space methods. Ariki [A] recently has extended this classification to Hecke algebras of classical type.

The main result of this paper is a complete classification of the representation type for the Schur superalgebras S(m|n, d). These algebras are of particular interest because the category of S(m|n, d)-supermodules is equivalent to the category of polynomial representations of degree d for the supergroup GL(m|n). Also, as with the classical Schur algebra, there is a Schur-Weyl duality between S(m|n, d) and the symmetric group Σ_{d} (see [BK]). We shall show that in most cases S(m|n, d) is not quasi-hereditary, which is contrary to a recent conjecture given by Marko and Zubkov. Since the Schur superalgebras are not quasi-hereditary the aforementioned filtration techniques cannot be used. In order to obtain information about the basic algebra for certain Schur superalgebras we will compute the endomorphism algebras of direct sums of signed Young modules. This will entail knowing the structures of certain signed Young modules, which is of independent interest from the viewpoint of the representation theory of symmetric groups.

1.2 We review below the results for the ordinary Schur algebra S(m, d), although the proof for S(m|n, d) does not rely on these results. The representation type of S(m, d) was determined in [E, DN, DEMN]^{1}.

(1.2.1) Theorem . Let S(m, d) be the Schur algebra over k where char k ≥ 0.

(a) S(m, d) is semisimple if and only if one of the following holds (i) char k = 0; (ii) d < p; (iii) p = 2, m = 2, d = 3.

(b) S(m, d) has finite representation type if and only if one of the following holds (i) p ≥ 2, m ≥ 3, d < 2p; (ii) p ≥ 2, m = 2, d < p^{2}; (iii) p = 2, m = 2, d = 5, 7.

REPRESENTATION TYPE OF S(m|n, d) 3

(c) S(m, d) has tame representation type if and only if one of the following holds (i) p = 3, m = 3, d = 7, 8; (ii) p = 3, m = 2, d = 9, 10, 11; (iii) p = 2, m = 2, d = 4, 9, 11.

1The statement that S(2, 11) for p = 2 has tame representation type was inadvertently omitted from the original statement of the theorem [DEMN, Thm. 1.2(A)]. We thank Alison Parker for pointing this out to us.

(d) In all other cases not listed above S(m, d) has wild representation type.

The classification of the representation type of the q-Schur algebras S_{q}(m, d) can be found in [EN1, Thm. 1.4(A)-(C)].

1.3 Throughout this paper we will assume that k is an algebraically closed field of characteristic p ≥ 0. The following theorem summarizes our results on the Schur superalgebras.

(1.3.1) Theorem . Let S(m|n, d) be the Schur superalgebra over k where char k ≠ 2.

(a) S(m|n, d) is semisimple if and only if one of the following holds (i) char k = 0; (ii) d < p; (iii) m = n = 1 and p ∤ d.

(b) S(m|n, d) has finite representation type but is not semisimple if and only if one of the following holds

(i) p ≤ d < 2p; (ii) m = n = 1 and p | d.

(c) In all other cases not listed above S(m|n, d) has wild representation type.

We should remark that when in characteristic two S(m|n, d) is equal to the ordinary Schur algebra S(m + n, d), and its representation type is classified by Theorem 1.2.1.

2 Comparing Module Categories for Superalgebras

2.1 Recall that k denotes our fixed ground field. All objects defined over k (algebras, modules, superalgebras, supermodules, etc.) are assumed to be finite-dimensional as kvector spaces.

A superspace is a ℤ_{2}-graded k-vector space V = V¯0 ⊕ V¯1. Given a homogeneous element of a superspace, v ∈ V, we write \overline{v} ∈ ℤ_{2} for the degree of the element. If V and W are superspaces, then Hom_{k}(V, W) is naturally a superspace with Hom_{k}(V, W)¯0 (resp. Hom_{k}(V, W)¯1) consisting of all linear maps which preserve (resp. reverse) grading.

A superalgebra is a ℤ_{2}-graded associative k-algebra A = A¯0 ⊕ A¯1 which satisfies A_{r}A_{s} ⊆ A_{r+s} for all r, s ∈ ℤ_{2}. An A-supermodule is a ℤ_{2}-graded A-module M = M¯0 ⊕ M¯1 which satisfies A_{r}M_{s} ⊆ M_{r+s} for all r, s ∈ ℤ_{2}. An A-supermodule homomorphism is a linear map f : M → N which satisfies f(am) = (−1)^{\overline{f} \overline{a}}af(m) for all homogeneous a ∈ A and all m ∈ M. As we will do later without comment, the given condition makes sense only for homogeneous elements. The general case is obtained by linearity. Note that if M and N are A-supermodules, then Hom_{A}(M, N) is naturally a superspace just as in the previous paragraph.

We can define the parity change functor Π on the category of A-supermodules as follows. On an object M we have that ΠM = M as a vector space, but we switch the ℤ_{2}-grading by setting (ΠM)_{r} = M_{r+¯1} for r ∈ ℤ_{2}. The action of A is defined via a.m = (−1)\overline{a}am for homogeneous a ∈ A and m ∈ M. On a morphism f, Πf is the same linear map as f. We refer the reader to [BKl, Sect. 2] for further discussion of general results on superalgebras, supermodules, etc.

Throughout the remainder of this subsection we assume char k ≠ 2. Given a superalgebra A there are three module categories which are natural to consider:

(i) the category A-smod of all A-supermodules and all (not necessarily graded) Asupermodule homomorphisms;

(ii) the underlying even category of A-smod where we take the objects of A-smod but

only the even (i.e. grading preserving) homomorphisms; (iii) the category A-mod of all A-modules.

Observe that an A-supermodule is indecomposable or irreducible regardless of whether we view it as an object of the category A-smod or its underlying even category. If M and N are indecomposable A-supermodules and f : M → N is an A-supermodule isomorphism, write f = f¯0 + f¯1 with f_{r} ∈ Hom_{A}(M, N)_{r} for r ∈ ℤ_{2}. Note that by considering the ℤ_{2}-grading it is straightforward to verify that f¯0 − f¯1 : M → N is also an isomorphism. Using this one can show that the map M ⊕ ΠM → N ⊕ ΠN given by

(m, n) ↦ (f¯0(m) + f¯1(n), f¯1(m) + f¯0(n))

is in fact an even isomorphism of A-supermodules. Consequently, by the graded version of the Krull-Schmidt theorem we see that M is isomorphic to either N or ΠN by an even isomorphism. Taken together the above discussion proves that the category A-smod is semisimple, finite, tame, or wild if and only if its underlying even category is semisimple, finite, tame, or wild, respectively. We use this observation without comment in what follows. The remainder of this section is devoted to studying the relationship between the representation type of these categories and the category A-mod.

