RANK VARIETIES AND PROJECTIVITY FOR A CLASS OF LOCAL ALGEBRAS

KARIN ERDMANN AND MILES HOLLOWAY

Abstract. We consider algebras K[X_{1}, …, X_{m}]/(X_{i}^{2}), where K is algebraically closed. To any module for this algebra we associate a rank variety. When char(K) = 2 we recover Carlson’s rank variety. The main result statess that the module is projective if and only if the variety vanishes. This has applications to other algebras, including tensor products of certain Brauer tree algebras and certain parabolic Hecke algebras. In addition, the result has implications for the graph structure of the stable Auslander-Reiten quiver.

1. Introduction

Given an algebraically closed field K, a major interest in the representation theory of finite dimensional K-algebras is in describing projective modules and even in detecting whether a given module is projective. One Approach in the context of finite dimensional group algebras kG (where char(K) = p and p||G|) is the construction of the cohomological variety (sometimes called support variety) for a given module (see [5]). These constructions can be done at the level of subalgebras of the form K[X_{1}, …, X_{m}]/(X_{i}^{p}). In [8] Carlson developed a rank variety that can be constructed for a module for such a subalgebra which has the virtue of being computationally explicit. It’s an important theorem of Avrunin and Scott that these two varieties are equivalent.

In attempting to generalize Carlson’s rank variety construction there are two possibilities. Firstly one might try to relax the commutativity of the X_{i} and deform the algebra with some quantum parameters. This has been taken up in [6], where it’s shown that through a Morita equivalence one can still use Carlson’s rank variety. Secondly one might relax the condition char(K) = p and consider arbitrary characteristic. It is this generalization that we address in this paper. Specifically we study algebras Λ_{m} := K[X_{1}, …, X_{m}]/(X_{i}^{2}) with char(K) arbitrary. For these algebras we are able to define a rank variety and moreover when we specialize to char(K) = 2 we recover Carlson’s original definition. Our main result, Theorem 4.3, is that the vanishing of this variety characterizes projectivity. In the case of group algebras and Carlson’s rank variety this has been known for a long time and can be traced to an argument due to Dade (§11 in [10]).

As well as calculating our rank variety for some examples we show that our projectivity characterization has implications for Hecke algebras associated to Coxeter groups, in particular the symmetric group. Here ℓ-parabolic subalgebras, HH_{q}(ℓ)^{⊗m}, are important and it is conjectured that, for type A, these may determine the representation theory of the Hecke algebra [ed. Is it possible to have a reference for this conjecture?]. In [12] connections are determined between HH_{q}(ℓ)^{⊗m} and Λ_{m} that enable us to exploit the results in this paper.

1991 Mathematics Subject Classification. 20C20.

Suitably reinterpreted our main result says that \underline{mod}- Λ_{m} has enough periodic modules. This has consequences for the tree strucutre of the underlying graph of the stable Auslander-Reiten quiver which we explore.

In considering M ∈ mod- Λ_{m} the following situation occurs. The X_{i} (i < m) act on M freely while X_{m} acts as a_{1}X_{1} + ⋯ + a_{m−1}X_{m−1}, where a_{i} are linear maps. In order for X_{m}^{2} = 0, a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j. It is this that explains the appearance of Clifford modules in our construction of a rank variety.

The paper is organized as follows. In §2 we recall the necessary background material on Clifford algebras and modules. In §3 we examine properties of the algebra Λ_{m} and develop the necessary constructions prior to our definition of the rank variety in §4. In §5 we calculate some examples while in §6 we develop some of the consequences for Hecke algebras and stable Auslander-Reiten quivers referred to above.

1.1. Notation. Throughout K will denote an algebraically closed field. If necessary we will indicate when restrictions are to be placed on its characteristic. All K-algebras considered will be finite-dimensional K-algebras. Given a K-algebra Λ, mod- Λ denotes the category of finite-dimensional Λ-modules. We denote by proj- Λ the full subcategory of projective Λ-modules and \underline{mod}- Λ denotes the stable module category.

We write ∑_{r} for the symmetric group on r letters and ⊗ for ⊗_{K}. We will use the ˆ notation to indicate omission in a list. That is {a_{1}, …, â_{t}, …, a_{m}} is understood to mean {a_{1}, …, a_{t−1}, a_{t+1}, … a_{m}} for 1 ≤ t ≤ m. Given a vector space W over K we write 1_{W} for the identity map in End_{K}(W).

1.2. Acknowledgement.

2. Clifford Algebras and Modules

In this section we recall various facts that we need for our constructions in §3 and §4. References for these facts can be found in [1] and [3].

2.1. Clifford Algebras. If X denotes an m-dimensional vector space over K, and Q a quadratic form on X, then the Clifford algebra, Cl(Q), over X with respect to Q is given by

T (X)/I(Q)

where T (X) is the tensor algebra over X and I(Q) is the two sided ideal generated by v ⊗ v − Q(v)1 for all v ∈ X. If {e_{i}} (1 ≤ i ≤ m) is a basis for X and we take Q given by

Q(∑ x_{i}e_{i}) = − ∑ x_{i}^{2}

then we will write C_{m} = Cl(Q) and simply call this the m generated Clifford algebra. It is generated by the {e_{i}} subject to the relations

e_{i}^{2} = −1, e_{i}e_{j} + e_{j}e_{i} = 0, for all i, j and i ≠ j.

If char(K) = 2 then C_{m} is isomorphic to the group algebra of an elementary abelian 2-group.

If char(K) ≠ 2 then C_{m} is a semisimple K-algebra and moreover

(2.1)  C_{2n} ≅ M_{2^{n}}(K), C_{2n+1} ≅ M_{2^{n}}(K) ⊕ M_{2^{n}}(K)

The centre of the algebra is

Z(C_{m}) = \begin{cases}\{±1\} & if m is even, \\ \{±1, ± ∏ e_{j}\} & if m is odd. \end{cases}

There is an automorphism α ∈ Aut(C_{m}) that is induced by −1 : X → X (minus the identity). So α(e_{j}) = −e_{j} and α(∏ e_{j}) = − ∏ e_{j} if m is odd.

The following is useful in constructing representations.

