RATIONAL REPRESENTATIONS OF GL_{2}.

VANESSA MIEMIETZ AND WILL TURNER

Abstract. Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra \underleftarrow{C}_{p}(F), with a transparent combinatorial structure, which we expect to control the rational representation theory of GL_{2}(F).

1. Introduction

In any first course on representation theory, the students will become familiar with representations of the algebraic group GL_{2}(ℂ), or with those of some close relation of this group. It is perhaps surprising therefore, over a century after the birth of group representation theory, that anything remains to be said about GL_{2}. However, development in the modular theory has been much slower than in characteristic zero, and the progress of its practitioners often resembles that of a tortoise chasing a hare. In this article, we wish to pursuade the reader that there is structure underlying the rational representation theory of GL_{2} over a field of positive characteristic, as simple as the structure appearing in characteristic zero, although quite different in nature.

Of course, even in positive characteristic, the usual hare-headed questions about GL_{2}-modules were answered long ago: irreducibles are parametrised by elements of the dominant region of the weight lattice, and have realisations as tensor products of Frobenius twists of socles of symmetric powers of the natural representation in small degrees, and powers of the determinant representation. However, the situation is more delicate than these easy truths imply. There are homological interactions between irreducible modules, and for a deeper understanding one ought to contemplate the manner in which these interactions occur. This is the concern of our paper.

We shall be more precise. Let A be an algebra with a self-dual bimodule T. Let B be the algebra whose category of ungraded representations is equivalent to the category of graded representations of the trivial extension algebra of A by T. Let C be the trivial extension algebra of B by its dual. Modulo the infinite dimensionality of C, we have a map

C ⟳ {algebras with a self-dual bimodule},

which takes an algebra A, with an A-A bimodule T, such that _{A}T_{A} ≅ _{A}T_{A}^{∗}, to a symmetric algebra C. The self-dual bimodule corresponding to C is the regular bimodule _{C}C_{C}.

For every n ∈ ℕ, there is a localisation

C_{n} ⟳ {algebras with a self-dual bimodule}

of C. There is a canonical epimorphism

A ↞ C_{n}(A).

Taking the inverse limit of the sequence

A ↞ C_{n}(A) ↞ C_{n}(C_{n}(A)) ↞ C_{n}(C_{n}(C_{n}(A))) ↞ ...,

we obtain an algebra \underleftarrow{C}_{n}(A).

Let S(2) = ⨁_{r≥0} S(2, r) be the Schur algebra associated to GL_{2}, defined over an algebraically closed field F of characteristic p > 0 [9]. There is a sequence of natural surjections

S(2, r) ↞ S(2, r + 2) ↞ S(2, r + 4) ↞ S(2, r + 6) ↞ ...

Let S(2, \underline{r}) be the inverse limit of this directed sequence of algebra epimorphisms. The category of rational representations of GL_{2}(F) is equivalent to the category of finite dimensional representations of the direct sum ⨁_{r∈ℤ} S(2, \underline{r}).

In the sequel, we define a certain filtration on S(2, r), refining the radical filtration, and denote by G(2, r) the graded ring associated to this filtration. There is a compatible sequence of surjections

G(2, r) ↞ G(2, r + 2) ↞ G(2, r + 4) ↞ G(2, r + 6) ↞ ...

Let G(2, \underline{r}) be the inverse limit of this directed sequence of algebra epimorphisms. Our main result is the following:

Theorem 1. Every block of G(2, \underline{r}) is Morita equivalent to \underleftarrow{C}_{p}(F).

Our proof of Theorem 1 is inductive. We apply results of K. Erdmann, A. Henke, and S. Koenig concerning S(2, r) ([8], [11]), to prove that certain Ringel self-dual blocks of G(2, r) are equivalent to C_{p}^{d}(F), for some d. Since every block of G(2, r) is a quotient of such a Ringel self-dual block, the theorem follows.