Given a superspace V, let δ : V → V be the linear involution defined by δ(v) = (−1)\overline{v}v, where v ∈ V is homogeneous. In particular, the involution δ : A → A is an algebra homomorphism and defines an action of ℤ_{2} on A. Consequently, we can define the smash product of kℤ_{2} and A as follows. Set

B = kℤ_{2} ⊗_{k} A

as a vector space. Identify kℤ_{2} with k[x]/(x^{2} − 1). Then the product in B is given by

(x^{r} ⊗ a)(x^{s} ⊗ b) = x^{r+s} ⊗ δ^{s}(a)b,  (2.1.1)

where r, s ∈ {0, 1} and a, b ∈ A. We identify kℤ_{2} and A as subalgebras via kℤ_{2}⊗1 and 1⊗A, respectively. Note that since we assume char k ≠ 2, x acts semisimply on any kℤ_{2}-module with eigenvalues ±1. The following result is proven in [Mui, Thm. 1.4b].

(2.1.2) Proposition . The category of B-modules is isomorphic to the underlying even category of A-supermodules.

Proof. Let M be an A-supermodule. We define an action of kℤ_{2} on M via xm = δ(m). It is straightforward to verify that this makes M into a B-module. Conversely, if M is a B-module, then we can view it as an A-module by restriction. To obtain a ℤ_{2}-grading we set M¯0 (resp. M¯1) to be the 1 (resp. −1) eigenspace of x acting on M. One can verify that this ℤ_{2}-grading makes M into an A-supermodule. It is straightforward to verify that a morphism in one category restricts or extends to a morphism in the other category. _{□} 2.2 Note that there is an involution ζ : B → B given by x^{r} ⊗ a ↦ (−1)^{r}x^{r} ⊗ a for r ∈ {0, 1} and a ∈ A. Given a B-module M we can twist it by ζ, which we denote by M^{ζ}. Namely, M^{ζ} = M as a vector space but with the action of B given by b.m = ζ(b)m for all b ∈ B and m ∈ M. Note that M^{ζ} is indecomposable if and only if M is indecomposable.

Since A is a subalgebra of B, there is a restriction functor res_{A}^{B} : B-mod → A-mod and an induction functor ind_{A}^{B} : A-mod → B-mod given by

ind_{A}^{B} N = B ⊗_{A} N

for any A-module N. Note that

ind_{A}^{B} N ≅ 1 ⊗ N ⊕ x ⊗ N  (2.2.1)

as A-modules. If M is a B-module, then we have

M ⊕ M^{ζ} ≅ ind_{A}^{B} res_{A}^{B} M.  (2.2.2)

Since x acts semisimply on M and M^{ζ}, it suffices to define the isomorphism on eigenvectors m ∈ M, m′ ∈ M^{ζ} of eigenvalue ε, ε′ ∈ {±1}, respectively. Then the map M ⊕ M^{ζ} → ind_{A}^{B} res_{A}^{B} M is given by

(m, m′) ↦ (1 + εx) ⊗ m + (1 − ε′x) ⊗ m′.

To verify that the map is B-linear, one uses the observation that if a ∈ A is an eigenvector for the action of x with eigenvalue α and m ∈ M is an eigenvector for the action of x with eigenvalue β, then am is an eigenvector with eigenvalue αβ.

(2.2.3) Proposition . Let M be an indecomposable B-module. Then M is either indecomposable as an A-module or the direct sum of two indecomposable A-modules. Furthermore, every indecomposable A-module N is a direct summand of ind_{A}^{B} N as A-modules.

On the other hand, if M is an indecomposable B-module which contains N as a direct summand, then M is unique up to isomorphism and twisting by ζ.

Proof. Let M be an indecomposable B-module and let

M = M_{1} ⊕ ⋯ ⊕ M_{k}  (2.2.4)

be a decomposition of M into indecomposable A-modules. By using (2.2.2) and applying ind_{A}^{B} one has

M ⊕ M^{ζ} ≅ ind_{A}^{B} M_{1} ⊕ ⋯ ⊕ ind_{A}^{B} M_{k}.

Since the left hand side is a decomposition as indecomposable B-modules, by the Krull-Schmidt theorem it follows that (2.2.4) cannot have more than two direct summands.

Now say N is an indecomposable A-module. By (2.2.1) we see that N is a direct summand of ind_{A}^{B} N. On the other hand, if M is an indecomposable B-module which contains N as a direct summand, then ind_{A}^{B} M ≅ M ⊕ M^{ζ} contains ind_{A}^{B} N. If ind_{A}^{B} N is indecomposable, then either M ≅ ind_{A}^{B} N or M ≅ (ind_{A}^{B} N)^{ζ}. If ind_{A}^{B} N is decomposable, then ind_{A}^{B} N ≅ M ⊕ M^{ζ}. This proves the desired result.

□

As is well known, if M is an irreducible A-supermodule then as an A-module M is either irreducible or the direct sum of two irreducibles. Combining the previous result with Proposition 2.2.3 shows this behaviour extends to indecomposables.

2.3 We can now prove the main result of this section: the representation type of A as a superalgebra coincides with that of A as an algebra.

(2.3.1) Theorem . The category A-smod is semisimple, finite, tame, or wild if and only if the same is true for A-mod.

Proof. By Proposition 2.1.2 it suffices to consider the category B-mod rather than A-smod.

Semisimple type: That A-smod is semisimple if and only if B-mod is semisimple is already known. See, for example, [BKl, Sect. 2] or [Mui, Thm. 1.4c].

Finite type: It is clear from Proposition 2.2.3 that A-mod has finitely many indecomposables if and only if B-mod does.

Tame type: Suppose A-mod has tame representation type. Let t be an indeterminant and d a positive integer. Then there are indecomposable A-modules D_{d,1}, …, D_{d,r_{d}} and indecomposable A − k[t]-bimodules E_{d,1}, …, E_{d,s_{d}} which are free as right k[t]-modules so that any indecomposable A-module of dimension d is either isomorphic to D_{d,i} for some i, or is isomorphic to E_{d,i} ⊗_{k[t]} L for some i and some irreducible k[t]-module L. Observe that the proof Proposition 2.2.3 applies verbatim to indecomposable A − k[t]-bimodules. Consequently, we can choose indecomposable B-modules D̃_{d,1}, …, D̃_{d,r_{d}} and indecomposable B −k[t]-bimodules Ẽ_{d,1}, …, Ẽ_{d,s_{d}} (which are free as k[t]-supermodules) where D̃_{d,i} contains D_{d,i} as a direct summand as A-modules and Ẽ_{d,i} contains E_{d,i} as a direct summand as A − k[t]-bimodules.