Lemma 2.1. The map

ψ : C_{m+2} → C_{m} ⊗ C_{2}

which is induced by

ψ : e_{j} → \begin{cases}e_{j−2} ⊗ ie_{1}e_{2} & for 2 < j ≤ m + 2, \\ 1 ⊗ e_{j} & for j = 1, 2. \end{cases}

(where i = \sqrt{−1} ∈ K) is an isomorphism.

2.2. Clifford modules. Here we describe irreducible modules for the Clifford algebras C_{m}. If char(K) = 2 then C_{m} is isomorphic to the group algebra of an elementary abelian 2-group and it is well known that, up to isomorphism, the only irreducible module is K, the trivial module. If char(K) ≠ 2 then by (2.1) we can expect, up to isomorphism, one irreducible module for m even and two for m odd. In either case the dimension of the irreducible module will be 2^{n} where m = 2n or m = 2n + 1.

We can give an explicit description of an irreducible C_{m}-module by choosing a suitable basis for a C_{2} -module and repeatedly using Lemma 2.1.

Definition 2.2. Define the following elements in M_{2}(K)

1 = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , E_{2} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , E_{1} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , j = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

For n a positive integer we define the following elements in M_{2^{n}}(K). (The notation in brackets emphasizes the dependence on n).

a_{2r−1} ( = a_{2r−1}^{n}) = \overbrace{1 ⊗ ⋯ ⊗ 1}^{n−r} ⊗E_{1} ⊗ \overbrace{j ⊗ ⋯ ⊗ j}^{r−1} for 1 ≤ r ≤ n,

a_{2r} ( = a_{2r}^{n}) = \overbrace{1 ⊗ ⋯ ⊗ 1}^{n−r} ⊗E_{2} ⊗ \overbrace{j ⊗ ⋯ ⊗ j}^{r−1} for 1 ≤ r ≤ n,

a_{0}(= a_{0}^{n}) = I_{2^{n}}, a_{2n+1}(= a_{2n+1}^{n}) = ⨂^{n} j.

It’s easy to see that

a_{2r−1}^{2} = a_{2r}^{2} = (−1)^{r−1}I_{2^{n}}, for 1 ≤ r ≤ n,
a_{2n+1}^{2} = (−1)^{n}I_{2^{n}}, a_{r}a_{s} + a_{s}a_{r} = 0, for r ≠ s.

We now suppose m = 2n or m = 2n + 1 and determine an explicit representation ρ : C_{m} → GL(K, 2^{n}), b_{r} := ρ(e_{r}). Setting b_{r} = λ_{r}a_{r}^{n}, λ_{r} ∈ K, it’s clear from the above that we only require b_{r}^{2} = −I_{2^{n}}. That is we need λ_{2r} = ±i^{r} (1 ≤ r ≤ n) and λ_{2r+1} = ±i^{r+1} (0 ≤ r ≤ n), where i = \sqrt{−1} ∈ K. Conjugating b_{r} by a_{r}a_{2n+1} when 1 ≤ r ≤ 2n, we can change the sign of λ_{r} without affecting the choice of sign for λ_{s} (s ≠ r, 1 ≤ s ≤ 2n). Hence for m = 2n we have an explicit description of the unique, up to isomorphism, irreducible C_{m}-module. For m = 2n + 1, we have a choice of sign for λ_{2n+1} once the others have been fixed and the two choices yield the two non-isomorphic irreducible C_{m}-modules. It’s worth noting, for m = 2n + 1, that if one of the irreducible modules is denoted by W, the other can be obtained by twisting by the automorphism α, written ^{α}W.

2.3. Restriction and Induction. If we denote by C_{m}^{t} the subalgebra of C_{m} generated by {e_{r}|r ≠ t} then clearly C_{m}^{t} ≅ C_{m−1}.

The following Lemma describes what happens to irreducible modules under restriction and induction from such subalgebras.

Lemma 2.3. If W is an irreducible C_{m}-module and V an irreducible C_{m}^{t}-module then

(1) If m is odd,

W ↓_{C_{m}^{t}}^{C_{m}} ≅ V
W ≅ V ↑_{C_{m}^{t}}^{C_{m}}

(2) If m is even,

W ↓_{C_{m}^{t}}^{C_{m}} ≅ V ⊕^{α}V
W ≅ V ↑_{C_{m}^{t}}^{C_{m}} ≅ ^{α}V ↑_{C_{m}^{t}}^{C_{m}}

Proof. Most of the statements follow immediately from the semisimplicity of C_{m} and the dimension of irreducible modules. We only have to rule out, for m even, the possibility W ↓_{C_{m}^{t}} ≅ 2V or 2^{α}V. But this follows from ^{α}W ≅ W and (^{α}W) ↓≅ ^{α}(W ↓), which show that W ↓_{C_{m}^{t}} must be invariant under twisting by α. □

3. The Algebra Λ_{m}

We now introduce the class of algebras studied in this paper. For m a positive integer we define

Λ_{m} := K[X_{1}, … X_{m}]/(X_{i}^{2}).

For char(K) = 2, Λ_{m} is isomorphic to the group algebra of an elementary abelian 2-group of rank m.

First let us record some facts about Λ_{m}. It is a symmetric, local, commutative finite dimensional K-algebra, with maximal ideal, also equal to the Jacobson radical, J = (X_{1}, …, X_{m}). It has infinite global dimension and has representation type that is finite if m = 1, tame if m = 2 and wild if m > 2.

Note that Λ_{m} has m + 1 subalgebras each isomorphic to Λ_{m−1} and generated by {X_{j}}, j ≠ t, for each 1 ≤ t ≤ m. More precisely let

Λ_{m}^{t} = K[X_{1}, …, X̂_{t}, …, X_{m}]/(X_{j}^{2}).

Clearly Λ_{m−1} ≅ Λ_{m}^{t}, for 1 ≤ t ≤ m and we’ll identify each Λ_{m}^{t} with its inclusion in Λ_{m}, so that

Λ_{m} = Λ_{m}^{t}[X_{t}]/(X_{t}^{2}).

3.1. Periodic modules. Λ_{m} is of course a Λ_{m}-module and, as Λ_{m} is commutative, X_{i} can act via multiplication as a Λ_{m}-module homomorphism. To keep this action distinct from X_{i} as an element in Λ_{m} we make the following definition.