In fact, we prove a rather stronger statement. Let S be a Ringel self-dual Schur algebra S(2, r). We demonstrate the existence of a filtration by ideals,

S ⊃ N ⊃ N^{2} ⊃ 0,

whose associated graded ring is Morita equivalent to C_{a}(A) ⊕ F^{⊕m}, where A is a smaller Ringel self-dual Schur algebra S(2, s), where 2 ≤ a ≤ p, and where m is some multiplicity.

In an earlier chapter, we give careful definitions of B, C, and C_{p}, and prove that under favourable conditions, they respect the quasi-heredity condition.

This is all very pleasing, but we believe more to be true. We predict that in fact S(2, r) ≅ G(2, r), for all r, and therefore S(2, \underline{r}) ≅ G(2, \underline{r}). In other words, we expect the following to be true:

Conjecture 2. Every block of rational representations of GL_{2}(F) is equivalent to \underleftarrow{C}_{p}(F) -mod.

In the final chapter of the paper, we consider this possibility in more detail. We demonstrate that the main obstacle to a proof by induction is a familiar one in modular representation theory: the lifting of a stable equivalence.

In his inductive approach to M. Broué’s abelian defect group conjecture [3], R. Rouquier has established that the main difficulty is the lifting of a stable equivalence to a derived equivalence [20]. In our microcosm, we give a similar inductive strategy to prove that S(2, r) ≅ G(2, r). We define a pair of infinite dimensional self-injective algebras, L_{1} and L_{2}, and prove the existence of a stable equivalence between these, sending simple modules to simple modules. If one could lift this stable equivalence to a Morita equivalence, an isomorphism S(2, \underline{r}) ≅ G(2, \underline{r}) would follow.

There are ramifications for the Temperley-Lieb algebra, which we briefly mention here. If F has characteristic p > 2, then an r-fold tensor product of the natural two dimensional GL_{2}(F)-module is a full tilting module for S(2, r). Therefore its endomorphism ring, known as the Temperley-Lieb algebra T L_{r}, is the Ringel dual of S(2, r). We have directed sequences of embeddings of idempotent subalgebras,

T L_{r} ↪ T L_{r+2} ↪ T L_{r+4} ↪ T L_{r+6} ↪ ...
A ↪ C_{n}(A) ↪ C_{n}(C_{n}(A)) ↪ C_{n}(C_{n}(C_{n}(A))) ↪ ....

Let T L\underline{r}, \underrightarrow{C}_{n}(A) denote the direct limits of these sequences of algebra monomorphisms. We expect any block of T L\underline{r} to be Morita equivalent to \underrightarrow{C}_{p}(F).

Acknowledgements: We have benefited from conversations with Anne Henke and Karin Erdmann concerning Schur algebras, with Joe Chuang, and with Raphaël Rouquier. We thank the EPSRC for financial support.

2. Setup

Throughout this paper, F will be a field and A an F -algebra. We denote by A -mod the category of finite dimensional left A-modules, and by mod- A the category of finite dimensional right A-modules. Given a finite dimensional left/right module M, we write the dual of M as M^{∗} = Hom_{F} (M, F), a right/left module.

We write A -proj for the category of finite dimensional projective left A-modules. Given a collection X ⊂ A -mod, we denote by F(X) the category of modules filtered by objects in X. Let J (A) denote the Jacobson radical of A.

We suppose that A is a locally finite dimensional algebra. In other words, there exists a set Λ, indexing a set of orthogonal idempotents {e_{λ}}_{λ∈Λ}, such that A ≅ ⨁_{λ,μ∈Λ} e_{λ}Ae_{μ}, and e_{λ}Ae_{μ} is finite dimensional. We assume that A/J (A) = ⨁_{λ∈Λ} M_{λ} is a direct sum of matrix rings M_{λ} over F, where e_{λ} is the unit of M_{λ}. Thus, Λ is an indexing set for isomorphism classes of simple A-modules. By the idempotent decomposition, simple modules have projective covers and injective hulls, providing 1 − 1-correspondences between isomorphism classes of simples, projectives and injectives.