Let M be an indecomposable B-module and N an indecomposable direct summand of M as an A-module. If N ≅ D_{d,i} for some d and i, then by the uniqueness statement in Proposition 2.2.3 it follows that M ≅ D̃_{d,i} or M ≅ (D̃_{d,i})^{ζ}. On the other hand, say N ≅ E_{d,i} ⊗_{k[t]} L for some d, i, and irreducible k[t]-module L. Since E_{d,i} is a direct summand of Ẽ_{d,i}, we have that N ≅ E_{d,i} ⊗_{k[t]} L is a direct summand of Ẽ_{d,i} ⊗_{k[t]} L. However, by [DEMN, Lemma 2.2] we know Ẽ_{d,i} ⊗_{k[t]} L is indecomposable. By uniqueness again we have M ≅ Ẽ_{d,i} ⊗_{k[t]} L or M ≅ (Ẽ_{d,i} ⊗_{k[t]} L)^{ζ} = (Ẽ_{d,i})^{ζ} ⊗_{k[t]} L. Therefore, if A-mod is tame, then B-mod is tame.

Wild type: Suppose that A-mod has wild representation type. By [DEMN, Theorem 2.2] and Proposition 2.2.3 it follows that B-mod has wild representation type.

□

3 Results on the Schur Superalgebra

3.1 The Schur Superalgebra: Fix nonnegative integers m, n, and d, and fix a superspace V with fixed basis v_{1}, …, v_{m+n} and ℤ_{2}-grading given by setting \overline{v}_{i} = ¯0 for i = 1, …, m and \overline{v}_{i} = ¯1 for i = m + 1, …, m + n. The tensor space V^{⊗d} is then naturally ℤ_{2}-graded by setting the degree of v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{d}} to be \overline{v}_{i_{1}} + ⋯ + \overline{v}_{i_{d}} for all 1 ≤ i_{1}, …, i_{d} ≤ m + n. The symmetric group Σ_{d} acts on V^{⊗d} on the right via

v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{j}} ⊗ v_{i_{j+1}} ⊗ ⋯ ⊗ v_{i_{d}}(j j + 1) := (−1)^{v̄_{i_{j}} v̄_{i_{j+1}}} v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{j+1}} ⊗ v_{i_{j}} ⊗ ⋯ ⊗ v_{i_{d}} for 1 ≤ i_{1}, …, i_{d} ≤ m + n and each 1 ≤ j < d.

The Schur superalgebra is defined to be

S(m|n, d) = End_{Σ_{d}}(V^{⊗d}),

with the ℤ_{2}-grading given by setting S(m|n, d)¯0 (resp. S(m|n, d)¯1) to be the set of all degree preserving (resp. degree reversing) maps. Observe that when p = 2 the above action coincides with the usual permutation action and so S(m|n, d) = S(m + n, d).

We also note that S(m|n, d) ≅ S(n|m, d) as superalgebras. Namely, let t : {1, …, m + n} → {1, …, m + n} be given by t(i) = i + n if i = 1, …, m and t(i) = i − m if i = m + 1, …, m + n. Fix a superspace Ṽ with fixed basis ṽ_{1}, …, ṽ_{m+n} and ℤ_{2}-grading given by setting \overline{ṽ}_{i} = ¯0 for i = 1, …, n and \overline{ṽ}_{i} = ¯1 for i = n + 1, …, m + n. Define a linear map τ : V^{⊗d} ⊗ sgn → Ṽ^{⊗d} via

τ(v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{d}} ⊗ 1) = (−1)^{(d−1)\overline{v}_{i_{1}}+(d−2)\overline{v}_{i_{2}}+···+\overline{v}_{i_{d−1}}} ṽ_{t(i_{1})} ⊗ ⋯ ⊗ ṽ_{t(i_{d})},

where v_{i_{1}}, …, v_{i_{d}} are elements of our fixed homogeneous basis for V. It is straightforward to verify on simple transpositions that in fact τ is a kΣ_{d}-module isomorphism (c.f. the proof of [BK, Theorem 6.4]). Consequently we have

S(m|n, d) = End_{Σ_{d}}(V^{⊗d}) ≅ End_{Σ_{d}}(V^{⊗d} ⊗ sgn) ≅ End_{Σ_{d}}(Ṽ^{⊗d}) = S(n|m, d),

and the isomorphisms preserve the ℤ_{2}-grading.

We now introduce certain modules which will be crucial in later calculations. Given a tuple of integers γ = (γ_{1}, …, γ_{t}) we denote the integer γ_{1} + ⋯ + γ_{t} by |γ|. Let

Λ(m|n, d) = {(λ_{1}, …, λ_{m}|μ_{1}, …, μ_{n}) ∈ ℤ^{m+n} | λ_{i}, μ_{i} ≥ 0 for all i and |λ| + |μ| = d} .

We write (λ|μ) for an element of Λ(m|n, d).

In general, given γ = (γ_{1}, …, γ_{t}), a sequence of nonnegative integers summing to r, let

Σ_{γ} = Σ_{γ_{1}} × Σ_{γ_{2}} × ⋯ × Σ_{γ_{t}}

viewed as a subgroup of Σ_{r} in the natural way by Σ_{γ_{1}} acting as permutations of the first γ_{1} letters, Σ_{γ_{2}} acting as permutations of the next γ_{2} letters, and so on.

Given (λ|μ) ∈ Λ(m|n, d) we define the signed permutation module M^{(λ|μ)} as

M^{(λ|μ)} = ind_{Σ_{λ}×Σ_{μ}}^{Σ_{d}} k ⊠ sgn,

where k denotes the trivial module for Σ_{λ_{1}} ×⋯×Σ_{λ_{m}} and sgn denotes the one dimensional sign representation for Σ_{μ_{1}} × ⋯ × Σ_{μ_{n}}.

By definition a Σ_{d}-module is a signed Young module if for some nonnegative integers m and n and (λ|μ) ∈ Λ(m|n, d) it is a direct summand of M^{(λ|μ)}. Signed permutation modules and signed Young modules were first considered by Donkin [Do]. Finally we observe the well known fact that as kΣ_{d} modules with the action from above we have:

V^{⊗d} ≅ ⨁_{(λ|μ)∈Λ(m|n,d)} M^{(λ|μ)}.  (3.1.1)

3.2 We begin by collecting a few preliminary results.

(3.2.1) Proposition . Let S be a finite-dimensional algebra containing an idempotent e, so eSe is a subalgebra of S.

(a) Every indecomposable eSe-module is a summand of the restriction to eSe of some indecomposable S-module.

(b) If eSe has infinite representation type then so does S.

(c) If eSe has wild representation type then so does S.

Proof. (a) Let N be an indecomposable eSe-module. Let U = S ⊗_{eSe} N. Then U is a nonzero S-module and:

U ≅ eU ⊕ (1 − e)U

as eSe modules. But eU ≅ N. Now writing U as a direct sum of indecomposable S modules and an easy argument with the Krull-Schmidt theorem gives the result. (b) This is clear from part (a). If S had only finitely many indecomposable modules then part (a) would imply eSe does as well. (c) This statement follows from the argument of Theorem 2.2 in [DEMN]. If the statement of (a) holds for an arbitrary algebra A and subalgebra B, the authors show that if A is tame then so is B. _{□}

3.3 One can now apply the previous result to specific situation of S(m|n, d): This allows us to descend from the superalgebra S(m′|n′, d) to S(m|n, d) for m ≤ m′, n ≤ n′.