Definition 3.1. For Λ_{m} = K[X_{1}, …, X_{m}]/(X_{i}^{2}), let

x_{i}(= x_{i}^{m}) ∈ Hom_{mod- Λ_{m}}(Λ_{m}, Λ_{m})

be the map induced by multiplication by X_{i}.

Our class of periodic modules will be parametrized by points in ℙ^{m}(K). In order to define these modules we will need to fix an irreducible representation of the m-generated Clifford algebra C_{m}.

Definition 3.2. Let ρ : C_{m} → GL(K, W) be an irreducible representation, and define b_{j} := ρ(e_{j}).

Viewing W ⊗ Λ_{m} as a Λ_{m}-module with X_{i} acting as 1_{W} ⊗ x_{i}^{m}, we can define the following Λ_{m}-module homomorphism. Let 0 ≠ \underline{λ} = (λ_{1}, …, λ_{m}) ∈ K^{m}, and define η = η(\underline{λ}) ∈ End_{Λ_{m}}(W ⊗ Λ_{m}) by

η = ∑_{j=1}^{j=m} λ_{j}b_{j} ⊗ x_{j}^{m}

We can now define our class of periodic modules.

Definition 3.3.

V (\underline{λ}) := im(η(\underline{λ}))

Remarks.

(i) It is clear that V (\underline{λ}) is independent, up to isomorphism, of the choice of representative from the isomorphism class of W.

(ii) V (\underline{λ}) only depends upon \underline{λ} ∈ ℙ^{m}(K), in fact later (see Lemma 3.6) we will see that

it only depends upon the orbit that \underline{λ} lies in under the action of a certain group. (iii) If m is odd, we have two non-isomorphic choices for W. Having chosen one irre-

ducible representation in Definition 3.2, the other can be obtained by twisting by the automorphism α. In terms of the map η, changing the choice of irreducible C_{m}- module is equivalent to replacing η(\underline{λ}) by η(−\underline{λ}). In any event, our V (\underline{λ}) remains unchanged.

As these class of modules are crucial for our construction of a rank variety, we will need to know more about them. The following proposition contains most of what we need.

Proposition 3.4. Let \underline{λ} ∈ ℙ^{m}(K), then (1) dim(V (\underline{λ})) = (1/2) dim W dim Λ_{m}.

(2) If λ_{t} ≠ 0 then V (\underline{λ}), restricted to Λ_{m}^{t}(≅ Λ_{m−1}), is free of rank dim(W) as a Λ_{m}^{t}-module.

(3) A minimal projective resolution for V (\underline{λ}) is

⋯ → W ⊗ Λ_{m} ⟶^{η(\underline{λ})} W ⊗ Λ_{m} ⟶^{η(\underline{λ})} W ⊗ Λ_{m}

(4) Ω(V (\underline{λ})) ≅ V (\underline{λ})

Proof. Clearly η^{2} = 0 and therefore im(η) ⊆ ker(η), and hence

dim(V (\underline{λ})) ≤ (1/2) dim(W) dim(Λ_{m}).

If λ_{t} ≠ 0 then to verify (2) it suffices to check that dim(soc(Λ_{m}^{t})V (\underline{λ})) = dim(W). To check this note that soc(Λ_{m}^{t}) = ∏_{j≠t} X_{j}, and for w ∈ W and f ∈ Λ_{m} we have

(3.1)  ∏_{j≠t} X_{j}(η(w ⊗ f)) = λ_{t}b_{t}(w) ⊗ (∏_{j=1}^{m} X_{j})f.

If f ∈ rad(Λ_{m}) this is zero. Since λ_{t}b_{t} is invertible, it follows that the space spanned by elements in (3.1) has dimension dim W, as required, and hence (2) follows. (1) now follows as we have seen that V (λ) as a Λ_{m−1}^{t}-module has a free direct summand of dimension dim(W) dim(Λ_{m}^{t}) = (1/2) dim(W) dim(Λ_{m}).

By dimension, it follows that im(η) = ker(η) and (3) and (4) now follow. □

Given \underline{λ} ∈ ℙ^{m}(K) we will write \underline{λ}^{t} for (λ_{1}, …, λ̂_{t}, …, λ_{m}) ∈ ℙ^{m−1}(K). With this notation V (\underline{λ}^{t}) has the obvious interpretation as a Λ_{m}^{t}-module.

Lemma 3.5. Let \underline{λ} ∈ ℙ^{m}(K) with λ_{t} = 0, then

(1) If m is odd,

V (\underline{λ}) ↓_{Λ_{m}^{t}}^{Λ_{m}} ≅ 2V (\underline{λ}^{t})
V (\underline{λ}) ≅ V (\underline{λ}^{t}) ↑_{Λ_{m}^{t}}^{Λ_{m}}

(2) If m is even,

V (\underline{λ}) ↓_{Λ_{m}^{t}}^{Λ_{m}} ≅ 4V (\underline{λ}^{t})
V (\underline{λ}) ≅ 2V (\underline{λ}^{t}) ↑_{Λ_{m}^{t}}^{Λ_{m}}

Proof. We will prove (1) and assume m is odd. We have the following exact sequence defining V (\underline{λ})

(3.2)  W ⊗ Λ_{m} ⟶^{η(\underline{λ})} W ⊗ Λ_{m} → V (\underline{λ}) → 0

Since λ_{t} = 0, by applying the map η(\underline{λ}) we can already view W as restricted to C_{m}^{t} and so by Lemma 2.3 isomorphic to V an irreducible C_{m}^{t}-module. By considering the map

Λ_{m} ⟶^{x_{r}^{m}} Λ_{m}

for r ≠ t, it’s clear that under restriction to Λ_{m}^{t}, this becomes

2Λ_{m}^{t} ⟶^{x_{r}^{m−1}⊕x_{r}^{m−1}} 2Λ_{m}^{t}

It now follows that restricting (3.2) to Λ_{m}^{t} we get

2V ⊗ Λ_{m}^{t} ⟶^{η(\underline{λ}^{t})⊕η(\underline{λ}^{t})} 2V ⊗ Λ_{m}^{t} → 2V (\underline{λ^{t}}) → 0

and hence V (\underline{λ}) ↓_{Λ_{m}^{t}} ≅ 2V (\underline{λ}^{t}).