Now let Λ be a poset which is interval-finite (i.e. for every μ ≤ λ ∈ Λ the set {ν|μ ≤ ν ≤ λ} is finite).

Recall that mod- A is a highest weight category in the sense of Cline, Parshall and Scott [5] if, for every λ ∈ Λ there exists an irreducible right module L^{r}(λ), a costandard right module ∇^{r}(λ), which embeds into the injective hull I^{r}(λ) of L^{r}(λ), such that the cokernel of this inclusion is filtered by ∇^{r}(μ) for μ ≥ λ, and ∇^{r}(λ)/ soc ∇^{r}(λ) consists of composition factors L^{r}(ν) for ν < λ. Dualizing with respect to F, we find this is equivalent to the corresponding projective indecomposable left modules P (λ) ∈ A -mod having standard filtrations. So, for every λ ∈ Λ there exists a standard module Δ(λ) and an epimorphism P (λ) ↠ Δ(λ), the kernel of which is filtered by modules Δ(μ) for μ > λ, and the kernel of the map Δ(λ) ↠ L^{l}(λ) consists of composition factors of the form L(ν) for ν < λ.

Let J ⊂ Λ be a nonempty finitely generated ideal. The subcategory mod- A[J] of objects which only have composition factors L(ν) for ν ∈ J is a highest weight category, whenever mod- A is a highest weight category ([5], Theorem 3.5). Let A^{J} = A/ ∑_{λ∉J} Ae_{λ}A. Then A^{J} is a locally finite-dimensional algebra and mod- A[J] ≅ mod- A^{J}.

Let I ⊂ Λ be a nonempty finitely generated coideal and define A_{I} := ⨁_{λ,μ∈I}e_{λ}Ae_{μ}.

Lemma 3. If mod- A is a highest weight category, then mod- A_{I} is a highest weight category.

Proof. We construct Δ-filtrations of projectives in A_{I} -mod. Projectives in A_{I} -mod are of the form P_{A_{I}} (λ) := Hom_{A}( ⨁_{μ∈I}Ae_{μ}, P_{A}(λ)). We define Δ_{A_{I}} (λ) :=

Hom_{A}( ⨁_{μ∈I}A_{μ}, Δ_{A}(λ)). Since Hom_{A}( ⨁_{μ∈I}A_{μ}, −) is exact we obtain a filtration of

P_{A_{I}} (λ) respecting the necessary conditions on orders. □

Let us define A_{I}^{J} := (A^{J})_{I}.

If I∩J is finite, then A_{I}^{J} is a finite dimensional quasi-hereditary algebra, whenever mod- A is a highest weight category ([5], Theorem 3.5).

Proposition 4. mod- A is a highest weight category if and only if A_{I}^{J} is quasihereditary for all finitely generated coideals I and finitely generated ideals J such that I ∩ J is finite.

Proof. As noted above, the “only if” statement is well known [5]. So suppose A_{I}^{J} is quasi-hereditary for all suitable I and J. By a standard argument of Dlab [6], the existence of a highest weight structure on mod- A is equivalent to the surjective multiplication map

\frac{Ae_{λ}}{∑_{μ>λ} Ae_{μ}Ae_{λ}} ⊗_{F} \frac{e_{λ}A}{∑_{μ>λ} e_{λ}Ae_{μ}A} ⟶ \frac{∑_{μ≥λ} Ae_{μ}A}{∑_{μ>λ} Ae_{μ}A}

being an isomorphism, for all λ ∈ Λ. But this can be checked on arbitrarily large finite truncations of Λ containing λ □

Corollary 5. For a locally finite-dimensional algebra A, A -mod is a highest weight category if and only if mod- A is a highest weight category.