(3.3.1) Corollary . Let m ≤ m′ and n ≤ n′. Then:

(a) If S(m|n, d) has infinite representation type then so does S(m′|n′, d).

(b) If S(m|n, d) has wild representation type then so does S(m′|n′, d).

Proof. Recall that Λ(m|n, d) is the set of pairs (λ|μ) where λ and μ are compositions with |λ| + |μ| = d and λ has at most m parts, μ has at most n parts. Thus Λ(m|n, d) is naturally a subset of Λ(m′|n′, d). Recall that:

S(m′|n′, d) ≅ End_{kΣ_{d}} ( ⨁_{(λ|μ)∈Λ(m′|n′,d)} M^{(λ|μ)}) .

So if we let e denote the projection onto the direct sum of all M^{(λ|μ)} with (λ|μ) ∈ Λ(m|n, d) then it is clear that eS(m′|n′, d)e ≅ S(m|n, d), so we can apply Proposition 3.2.1 to prove the result. _{□}

3.4 The classical (and infinitesimal) Schur algebra S(n, n) has a natural one-dimensional representation corresponding to the determinant for GL(n). This allows one to embed the module category for S(n, d) into that of S(n, d + n), which can be used to study representation type (e.g. [DNP, Thm. 2.2]). Unfortunately the analogue of the determinant representation for the supergroup GL(m|n), the one-dimensional Berezinian representation, is not polynomial and so it does not give a natural one-dimensional S(m|n, m + n)-module.

We instead use the existence of a one-dimensional representation for S(2|1, p). Namely, let L(0, 0|p) denote one-dimensional supermodule which is the Frobenius twist of the onedimensional GL(2) × GL(1)-module L(0, 0) ⊠ L(1) (see [BK, Remark 4.6]). Since the polynomial representations of GL(m|n) correspond with the representations of the supermonoid M(m|n) one can follow the arguments given in [DNP, Prop. 2.1, Prop. 2.2] by replacing the determinant representation with L(0, 0|p) to deduce the following result.

(3.4.1) Proposition . If S(2|1, d) has wild representation type then so does S(2|1, d + cp) for any c ∈ N.

3.5 We will assume familiarity with basic modular representation theory of the symmetric group. In the next section we will prove the main result of the paper. Surprisingly, we will use virtually no representation theory of the superalgebra. However, we will need to know that Young modules Y^{(a,b,1^{c})} are actually signed Young modules when c < p and m = 2, n = 1.

(3.5.1) Proposition . Let 1 ≤ c < p. Then the Young module Y^{(a,b,1^{c})} is a summand of the signed permutation module M^{(a,b|c)}.

Proof. Since c < p we have

M^{(1^{c})} ≅ kΣ_{c} ≅ ind_{Σ_{1^{c}}}^{Σ_{c}} k ≅ ⨁_{λ⊢c}(S^{λ})^{dim S^{λ}}.  (3.5.2)

But the Specht module S^{λ} has dimension greater than one except when λ = (1^{c}) or λ = c. Thus using (3.5.2) we obtain:

M^{(a,b,1^{c})} ≅ M^{(a,b,c)} ⊕ M^{(a,b|c)} ⨁_{λ⊢c, λ≠(1^{c}), (c)} ind_{Σ_{a}×Σ_{b}×Σ_{c}}^{Σ_{d}}(k ⊠ k ⊠ S^{λ})^{d_{λ}} (3.5.3)

with all the d_{λ} > 1. Since Y^{(a,b,1^{c})} occurs exactly once as a direct summand of M^{(a,b,1^{c})} and Hom_{Σ_{d}}(Y^{(a,b,1^{c})}, M^{(a,b,c)}) = 0, the result follows from (3.5.3).

□

3.6 The significance of Proposition 3.5.1 is that it gives us a Young module Y^{λ} as a summand of the tensor space V^{⊗d}, even though λ has more than three parts and m+n = 3. We need information about one other special signed Young module for the p = 3 case.

(3.6.1) Proposition . Suppose p does not divide a. Then:

M^{(a,0|p)} ≅ S^{(a+1,1^{p−1})} ⊕ S^{(a,1^{p})}.

The two Specht modules in the decomposition are both irreducible and:

S^{(a,1^{p})} ≅ D^{(a,2,1^{p−2})}.

Proof. The Specht module decomposition follows from Young’s rule (see [J2, p.51]). The irreducibility of the “hook” Specht modules when p ∤ d is well known work of Peel [P]. Finally knowing which irreducible S^{(a,1^{p})} corresponds to can be calculated directly, or also follows as a very special case of a theorem of James [J1, Thm. A]. _{□} 4 Proof of the main result

In this section we prove Theorem 1.3.1. One of the main ideas will be to identify a subalgebra eSe with kQ/I for some quiver Q. Then we can use Gabriel’s Theorem, since if the separated quiver is not a Dynkin or Euclidean diagram then the algebra is of wild type.

4.1 S(1|1, d) :

If n = 0 or m = 0 the superalgebra S(m|n, d) is isomorphic to the Schur algebra S(m, d), so the smallest nontrivial case for us is the superalgebra S(1|1, d). This has been studied by Marko and Zubkov:

(4.1.1) Proposition . [MZ, Prop. 2.1] Let S = S(1|1, d). If p does not divide d then S is semisimple. If p divides d then S has d + 1 pairwise non-isomorphic one-dimensional modules, labelled by the weights 0, 1, …, d. The algebra is basic, connected, and has left regular representation given by:

SS = {0 \atop 1} ⊕ 1 0 2 1 ⊕ 2 1 3 2 ⊕ ⋯ ⊕ \begin{matrix}r − 1 \\ r − 2  r \\ r − 1 \end{matrix} ⊕ \begin{matrix}r \\ r − 1 \end{matrix}  (4.1.2)

From here it is easy to obtain the quiver. In particular, for m ≥ 1 we define an algebra Ã_{m} to be the algebra kQ/I where Q is the quiver with m + 1 vertices

[figure: α_{1}; β_{1}; α_{2}; β_{2}; α_{m−1}; β_{m−1}; α_{m}; β_{m}]

and I is the ideal generated by the relations:

α_{i}α_{i+1} = 0, β_{i+1}β_{i} = 0 α_{1}β_{1} = β_{m}α_{m} = 0 and β_{i}α_{i} = α_{i+1}β_{i+1} for 1 ≤ i ≤ m − 1.

We remark that the quiver for Ã_{m} is the same as the quiver for Erdmann’s algebra (defined in [E, 3.1]) A_{m+1} with one additional relation, namely that β_{m}α_{m} = 0. As we will see in Section 5, this extra relation is crucial in that it makes the algebra have infinite global dimension.

(4.1.3) Theorem . When p divides d the algebra S(1|1, d) is isomorphic to the algebra Ã_{d} and has finite representation type.