Going in the other direction we have the following exact sequence defining V (\underline{λ}^{t}).

(3.3)  V ⊗ Λ_{m}^{t} ⟶^{η(\underline{λ}^{t})} V ⊗ Λ_{m}^{t} → V (\underline{λ}^{t}) → 0

Because λ_{t} = 0, as far as η(\underline{λ}) is concerned, we may already view, by Lemma 2.3, V as W ↓_{C_{m}^{t}} where W is an irreducible C_{m}-module. Considering the map

Λ_{m}^{t} ⟶^{x_{r}^{m−1}} Λ_{m}^{t}

it’s clear that under − ⊗_{Λ_{m}^{t}} Λ_{m} this becomes x_{r}^{m}. Hence applying − ⊗_{Λ_{m}^{t}} Λ_{m} to (3.3) we obtain (3.2) and so V (\underline{λ}^{t}) ⊗_{Λ_{m}^{t}} Λ_{m} ≅ V (\underline{λ}).

For m even, we can use exactly the same arguments together with (2) of Lemma 2.3. □

We finish the list of properties of the V (\underline{λ}) modules with one final observation. The group G := ∏^{m} _{∑2} acts on ℙ^{m}(K) as follows. If \underline{λ} ∈ ℙ^{m}(K) and π = (π_{1}, …, π_{m}) ∈ G then the action is given by

π.\underline{λ} = (sgn(π_{1})λ_{1}, …, sgn(π_{m})λ_{m}).

Lemma 3.6. Given \underline{λ} ∈ ℙ^{m}(K) and π ∈ G then

V (\underline{λ}) ≅ V (π.\underline{λ})

Proof. Clearly it suffices to show that if \underline{λ}′ differs from \underline{λ} in the sign of λ_{t} say (for some 1 ≤ t ≤ m) then V (\underline{λ}′) ≅ V (\underline{λ}). But we have seen that by conjugating by a suitable element in GL(K, W) we can change the sign of b_{t} only. Conjugating by such an element is giving a new representation ρ′ : C_{m} → GL(K, W). Constructing ‘V (\underline{λ})’ with this new representation is clearly V (\underline{λ}′) with the original. But by remark (i) following Definition 3.3 V (\underline{λ}) is independent, up to isomorphism, of the choice of representation of the irreducible C_{m}-module and hence V (\underline{λ}′) ≅ V (\underline{λ}). □

3.2. Arbitrary modules. In §§3.1 the definition of η(\underline{λ}) was that of a certain map

W ⊗ Λ_{m} → W ⊗ Λ_{m}.

As we have already noted, Λ_{m} is a local K-algebra with Λ_{m} being the only, up to isomorphism, indecomposable projective Λ_{m}-module. Through additivity we could construct an appropriate map

W ⊗ P → W ⊗ P

where P ∈ proj- Λ_{m} is arbitrary. Using Proposition 3.4 we would be able to describe various properties of this map for example its rank. We can now ask the question, to what extent do the properties of this map characterise P as a projective Λ_{m}-module?

Our definition of rank variety in §4 will answer this question, but in order to describe the variety we will need to examine the above map when P is replaced by some arbitrary Λ_{m}-module.

Let M ∈ mod- Λ_{m} and denote by y_{i} the Λ_{m}-module homomorphism that is induced by multiplication by X_{i} (for 1 ≤ i ≤ m). Given \underline{λ} ∈ ℙ^{m}(K) define σ(= σ_{M}(\underline{λ})) ∈ End_{Λ_{m}}(W ⊗ M) by

σ = ∑_{j=1}^{m} λ_{j}b_{j} ⊗ y_{j}.

Remarks.

(i) η(\underline{λ}) = σ_{Λ_{m}}(\underline{λ}).

(ii) It is clear by construction that σ^{2} = 0 so that rank(σ) ≤ (1/2) dim(W) dim(M). In the light of remark (ii) the following Lemma will be useful.

Lemma 3.7. For \underline{λ} ∈ ℙ^{m}(K), M ∈ mod- Λ_{m} and σ = σ_{M}(\underline{λ}), the following are equivalent. (1) rank(σ) = (1/2) dim(W) dim(M).

(2) \underline{Hom}_{Λ_{m}}(V (\underline{λ}), M) = 0.

(3) The sequence

W ⊗ M ⟶^{σ} W ⊗ M ⟶^{σ} W ⊗ M

is exact.

Proof. We have already noted that σ^{2} = 0 so that im(σ) ⊆ ker(σ) and hence (1) is equivalent to nullity(σ) = rank(σ). It is also clear that (1) is equivalent to (3).

Applying Hom_{Λ_{m}}(−, M) to the exact sequence

W ⊗ Λ_{m} ⟶^{η(\underline{λ})} W ⊗ Λ_{m} → V (\underline{λ}) → 0

and using the commutative square

Hom_{Λ_{m}}(W ⊗ Λ_{m}, M) ⟶^{Hom(η(\underline{λ}),M)} Hom_{Λ_{m}}(W ⊗ Λ_{m}, M)
∥  ∥
W ⊗ M  ⟶^{σ}  W ⊗ M

we obtain

(3.4)  Hom_{Λ_{m}}(V (\underline{λ}), M) ≅ ker(σ)

If we now apply Hom_{Λ_{m}}(−, M) to the short exact sequence

0 → V (\underline{λ}) ⟶^{j} W ⊗ Λ_{m} ⟶^{p} V (\underline{λ}) → 0

where j is the inclusion map and p the natural map induced by η(\underline{λ}) and use (3.4) we obtain the following exact sequence

0 → ker(σ) → W ⊗ M → ker(σ) → \underline{Hom}_{Λ_{m}}(V (\underline{λ}), M) → 0

It follows by considering dimensions together with the rank-nullity theorem that

nullity(σ) − rank(σ) = dim \underline{Hom}_{Λ_{m}}(V (\underline{λ}), M)

Hence (1) and (2) are equivalent. □

Corollary 3.8. Let \underline{λ} ∈ ℙ^{m}(K), M ∈ mod- Λ_{m} and σ = σ_{M}(\underline{λ}). If

rank(σ) = (1/2) dim(W) dim(M)

then the induced map

\overline{σ} : W ⊗ (M/JM) → W ⊗ (M/JM)

is injective.