Proof. The proof of the algebra isomorphism is essentially the same as Proposition 3.2 in [E], so we do not give it here. The algebra Ã_{m} is a three nilpotent algebra, but the only indecomposable modules of radical length three are projective. Hence, the representation type of Ã_{m} is the same as the representation type of Ã_{m} modulo its radical squared (which is two nilpotent). The separated quiver is a disjoint union of two copies of the Dynkin diagram A_{m+1}, and hence by Gabriel’s Theorem the algebra is of finite type. _{□}

4.2 S(m|n, d), d < 2p : We first consider the case where p ≤ d < 2p. It turns out this case is essentially the same as for the Schur algebra S(2, d). Recall that the summands of the signed permutation modules M^{(α|β)} are called signed Young modules. We recall the following theorem of Donkin.

(4.2.1) Theorem . [Do, 2.3 (6)] Let m, n ≥ d. The isomorphism classes of indecomposible signed Young modules are labelled by the set

Λ^{++}(m|n, d) := {(λ|pμ) | |λ| + p|μ| = d}

where λ and μ are partitions.

Following [Do], we write Y^{(λ|pμ)} for the indecomposible signed Young module labelled by (λ|pμ) ∈ Λ^{++}(m|n, d).

We wish to show that for d < 2p the signed Young modules are exactly the ordinary Young and twisted Young modules. To begin we will show that the nonprojective signed Young modules are all distinct.

Given a partition λ ⊢ d, let λ′ denote the transpose partition of λ. For λ ⊢ d p-regular we let m(λ) be the Mullineux conjugate of λ, so D^{λ} ⊗ sgn ≅ D^{m(λ)}. It is well known that the Young module Y^{m(λ)′} is the projective cover of D^{λ}, a fact we will use several times below.

(4.2.2) Proposition . If the Young module Y^{λ} is isomorphic to a twisted Young module Y^{μ} ⊗ sgn, then λ is p-restricted, in which case λ = m(μ′)′.

Proof. Applying the adjoint Schur functor to Y^{λ} and Y^{μ} ⊗sgn (see [HN, Thms 3.4.2, 3.8.2]) tells us that P (λ) ≅ T (μ′). But the tilting modules are self dual, so P (λ) is self-dual and hence has simple socle L(λ). But V (μ′) is a submodule of T (μ′) and the socle of V (μ′) is L(m(μ′)′) so λ = m(μ′)′ is p-restricted. _{□}

For d < 2p it is clear that the p-singular partitions are exactly those of the form (μ_{1} + p, μ_{2}, …) for μ a partition of d − p. Thus the number of signed Young modules for d is the total number of partitions of d plus the number of p-singular partitions, which by Prop. 4.2.2 is precisely the number of Young and twisted Young modules. We have shown the following result.

(4.2.3) Proposition . For p ≤ d < 2p the signed Young modules are exactly the set of Young modules Y^{λ} together with the nonprojective twisted Young modules Y^{μ} ⊗ sgn where μ is p-singular.

We also remark that the non-projective Young modules are actually irreducible and are in distinct blocks. Symmetric group blocks of defect one are well understood and we have the following. Let τ ⊢ d − p and let B be the block of kΣ_{d} with p-core τ. Let

λ_{1} = (τ_{1} + p, τ_{2}, …, τ_{s}) ⊵ λ_{2} ⊵ ⋯ ⊵ λ_{p} = (τ, 1^{p})

be the p partitions of d with p-core τ.

Let μ = τ′ + (p). Then μ is p-singular and an easy calculation shows Y^{μ} ⊗ sgn ≅ D^{λ_{p−1}}, i.e. this is the nonprojective twisted Young module in the block. So the signed Young modules in the block are:

Y^{λ_{1}} ≅ D^{λ_{1}}, Y^{λ_{2}} ≅ \begin{matrix}D^{λ_{1}} \\ D^{λ_{2}} \\ D^{λ_{1}} \end{matrix} , Y^{λ_{i}} ≅ \begin{matrix}D^{λ_{i−1}} \\ D^{λ_{i}} D^{λ_{i−2}} \\ D^{λ_{i−1}} \end{matrix} , Y^{λ_{p}} ≅ \begin{matrix}D^{λ_{p−1}} \\ D^{λ_{p−2}} \\ D^{λ_{p−1}} \end{matrix} , Y^{μ} ⊗ sgn ≅ D^{λ_{p−1}}, (4.2.4)

where 3 ≤ i < p. Thus the basic algebra for S(m|n, d) is a direct sum of two-sided ideals which are either semisimple (corresponding to blocks of kΣ_{d} of defect 0), or which are isomorphic to

End_{kΣ_{d}} (⨁_{i=1}^{p} Y^{λ_{i}} ⊕ (Y^{μ} ⊗ sgn)) .  (4.2.5)

The quiver and relations for the algebra in (4.2.5) is just Ã_{p} which we saw earlier is of finite type. So we can now prove the first half of Theorem 1.3.1.

(4.2.6) Theorem . Let S(m|n, d) be the Schur superalgebra where 0 ≤ d < 2p.

(a) S(m|n, d) is semisimple if and only if one of the following holds (i) char k = 0; (ii) d < p; (iii) m = n = 1 and p ∤ d.

(b) S(m|n, d) has finite representation type but is not semisimple if and only if one of the following holds

(i) p ≤ d < 2p; (ii) m = n = 1 and p | d.

Proof. (a) If d < p or char k = 0 then kΣ_{d} is semisimple, hence so is the algebra S(m|n, d) := End_{kΣ_{d}}(V^{⊗d}). Part (iii) follows by Proposition 4.1.1.

(b) The calculation above shows that S(m|n, d) when max(m, n) ≥ 2 and p ≤ d < 2p has finite representation type (and is not semisimple). The same is true for S(1|1, d) when p | d by Proposition 4.1.1. _{□}

4.3 S(2|1, d), d ≥ 2p :

In this section we begin to show that for d ≥ 2p the algebra S(2|1, d) has wild type. By Proposition 3.4.1 it will suffice to prove this for 2p ≤ d < 3p. Recall that:

S(2|1, d) ≅ End_{kΣ_{d}} ( ⨁_{(λ_{1},λ_{2}|μ_{1})∈Λ(2|1,d)} M^{(λ_{1},λ_{2}|μ_{1})}) .

In each case we will find some collection {Y_{i}} of indecomposable signed Young modules such that the algebra End_{kΣ_{d}}(⊕Y_{i}) is wild type. But this will imply S(2|1, d) is wild type by Proposition 3.2.1.

To show that the algebra End_{kΣ_{d}}(⊕Y_{i}) has wild type we will show, in each case, that it is isomorphic to an algebra kQ/I where Q is a quiver and I is an ideal contained in the ideal generated by paths of length at least two. In each case the separated quiver of Q will not be a union of Dynkin or extended Dynkin diagrams, which by Gabriel’s theorem means the algebra is of wild type. The main difficulty will be to make an appropriate choice of the {Y_{i}}.