Proof. The condition on the rank implies, by Lemma 3.7, the exactness of

W ⊗ M ⟶^{σ} W ⊗ M ⟶^{σ} W ⊗ M

Also by construction of σ we have ker(σ) = im(σ) ⊆ W ⊗ JM. It follows that \overline{σ} must be injective. □

4. Rank Variety

Let us recall that W is some fixed irreducible representation as in Definition 3.2 and so for m = 2n or m = 2n + 1, dim(W) = 2^{n}. Also we noted in subsection 3.1 that the group G(= ∏^{m} _{∑2}) acts on ℙ^{m}(K). Denote the orbit space by ℙ^{m}(K)/G and write [\underline{μ}] for the orbit containing \underline{μ} ∈ ℙ^{m}(K). With this in mind we make the following definition of a rank variety associated to a Λ_{m}-module.

Definition 4.1. For M ∈ mod- Λ_{m}, let

V^{r}(M) = \{[\underline{μ}] ∈ ℙ^{m}(K)/G | rank(σ_{M}(\underline{μ})) < (1/2) dim(W) dim(M)\}

Remarks.

(i) By Lemma 3.7 rank(σ_{M}(\underline{μ})) < (1/2) dim(W) dim(M) is equivalent to \underline{Hom}(V (\underline{μ}), M) ≠ 0. If we have [\underline{μ}′] = [\underline{μ}], so that \underline{μ}′ = π.\underline{μ} for some π ∈ G, then by Lemma 3.6 V (\underline{μ}′) ≅ V (\underline{μ}) and so \underline{Hom}(V (\underline{μ}′), M) ≠ 0. Hence the definition is well defined.

(ii) If char(K) = 2, then G acts trivially on ℙ^{m}(K) and W = K. Hence σ(\underline{μ}) = ∑_{j=1}^{m} μ_{j}y_{j}. Asking for rank(σ) < (1/2) dim(M) is equivalent to requiring M ↓_{<1+v>} not to be a free module, where v = ∑_{j=1}^{m} μ_{j}X_{j} and < 1 + v > is a shifted cyclic 2-group. Hence we recover Carlson’s original definition (see [8]) of rank variety in the case of an elementary abelian 2-group.

Proposition 4.2. V^{r}(M) is a projective variety.

Proof. By choosing a basis for M, the matrix of σ will have entries that are homogeneous in the μ_{i} of degree 1. Asking that the rank of this matrix be less than (1/2) dim(W) dim(M) is equivalent to asking that all (1/2) dim(W) dim(M) × (1/2) dim(W) dim(M) minor determinants of this matrix are zero. These are a collection of homogeneous polynomials in the μ_{i}. □

We have the following elementary properties.

(A) V^{r}(M ⊕ N) = V^{r}(M) ∪ V^{r}(N) for M, N Λ_{m}-modules (B) V^{r}(Ω^{n}M) = V^{r}(M) for any n and M a Λ_{m}-module.

(C) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of Λ_{m}-modules then

V^{r}(M_{i}) ⊆ V^{r}(M_{j}) ∪ V^{r}(M_{s}) for any {i, j, s} = {1, 2, 3}. We can now answer the question posed at the beginning of subsection 3.2.

Theorem 4.3. For M a Λ_{m}-module, we have

V^{r}(M) = ∅ ⟺ M is a projective Λ_{m}-module

Proof. If M is a projective (actually free) Λ_{m}-module then \underline{Hom}_{Λ_{m}}(−, M) is the zero functor. It follows from Lemma 3.7 that V^{r}(M) = ∅.

Suppose then that M ∈ mod- Λ_{m}, is such that V^{r}(M) = ∅. We’ll use induction on m to show that this implies M is a projective Λ_{m}-module. The case m = 1 is simple. An indecomposable Λ_{1}-module is either K or Λ_{1}. As dim(W) = 1, σ = λy_{1} (λ ∈ K) and

rank(σ) = \begin{cases}0 & if M = K, \\ 1 = (1/2) dim(M) & if M = Λ_{1}. \end{cases}

It’s now immediate that V^{r}(M) = ∅ implies M a projective Λ_{1}-module.

We now assume the result for Λ_{t}, 1 ≤ t ≤ m − 1. Without loss of generality we may assume M is indecomposable and non-projective. We’ll assume M ≠ 0 and aim for a contradiction. We claim M ↓_{Λ_{m}^{t}} is projective as a Λ_{m}^{t}-module for all 1 ≤ t ≤ m. By the Eckmann-Shapiro Lemma we have, for \underline{λ} ∈ K^{m} arbitrary subject to λ_{t} = 0,

\underline{Hom}_{Λ_{m}^{t}}(V (\underline{λ}^{t}), M ↓_{Λ_{m}^{t}}) ≅ \underline{Hom}_{Λ_{m}}(V (\underline{λ}^{t}) ↑^{Λ_{m}}, M).

By Lemma 3.5 V (\underline{λ}^{t}) ↑^{Λ_{m}} is either V (\underline{λ}) or a summand of it. By assumption V^{r}(M) = ∅ and so by Lemma 3.7 \underline{Hom}_{Λ_{m}}(V (\underline{λ}), M) = 0. Hence \underline{Hom}_{Λ_{m}^{t}}(V (\underline{λ}^{t}), M ↓_{Λ_{m}^{t}}) = 0 for arbitrary \underline{λ}^{t} ∈ K^{m−1}. Again by Lemma3.5 this implies V^{r}(M ↓_{Λ_{m}^{t}}) = ∅. By the inductive hypothesis we then have that M ↓_{Λ_{m}^{t}} is a projective Λ_{m}^{t}-module as claimed.

Now choose \underline{λ} arbitrary subject to λ_{m} = 1. It will be convenient to compose σ with the invertible map −b_{m} ⊗ 1_{M} : W ⊗ M → W ⊗ M. This will not change any arguments and so we’ll call this new map σ from now on. It is

σ = 1_{W} ⊗ y_{m} − ∑_{j=1}^{m−1} λ_{j}a_{j} ⊗ y_{j}

where a_{j} := b_{m}b_{j} for 1 ≤ j ≤ m − 1. Note that a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j. We can now use the fact that M is projective on restriction to subalgebras to provide a suitable basis with which to describe the matrices representing the maps y_{j}. We will identify Λ_{m}^{m} with Λ_{m−1}. As M ↓_{Λ_{m−1}} is projective, say free of rank r, with respect to a suitable basis (we identify M ↓_{Λ_{m−1}} with K^{r} ⊗ Λ_{m−1}) we have

y_{j} = I_{r} ⊗ x_{j}^{m−1} for 1 ≤ j ≤ m − 1

y_{m} = a ⊗ 1_{Λ_{m−1}} + ∑_{i=1}^{m−1} e_{i} ⊗ x_{i}^{m−1} + z

where z ∈ M_{r}(J^{2}Λ_{m−1}), I_{r} is the identity r × r-matrix and e_{i} and a are r × r-matrices.

By assumption M is assumed not to be projective as a Λ_{m}-module, so that soc(Λ_{m}) = KX_{1} … X_{m} annihilates M. That is 0 = y_{m}y_{m−1} … y_{1} = a ⊗ x_{1} … x_{m−1}. But x_{1} … x_{m−1} acts on Λ_{m−1} as soc(Λ_{m−1}) and is clearly non-zero and so in fact a = 0. We know that M ↓_{Λ_{m}^{t}} is free of rank r as a Λ_{m}^{t}-module for 1 ≤ t ≤ m, and have already exploited this fact when t = m. For the other cases we can say that the action of soc(Λ_{m}^{t}) on M must have rank r. That is y_{m} … y_{t+1}y_{t−1} … y_{1} must have rank r. From this it follows that e_{t} should be non-singular, for 1 ≤ t ≤ m − 1. In order for y_{m} to be an action of X_{m}, we must have y_{m}^{2} = 0 and it follows that e_{t}e_{s} + e_{s}e_{t} = 0 for t ≠ s.

Let \overline{x}_{i} denote the map

Λ_{m−1}/J → J/J^{2}

that is induced from x_{i}^{m−1}. So \overline{x}_{i} is the i^{th} unit column vector in K^{m−1}. Similarly let \overline{y}_{i} denote the map

M/JM → JM/J^{2}M

that is induced from y_{i}.

With respect to our chosen basis the matrix for \overline{σ} is given by

I_{k} ⊗ \overline{y}_{m} − ∑_{j=1}^{m−1} λ_{j}b_{m}b_{j} ⊗ \overline{y}_{j} = ∑_{j=1}^{m−1}(I_{k} ⊗ e_{j} − λ_{j}a_{i} ⊗ I_{r}) ⊗ \overline{x}_{j}

where k = dim(W), and is an rk(m − 1) × rk-matrix. By assumption V^{r}(M) = ∅ and so by Corollary 3.8 this matrix should be injective and hence have rank rk. We’ll reach a contradiction by finding suitable λ_{t} ∈ K so that a non-zero vector in K^{rk} can be found that lies in the kernel of \overline{σ}.

By permuting the rows of \overline{σ}, we can replace the above matrix by the following matrix, without changing the rank.

(\begin{matrix}α_{1} \\ α_{2} \\ · \\ · \\ · \\ α_{m} \end{matrix})

where

α_{t} = (I_{k} ⊗ e_{t} − λ_{t}a_{t} ⊗ I_{r}), for 1 ≤ t ≤ m,

are rk × rk matrices. We can compose each matrix α_{t} by the invertible matrix (a_{t} ⊗ I_{r}) without affecting our search for a non-zero vector that is in the kernel of each α_{t}. But

(a_{t} ⊗ I_{r})α_{t} = (a_{t} ⊗ e_{t} − λ_{t}I_{rk})

and {a_{t} ⊗ e_{t}} are a commuting set of matrices. As K is algebraically closed we can choose λ_{t} so as to find a non-zero vector ν such that α_{t}^{n}ν = 0 for 1 ≤ t ≤ m. Hence we have our contradiction and so in fact M = 0 as required. □

5. Examples

5.1. Carlson’s favourite example. Here we borrow an example used to illustrate the rank variety in the case of an elementary abelian group of order 8 (see [5]), and see what variety we get when char(K) ≠ 2.

We assume char(K) ≠ 2 and consider M ∈ mod- Λ_{3}, a 4-dimensional space, where the action of X_{i}, denoted by y_{i}, is given by

y_{1} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) , y_{2} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \end{matrix}) ,

y_{3} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & c & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix})

where a, b, c ∈ K. Here m = 3 = 2.1+1 and so, since char(K) ≠ 2, W = K^{2} as a K-space. By absorbing the scalars within μ_{i} we may take b_{j} = a_{j}^{1} as in Definition 2.2. We have, for \underline{μ} ∈ K^{3},

σ(\underline{μ}) = μ_{1}b_{1} ⊗ y_{1} + μ_{2}b_{2} ⊗ y_{2} + μ_{3}b_{3} ⊗ y_{3}.

By definition [\underline{μ}] ∈ V^{r}(M) if and only if rank(σ(\underline{μ})) < (1/2) dim(W) dim(M) = 4. The matrix representing σ(\underline{μ}) is given by

(\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ aμ_{2} & 0 & 0 & 0 & μ_{1} & −cμ_{3} & 0 & 0 \\ 0 & bμ_{2} & 0 & 0 & −μ_{3} & μ1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ μ_{1} & cμ_{3} & 0 & 0 & −aμ_{2} & 0 & 0 & 0 \\ μ_{3} & μ_{1} & 0 & 0 & 0 & −bμ_{2} & 0 & 0 \end{matrix})

The only non-trivial 4 × 4 minor determinant is the one formed by rows 3,4,7 and 8 and columns 1,2,5 and 6. Requiring that this determinant is zero works out to be (μ_{1}, μ_{2}, μ_{3}) satisfying the following homogeneous equation of degree 4.

(a^{2} + b^{2})X_{1}^{2}X_{2}^{2} + a^{2}b^{2}X_{2}^{4} − 2abcX_{2}^{2}X_{3}^{2} − 2cX_{1}^{2}X_{3}^{2} + X_{1}^{4} + c^{2}X_{3}^{4} = 0.