Our proof will split into three cases; each following the same logic. First we handle the case 2p ≤ d < 3p − 2. Then we handle d = 3p − 2 and d = 3p − 1 separately. That this happens is not surprising, as the blocks of defect two are slightly different in the three cases, see for instance Table 1 in [S]. Essentially everything is understood for symmetric group blocks of defect two. The decomposition numbers are all known by [R]. The structure of the projective indecomposable modules can be determined from [S]. Finally Chuang and Tan [CT] have determined the structure of the nonprojective Young modules. Thus we just need to judiciously select signed Young modules to get a quiver of wild type. We will not give the details of determining the module structures we need, they can be found in the works cited above.

Recall that if λ is a p-regular partition of d, we write m(λ) for the Mullineux conjugate of λ, so D^{λ} ⊗ sgn ≅ D^{m(λ)} and the Young module Y^{m(λ)′} is the projective cover of D^{λ}.

4.4 S(2|1, d), 2p ≤ d < 3p − 2 : Let d = 2p + t for 0 ≤ t < p − 2. Using the Chuang-Tan notation we have the following partitions in the principal block of Σ_{d}:

λ^{(0)} = (2p + t), λ^{(1)} = (2p − 1, t + 1), λ^{(2)} = (2p − 2, t + 1, 1), λ^{(∗)} = (p + t, p).

If we let σ = (p + t, p − 1, 1) then σ is p-restricted and Y^{σ} is the projective cover of D^{λ^{(1)}}. Using i to denote D^{λ^{(i)}} we have the following Loewy structures [CT, Thm. 2.4]:

Y^{λ^{(0)}} ≅ 0, Y^{λ^{(1)}} ≅ 0 1 0 , Y^{λ^{(2)}} ≅ 1 0 2 1 , Y^{λ^{(∗)}} ≅ \begin{matrix}1 \\ 0 ∗ \\ 1 \end{matrix} ,  (4.4.1)

Y^{σ} ≅ \begin{matrix}1 \\ ∗ 0 2 \\ 1 μ 1 \\ ∗ 0 2 \\ 1 \end{matrix} .

Since all the partitions above have third part zero or one and since by Proposition 3.5.1 Y^{(a,b,1)} is a summand of M^{(a,b|1)}, we know all the modules in (4.4.1) are signed Young modules for m = 2, n = 1. Consider the following quiver L given below:

[figure: 0; 1; *; μ; 2]

Let U be the direct sum of the five modules in (4.4.1). We have:

(4.4.2) Theorem .

(a) The algebra End_{kΣ_{2p+t}}(U) is isomorphic to the algebra kL/I where I is in the ideal generated by paths of length two. In particular it is of wild representation type.

(b) The superalgebra S(2|1, d) has wild representation type for 2p ≤ d < 3p − 2.

Proof. (a) The signed Young modules making up U have the same structure as the modules E_{i} for 1 ≤ i ≤ 5 in Prop. 3.10 of [E] and the determination of the quiver proceeds in exactly the same way.

(b) This follows immediately from part (a) by calculating the separated quiver of L. The separated quiver is two copies of:

[figure]

which is not a Dynkin or Euclidean diagram. _{□}

4.5 S(2|1, 3p − 2) : Let d = 3p−2. Using the Chuang-Tan notation we have the following partitions in the principal block of Σ_{d}:

λ^{(0)} = (3p − 2), λ^{(1)} = (2p − 1, p − 1), λ^{(2)} = (2p − 2, p − 2, 1^{2}),
λ^{(3)} = (2p − 2, p − 3, 1^{3}), λ^{(∗)} = (2p − 2, p), δ = (2p − 2, p − 1, 1).

It is also an easy calculation that:

m(δ)′ = (2p − 3, p − 1, 1^{2})

so Y^{(2p−3,p−1,1^{2})} is the projective cover of D^{δ}. Let τ = (2p − 3, p − 1, 1^{2}). Using i to denote D^{λ^{(i)}} we have the following structures for Young modules [CT]:

[figure: Y^{λ^{(2)}} ≅; 1; 2; *; 0; 0; δ; 1; δ; 2; *; 0; δ; τ; 1; 2; *; 0; δ; δ; Y^{τ} ≅; Y^{δ} ≅; *; 0; 1; δ; *; 0; 1; 1]

Y^{λ^{(0)}} ≅ 0, Y^{λ^{(1)}} ≅ 0 1 0 , Y^{λ^{(3)}} ≅ \begin{matrix}2 \\ δ 3 \\ 2 \end{matrix} .  (4.5.1)

All the Young modules in (4.5.1) are either for two-part partitions or partitions of the form (a, b, 1^{c}) for c < p, so by Proposition 3.5.1 they are all signed Young modules. Now consider the following quiver which we denote by Q.

[figure: 0; 1; 2; 3; δ; τ; f; g; h; k; α_{1}; α_{2}; α_{3}; β_{1}; β_{2}; β_{3}]

Let T be the direct sum of the six signed Young modules in (4.5.1). Then we have the following result.

(4.5.2) Theorem .

(a) The algebra Λ = End_{kΣ_{3p−2}}(T) is isomorphic to the algebra kQ/I where I is the ideal generated the relations:

\begin{matrix}α_{1}α_{2} = α_{2}α_{3} = α_{2}h = β_{3}β_{2} = β_{2}β_{1} = β_{3}g = fα_{3} = α_{2}h = 0 \\ hk = gf = β_{1}α_{2} + α_{3}β_{3} \end{matrix}  (4.5.3)

(b) The superalgebra S(2|1, d) has wild representation type for d = 3p − 2.

Proof. (a) Let {e_{0}, e_{1}, e_{2}, e_{3}, e_{δ}, e_{τ} } ∈ Λ be the canonical projections onto the Young module summands of T. Choose nonzero maps α_{i} and β_{i}. (The module structures leave no choice for the kernels and images of these maps). The image and kernels of the maps f, g, h, k can also be determined by the structure of the modules given in (4.5.1). In particular we have the following Loewy structures for the kernels (from which the images are easily determined):

ker f ≅ \begin{matrix}∗ 0 \\ 1 \end{matrix} , ker k ≅ \begin{matrix}τ \\ 2 ∗ 0 \\ δ \end{matrix} , ker g ≅ \begin{matrix}2 0 \\ δ \end{matrix} , ker h ≅ {0 \atop 1}

Knowing the kernels and images of the maps is sufficient to check that the zero relations in (4.5.3) are satisfied. We need only show we can choose the maps so that hk = gf = β_{1}α_{2} + α_{3}β_{3}.

Notice that the image of β_{2}α_{2} is the simple module 1 and the image of α_{3}β_{3} is the simple module δ while both hk and gh have image equal to the socle of Y^{λ^{(2)}}. Since End(Y^{λ^{(2)}}) is three-dimensional, we can take as a basis {e_{2}, α_{3}β_{3}, β_{2}α_{2}}. Write:

hk = aβ_{2}α_{2} + bα_{3}β_{3}
gf = cβ_{2}α_{2} + cα_{3}β_{3}

where the images ensure a, b, c, d are all nonzero.

Now just do the substitutions:

β′_{2} = aβ_{2}, α′_{2} = cα_{2}, β′_{3} = bβ_{3}, α′_{3} = dα_{4}.

This will give the desired relation without effecting the earlier zero relations. The maps {α_{i}, β_{i}, h, k, f, g} generate rad Λ and are independent modulo I^{2}, so there is an epimorphism kQ/I onto Λ. Now one can simply check the dimensions agree to obtain the isomorphism.

(b) This follows immediately from (a) by calculating the separated quiver of Q and verifying that it is not a Dynkin or Euclidean diagram. _{□}

In the special case where p = 3 (i.e. S(2|1, 7)) the partition λ^{(3)} above does not exist. In this case to obtain the same quiver just replace Y^{λ^{(3)}} by the irreducible module D^{δ}, which is a signed Young module by Proposition 3.6.1. The quiver and relations are easily seen to be identical.

4.6 S(2|1, 3p − 1) :

Let d = 3p − 1. Using the Chuang-Tan notation we have the following partitions in the principal block of Σ_{d}:

λ^{(0)} = (3p − 1), λ^{(1)} = (2p − 1, p − 1, 1), λ^{(2)} = (2p − 1, p − 2, 1^{2}),
λ^{(3)} = (2p − 1, p − 3, 1^{3}), λ^{(∗)} = (2p − 1, p), ρ = (2p − 2, p, 1),

where ρ = m(λ^{(∗)})′, and hence Y^{ρ} is the projective cover of D^{λ^{(∗)}}.

Using i to denote D^{λ^{(i)}} we have the following structures for Young modules [CT]:

Y^{λ^{(0)}} ≅ 0, Y^{λ^{(∗)}} ≅ ∗, Y^{λ^{(1)}} ≅ \begin{matrix}0 ∗ \\ 1 \\ 0 ∗ \end{matrix} , Y^{λ^{(2)}} ≅ 1 0 2 1 ,  (4.6.1)

Y^{λ^{(3)}} ≅ 2 1 3 2 , Y^{ρ} ≅ \begin{matrix}∗ \\ 1 \\ ∗ 0 ρ \\ 1 \\ ∗ \end{matrix} .

Now consider the following quiver called H:

[figure: 3; 2; 1; 0; ∗; ρ]

Let W be the direct sum of the six signed Young modules in (4.6.1). Then we have the following result.

(4.6.2) Theorem .

(a) The algebra End_{kΣ_{3p−1}}(W) is isomorphic to the algebra kH/I where I is in the ideal generated by paths of length two. In particular it is of wild representation type.

(b) The superalgebra S(2|1, d) has wild representation type for d = 3p − 1.

Proof. (a) We leave the calculations here for the reader. They are similar to the proof of Theorem 4.5.2. (b) Once again the separated quiver for H is not Dynkin or Euclidean so S(2|1, 3p − 1) is of wild representation type. _{□}

Just as in the 3p − 2 case, we remark that the partition λ^{(3)} does not exist when p = 3 (i.e. S(2|1, 8)). Again, replacing Y^{λ^{(3)}} with the simple module D^{(5,2,1)} ≅ D^{λ^{(1)}} gives the same quiver. D^{(5,2,1)} is a signed Young module by Proposition 3.6.1.

4.7 We can now complete the proof of Theorem 1.3.1 by proving the following result.

(4.7.1) Theorem . The algebra S(m|n, d) has wild representation type when max(m, n) ≥ 2 and d ≥ 2p.

Proof. Since S(m|n, d) ≅ S(n|m, d) we may assume that m ≥ n. By assumption m ≥ 2 and n ≥ 1. The algebra S(2|1, d), 2p ≤ d < 3p has wild representation type from Theorems 4.4.2(b), 4.5.2(b), 4.6.2(b). Hence, by Proposition 3.4.1, S(2|1, d) has wild type for d ≥ 2p. One can now apply Corollary 3.3.1(b) to conclude that S(m|n, d) has wild representation type for d ≥ 2p. _{□}

5 Global Dimension

5.1 In [MZ, Conj. 1] it is conjectured that S(m|n, d) is quasi-hereditary whenever d is coprime to p. In this section we use our previous calculations to show this is far from true. In fact, we will prove the following theorem.

(5.1.1) Theorem . For d ≥ p ≥ 5 and m, n ≥ d the superalgebra S(m|n, d) has infinite global dimension.

This result suggests the following question.

(5.1.2) Question . Is it true that S(m|n, d) is either semisimple or has infinite global dimension? In particular, is it true that non-semisimple Schur superalgebras are never quasi-hereditary?

5.2 Before proving Theorem 5.1.1 we first need to determine which block of kΣ_{d} contains a given signed Young module. Throughout the remainder of this section we assume m, n ≥ d.

Recall the classification of irreducible S(m|n, d)-supermodules when m, n ≥ d given in [Do, 2.3 (3)] (for the statement when m, n, and d are arbitrary, see [BK, Thm. 6.5]).

(5.2.1) Theorem . Let m, n ≥ d. The irreducible supermodules of S(m|n, d) are labeled by heighest weight by the set Λ^{++}(m|n, d) = {(λ|pμ) | |λ| + p|μ| = d} where λ and μ are partitions.

Write L(λ|pμ) for the irreducible S(m|n, d)-supermodule of highest weight (λ|pμ) and let P (λ|pμ) denote the projective cover of L(λ|pμ).

Just as with the ordinary Schur algebra, one can define a Schur functor

F : S(m|n, d)-supermodules → kΣ_{d}-modules.

Like with the ordinary Schur functor, F is exact and if L is an irreducible S(m|n, d)- supermodule then F L is either zero or an irreducible kΣ_{d}-module. All irreducible kΣ_{d}- modules appear in this way. In fact, by [BK, (5.13)] we have

F L(λ|pμ) ≅ \begin{cases}D^{m(λ′)}, & if |μ| = 0 and λ′ is p-regular; \\ 0, & otherwise. \end{cases}  (5.2.2)

Furthermore, by [Do, p. 662] we have

F P (λ|pμ) ≅ Y^{(λ|pμ)}.  (5.2.3)

Thus to determine which block contains Y^{(λ|pμ)} it will suffice to obtain information about the composition factors of P (λ|pμ).

Before proceeding, we require additional notation. Let

X = {ξ = (ξ_{1}, …, ξ_{m}|ξ_{m+1}, …, ξ_{m+n}) | ξ_{1}, …, ξ_{m+n} ∈ ℤ} .

We view (λ|pμ) ∈ Λ^{++}(m|n, d) as an element of X in the natural way as (λ, 0, …, 0|pμ, 0, …, 0). For i = 1, …, m + n, let ε_{i} ∈ X be the element which has a 1 in the ith position and 0’s elsewhere. Let ϑ ∈ X be the element

ϑ = (−1, −2, …, −m|m − 1, m − 2, …, m − n + 1, m − n).