It’s interesting to note that if we now assume char(K) = 2, the above equation can be rearranged to be

((X_{1} + aX_{2})(X_{1} + bX_{2}) − cX_{3}^{2})^{2} = 0,

which is the square of the equation defining the variety obtained in [5].

5.2. The rank variety of V (\underline{λ}). One of the geometrically appealing results of Carlson’s rank variety, for elementary abelian p-groups, is the characterization of periodic modules as those whose rank variety is a line inside affine space, equivalently a point in projective space. This result relies upon the Hopf structure of the group algebra. Lacking such a structure for Λ_{m} we cannot hope to replicate the argument for our rank variety but we do at least have the following.

Proposition 5.1. Let \underline{λ} ∈ K^{m}, then

V^{r}(V (\underline{λ})) = \{[\underline{λ}]\}.

Proof. Note that it is immediate from Lemma 3.7 that [\underline{λ}] ∈ V^{r}(\underline{λ}).

Let \underline{μ} ∈ K^{m}, then \{[\underline{μ}]\} ∈ V^{r}(V (\underline{λ})) precisely when σ = σ_{V (\underline{λ})}(μ) has rank less than (1/2) dim(W) dim(V (\underline{λ})). Recall that

σ = ∑_{i=1}^{m} μ_{i}b_{i} ⊗ y_{i}

where y_{i} is the action of X_{i} on V (\underline{λ}). In order to calculate the rank of σ we need to understand the actions y_{i}.

Without loss of generality we may assume λ_{m} = 1. By Proposition 3.4 V (\underline{λ}) ≅ W ⊗Λ_{m−1} as a Λ_{m−1} module (we have identified Λ_{m−1} with Λ_{m}^{m}). We will make this an identification. Recall that, again by Proposition 3.4, we have an inclusion j : V (\underline{λ}) → W ⊗ Λ_{m} of Λ_{m}- modules and in fact

(5.1)  V (\underline{λ}) = ker(η(\underline{λ})).

It’s clear, by considering restrictions to Λ_{m−1}, that we can identify j by the following commutative square

V (\underline{λ})  ⟶^{j} W ⊗ Λ_{m}
∥  ∥
W ⊗ Λ_{m−1} ⟶^{1_{W} ⊗j_{m−1}} W ⊗ Λ_{m}

where j_{m−1} : Λ_{m−1} → Λ_{m} is the natural inclusion. We will also write 1_{m} for the identity map in End(Λ_{m}). It’s clear that under this identification we have y_{r} = 1_{W} ⊗ x_{r}^{m−1} for 1 ≤ r ≤ m − 1. Using (5.1) we have for v ∈ V (\underline{λ})

(b_{m} ⊗ 1_{m})X_{m}j(v) = − ∑_{r=1}^{m−1} λ_{r}(b_{r} ⊗ 1_{m})X_{r}j(v).

Now j is a Λ_{m}-module homomorphism, so rearranged we have

j(y_{m}v) = ∑_{r=1}^{m−1} λ_{r}(b_{m}b_{r} ⊗ 1_{m})j(y_{r}v)

Using the identification of j above we have j∘(f ⊗1_{m−1}) = (f ⊗1_{m})∘j for any f ∈ End(W). Hence

j(y_{m}v) = j(∑_{r=1}^{m−1}(λ_{r}(b_{m}b_{r} ⊗ 1_{m−1}) ∘ y_{r})v)

Since j is an inclusion we can conclude that

y_{m} = ∑_{r=1}^{m−1} λ_{r}b_{m}b_{r} ⊗ x_{r}^{m−1}.

We can now rewrite the map σ as

σ : W ⊗ W ⊗ Λ_{m−1} → W ⊗ W ⊗ Λ_{m−1}

σ = ∑_{r=1}^{m−1} β_{r}⊗_{r}^{m−1}

where

β_{r} = μ_{r}b_{r} ⊗ 1_{W} − λ_{r}b_{m} ⊗ b_{m}b_{r} ∈ End(W ⊗ W)

Note that the β_{r} satisfy β_{r}β_{s} + β_{s}β_{r} = 0 and β_{r}^{2} = −(μ_{r}^{2} − λ_{r}^{2})1_{W ⊗W}.

If some β_{r} were non-singular then by examining im(σ) ↓_{Λ_{m−1}^{r}} we could repeat the argument used in the proof of (2) in Proposition 3.4 and find that soc(Λ_{m−1}^{r}) im(σ) had dimension equal to rank(β_{r}) = dim(W ⊗W). In that case im(σ) ≅ W ⊗W ⊗Λ_{m−1}^{r} as Λ_{m−1}^{r}- modules and in particular rank(σ) = dim(W) dim(W) dim(Λ_{m−1}^{r}) = (1/2) dim(W) dim(V (\underline{λ})). So if [\underline{μ}] ∈ V^{r}(V (\underline{λ})) then all β_{r} must be singular, that is μ_{r} = ±λ_{r} for all r, and hence [\underline{μ}] = [\underline{λ}]. □

6. Related Algebras and Applications

In this section we wish to explore some applications of our characterization of projectivity for Λ_{m}-modules.

Given some finite-dimensional K-algebra B if we wish to use some of the results from §4 we might hope to find some subalgebra isomorphic to Λ_{m} for some m. But we would also need this subalgebra to behave well inside B if we wish to relate projectivity.

In §§6.1 we consider a fairly general set up between two algebras A and B and recall some conditions under which we can relate projectivity between them.

In §§6.2 we examine two applications where a subalgebra isomorphic to some Λ_{m} occurs. The first concerns Hecke algebras of symmetric groups and projectivity. The second concerns the structure of the underlying tree occurring in components of the stable Auslander-Reiten quiver of an algebra.

6.1. Let A and B be finite-dimensional K-algebras and suppose we have a K-algebra monomorphism

i : A → B.

Here we recall some standard facts about relating proj- A and proj- B. Through i, B becomes an A-bimodule and we have the induction functor

F =↑^{B} − := B ⊗_{A} − : mod- A → mod- B.

It has a right adjoint H =↓_{A} −, the restriction functor. In seeking to relate projectivity between these algebras we would like the functors F and H to send the full subcategories proj- A and proj- B to each other. It’s clear that F already does and in order for H to do so we need HA = _{B}A ∈ proj- B.