Note that our definition of ϑ differs from the one in [K, (2.8)] by a multiple of the GL(m|n) analogue of the determinant representation. This has no significant effect on our arguments.

Given i = 1, …, m + n we define r_{i} : X → ℤ by

r_{i}(ξ) = \begin{cases}ξ_{i} + ϑ_{i}, & if i = 1, …, m; \\ −(ξ_{i} + ϑ_{i}), & if i = m + 1, …, m + n. \end{cases}

For r ∈ ℤ_{p} and ξ ∈ X, define

A_{r}(ξ) = |{i = 1, …, m + n | r_{i}(ξ + ε_{i}) ≡ r (mod p)}|,
B_{r}(ξ) = |{i = 1, …, m + n | r_{i}(ξ) ≡ r (mod p)}|.

For a partition λ we let λ̃ denote its p-core. Recall that the blocks of kΣ_{d} are parameterized by the p-cores of the partitions of d [JK, Sect. 2.7]. Also recall that the category of S(m|n, d)-supermodules is equivalent to the category of polynomial representations of degree d for the supergroup GL(m|n). That is we can view the category of S(m|n, d)-supermodules as a full subcategory of the category of representations of GL(m|n) and, in turn, as a full subcategory of the category of supermodules for the superalgebra Dist(GL(m|n)) of distributions for GL(m|n). See [BK] for a full discussion of these matters.

(5.2.4) Lemma . If L(λ|pμ) and L(ν|pη) have the same central character as Dist(GL(m|n))- supermodules, then λ̃ = ν̃.

Proof. If L(λ|pμ) and L(ν|pη) have the same central character, then by [K, Lem. 3.3] we have

A_{r}(λ|pμ) − B_{r}(λ|pμ) = A_{r}(ν|pη) − B_{r}(ν|pη)  (5.2.5)

for all r ∈ ℤ_{p}. However, note that if i = m + 1, …, m + n, then

r_{i}(λ|pμ) ≡ −ϑ_{i} (mod p)
≡ r_{i}(ν|pη) (mod p),

and

r_{i}((λ|pμ) + ε_{i}) ≡ −ϑ_{i} − 1 (mod p)
≡ r_{i}((ν|pη) + ε_{i}) (mod p).

Consequently, we see that (5.2.5) holds if and only if

A′_{r}(λ|pμ) − B′_{r}(λ|pμ) = A′_{r}(ν|pη) − B′_{r}(ν|pη)  (5.2.6)

for all r ∈ ℤ_{p}, where for ξ ∈ X we define

A′_{r}(ξ) = |{i = 1, …, m | r_{i}(ξ + ε_{i}) ≡ r (mod p)}|,
B′_{r}(ξ) = |{i = 1, …, m | r_{i}(ξ) ≡ r (mod p)}|.

Given (λ|pμ) and r ∈ ℤ_{p}, for brevity let us write A′_{r} for A′_{r}(λ|pμ) and B′_{r} for B′_{r}(λ|pμ). By definition, we have A′_{r} = B′_{r−1} for all r ∈ ℤ_{p}. We then have the following equations in the B′_{r}’s:

B′_{1} + B′_{2} + ⋯ + B′_{p} = m
B′_{r−1} − B′_{r} =: b_{r} (for r = 1, …, p).

However, a straightforward check verifies that the determinant for this system of linear equations is nonzero so one can solve this system for the B′_{r}’s. That is, (5.2.6) holds if and only if

B′_{r}(λ|pμ) = B′_{r}(ν|pη)  (5.2.7)

for all r ∈ ℤ_{p}. At this point we assume without loss that |λ| = |ν| since B′_{r}(λ, 1^{p}|pμ) = B′_{r}(λ|pμ) and since the p-cores of (λ, 1^{p}) and λ coincide.

Since for i = 1, …, m we have r_{i}(λ|pμ) = λ_{i} − i, it follows that (5.2.7) holds if and only if there is an element σ ∈ Σ_{m} so that

λ_{i} − i ≡ ν_{σ(i)} − σ(i) (mod p)  (5.2.8)

for all i = 1, …, m. Finally, by the Nakayama Rule [Mar, Thm. 5.1.1] we have that (5.2.8) holds if and only if λ̃ = ν̃. This proves the desired result. _{□}

(5.2.9) Corollary . The signed Young module Y^{(λ|pμ)} is in the kΣ_{d} block with p-core λ̃.

Proof. Let L(ν|pη) be a composition factor of P (λ|pμ). Then since both L(λ|pμ) and L(ν|pη) are composition factors of an indecomposible Dist(GL(m|n))-supermodule, they must have the same central characters (c.f. [K, Sect. 2.8]). By the previous lemma, we have that λ̃ = ν̃. The result then follows by exactness of the Schur functor along with (5.2.2) and (5.2.3). _{□} 5.3 We can now prove Theorem 5.1.1. Since p ≥ 5 there must be a partition τ ⊢ d − p which is a p-core by the work of Granville and Ono [GO]. Consequently, Σ_{d} has a p-block of defect one. Let B denote this block.

(5.3.1) Lemma . There are p + 1 signed Young modules in B. They are the p distinct Young modules Y^{(λ|∅)} ≅ Y^{λ} with λ̃ = τ and one non-projective twisted Young module Y^{(τ|p)} ≅ Y^{(τ′+p)} ⊗ sgn .

Proof. By Corollary 5.2.9 the signed Young modules which occur in B are those of the form Y^{(λ|pμ)} with λ̃ = τ. That is, precisely those listed. Now, since twisted Young modules are signed Young modules, we have Y^{(τ′+p)} ⊗ sgn is a signed Young module in the block B, and it must be Y^{(τ|p)}. _{□}

Now we have the direct sum decomposition

S(m|n, d) = End_{kΣ_{d}} (V^{⊗d}) ≅ End_{kΣ_{d}} ( ⨁_{Y^{(λ|μ)}∈B} a_{(λ|μ)}Y^{(λ|μ)}) ⊕ End_{kΣ_{d}}(U)

where a_{(λ|μ)} denotes the multiplicity of Y^{(λ|μ)} in V^{⊗d} and U has no summands in the block B. That is, we have direct sum decomposition of S(m|n, d) into graded two-sided ideals, one of which is Morita equivalent to

End_{Σ_{d}} ( ⨁_{Y^{(λ|μ)}∈B} Y^{(λ|μ)}) .

This is precisely the situation of (4.2.4) and the quiver is just Ã_{p}. It is easy to see this (super)algebra has infinite global dimension. The projective resolutions of the simple modules are all periodic and do not terminate. Since a two-sided graded ideal has infinite global dimension, so does S(m|n, d), proving Theorem 5.1.1.

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Department of Mathematics, University of Toledo, 2801 W. Bancroft, Toledo, OH 43606, USA

   E-mail address: david.hemmer@utoledo.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

   E-mail address: kujawa@math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

   E-mail address: nakano@math.uga.edu