We’ll now assume that HA ∈ proj- B, and note that F is then an exact functor. A natural question is under what conditions does it follow that, for M ∈ mod- B, HM ∈ proj- A implies M ∈ proj- B ? Similarly for the functor F.

The following conditions are equivalent and we’ll denote by (H) any of these conditions holding.

(H1) If X ∈ mod- B, then X ∈ proj- B ⟺ HX ∈ proj- A .

(H2) The counit of the adjunction ϵ : F H → Id_{mod- B} is a split epimorphism.

(H3) The natural map, induced by multiplication, B ⊗_{A} B → B is a split epimorphism

of B-bimodules. Similarly the following conditions are also equivalent and we denote by (F) their satis-

faction.

(F1) If X ∈ mod- A, then X ∈ proj- A ⟺ F X ∈ proj- B .

(F2) The unit of the adjunction η : Id_{mod- A} → HF is a split monomorphism.

(F3) The map i : A → B is a split monomorphism of A-bimodules. That is A is a

summand of B as an A-bimdoule. Two examples where the above situation can occur are the following.

(1) Skew group rings. In keeping with traditional notation we take A = Λ a finitedimensional K-algebra and G a finite group which acts on Λ and whose order is invertible in K. We can then construct the skew group ring ΛG , see [15], and consider the natural ring monomorphism i : Λ → ΛG. Both conditions (H) and (F) are satisfied.

(2) Hecke algebras. If B is a finite dimensional K-algebra and e an idempotent, then taking A = End(Be)(≅ eBe) and i : A → B the natural map then condition (F) holds.

By combining these two examples we obtain the following lemma.

Lemma 6.1. Suppose Λ is a finite dimensional K-algebra and G a finite group that acts on Λ and whose order is invertible in K. Let e be an idempotent in the skew group ring ΛG. Let H : mod- ΛG → mod- Λ be the forgetful functor. Then for M ∈ mod- Λ we have

M ∈ proj- eΛGe ⟺ H(ΛG ⊗_{eΛGe} M) ∈ proj- Λ

6.2. Applications.

6.2.1. The Hecke algebra of a symmetric group. Suppose q is a primitive ℓ^{th} root of unity in K, then an ℓ-parabolic Hecke algebra is defined to be an algebra of the form H_{q}(ℓ)^{⊗m} where H_{q}(ℓ) is the Iwahori-Hecke algebra of the symmetric group ∑_{ℓ}. In [12][ed. Is this correct?] it is proved that any block C of an ℓ-parabolic Hecke algebra is Morita equivalent to an algebra eRe where R = Λ_{s}G [ed. What is s?] and e is an idempotent. Moreover the order of G is invertible in K.

By Lemma 6.1 the algebra Λ_{s} controls projectivity of modules for ℓ-parabolic Hecke algebras. Furthermore, when K has characteristic zero and H is any Hecke algebra of a symmetric group then ℓ-parabolic subalgebras of H control projectivity of H-modules. (See [11]). [ed. Do you think it’s worthwhile summarizing this last paragraph as say a Corollary putting all the bits together?]

6.2.2. Stable Auslander-Reiten quivers. Let A be a finite-dimensional symmetric algebra, and let Γ_{s}(A) be the stable Auslander-Reiten quiver of A. Recall (see [4]) this is a locally finite graph, with vertices labelled by the indecomposable non-projective A-modules (up to isomorphism). The number of arrows between two vertices being given by the dimension of the space of irreducible maps between the two modules representing the two vertices.

It is known that any connected component is isomorphic to ZT/Π where T is a tree and Π an admissible group of automorphisms of ZT. The underlying graph, the tree class T̄ [ed. what’s the difference between T̄ and T?], is an invariant of the component. One would like to know which T̄ can occur.

We say that \underline{mod}- A has enough periodic modules if for any 0 ≠ M ∈ \underline{mod}- A there is an Ω-periodic module V such that \underline{Hom}_{A}(V, M) ≠ 0. This condition is clearly satisfied when A has finite representation type, but in general it is not even known whether there are always non-trivial Ω-periodic modules.

Proposition 6.2. Suppose A is connected and of infinite representation type. If \underline{mod}- A has enough periodic modules then any component of the stable Auslander-Reiten quiver of A has tree class either one of the infinite Dynkin diagrams A_{∞}, A_{∞}^{∞}, D_{∞}, or a Euclidean diagram.

Proof. If M itself is periodic then,by [HPR] [ed This needs to be a bibitem], the component containing M is a tube and has tree class A_{∞}. Otherwise consider d_{V} = dim \underline{Hom}_{A}(V, −) which induces an additive function, in the sense of [HPR], on the associated tree. The Proposition now follows from [HPR]. □

Corollary 6.3. Suppose A and B are algebras as in §§6.1, with HB ∈ proj- A and assume condition (H) holds. If A ≅ Λ_{m}, for some m, then any tree class of Γ_{s}(B) is either Dynkin or Euclidean.

Proof. If V is an Ω-periodic A-module then F V is an Ω-periodic A-module. For M ∈ mod- B, the Eckmann-Shapiro lemma gives

\underline{Hom}_{B}(F V, M) ≅ \underline{Hom}_{A}(V, HM)

By Theorem 4.3 and Lemma 3.7 together with condition (H) we see that \underline{mod}- B has enough periodic modules. The result now follows from Proposition 6.2. □

Remark. For char(K) = 2 this result is due to Webb [ed. Reference?]. For char(K) ≠ 2 and m = 2 it is known that the stable Auslander-Reiten quiver consists of one Euclidean component and otherwise only tubes. For m > 2 this is a new result.

References

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[8] J.F. Carlson, Varieties for modules, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 37-44.

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[11] K. Erdmann, On Auslander-Reiten components for group algebras, J. Pure Appl. Algebra 104(2) (1995), pp 149–160.

[12] K. Erdmann, Subalgebras of Hecke algebras, preprint 2002.

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[15] I. Reiten, C. Riedtmann Skew Group Algebras in the Representation Theory of Artin Algebras J. Algebra 92 (1985), pp. 225-282.

University of Oxford, Mathematics Institute, 24-29 St.Giles, Oxford, ENGLAND

   E-mail address: erdmann@maths.ox.ac.uk, holloway@maths.ox.ac.uk