Proof. Follows immediately from Proposition 4 and the same statement for finite-dimensional algebras ([16], 4.3(b)).□

Definition 6. A locally finite-dimensional algebra A is quasi-hereditary if A -mod and mod- A are highest weight categories.

Note that by corollary 5, we can now move freely between left and right modules, standard and costandard filtrations and we have the usual duality relations between standard modules on one side and costandard modules on the other: Δ^{r}(λ) ≅ ∇(λ)^{∗}, ∇^{r}(λ) ≅ Δ(λ)^{∗}.

For the rest of this chapter, let A be a locally finite-dimensional quasi-hereditary algebra with poset Λ of weights, left standard modules Δ(λ), left costandard modules ∇(λ), right standard modules Δ^{r}(λ) and right costandard modules ∇^{r}(λ). The remaining propositions in this chapter are all proved by cutting down to a suitable finite-dimensional subquotient and applying Ringel’s tilting theory for finitedimensional quasihereditary algebras there [18]. We therefore omit the proofs.

Definition 7. T ∈ A -mod is called tilting if it is filtered by standard and by costandard modules.

Proposition 8. There is a one-to-one correspondence between Λ and the set of indecomposable tilting modules in A -mod.

We denote by T (λ) the unique indecomposable tilting module such that [T (λ) : L(λ)] = 1, and [T (λ) : L(μ)] ≠ 0 implies μ ≤ λ.

Definition 9. We say that A′ is Ringel dual to A if there exist multiplicities

n_{λ} ∈ ℤ_{≥1}, such that A′ ≅ ⨁_{λ,μ∈Λ} Hom_{A}(T (λ)^{n_{λ}}, T (μ)^{n_{μ}}).

If A, A′ are Ringel dual, then T = ⨁_{λ∈Λ} T (λ)^{n_{λ}} is an A-A′ bimodule. In these circumstances, we call it a tilting bimodule.

For any subset Γ of Λ let Γ′ equal Γ as a set, but with the opposite order. Thus, for an ideal J ⊆ Λ we obtain a coideal J′ ⊆ Λ′, for a coideal I ⊆ Λ we obtain an ideal I′ ⊆ Λ′.

Proposition 10. A′ is quasi-hereditary with poset Λ′.

Proposition 11. A′′ ≅ A.

Proposition 12.

(i) (A^{J})′ ≅ A′_{J′}

(ii)(A_{I})′ ≅ (A′)^{I′}.

3. Algebraic constructions

Throughout this chapter A will be a finite-dimensional algebra, endowed with an A-A-bimodule T.

Define B_{0} := ⨁_{i∈ℤ}A_{i} where A_{i} ≅ A for all i ∈ ℤ. We define B_{1} := ⨁_{i∈ℤ}_{i}T_{i+1} as a B_{0}, B_{0}-bimodule, where each _{i}T_{i+1} is isomorphic to T but with action of A_{i} on the left and of A_{i+1} on the right.

Let B be the trivial extension of B_{0} by B_{1}; we can think of this as a matrix

B = ( \begin{matrix}... & _{i−2}T_{i−1} & 0 & ⋯ \\ 0 & A_{i−1} & _{i−1}T_{i} & 0 & ⋯ \\ ⋯ & 0 & A_{i} & _{i}T_{i+1} & 0 & ⋯ \\ & ⋯ & 0 & A_{i+1} & _{i+1}T_{i+2} & 0 \\ & & & & A_{i+2} & ... \\ . & . & . \end{matrix} )

where the A_{i} are on the leading diagonal. Let

B^{∗} = ⨁_{i∈ℤ} Hom_{F} (B1_{A_{i}}, F),

a B-B-bimodule. Let C be the trivial extension of B by B^{∗}. Then C is a locally finite dimensional, symmetric algebra.

Let C^{n} denote the quotient C/ ∑_{k>n} C1_{A_{i}}C of C. Let C_{1}^{n} denote the subalgebra ∑_{i,j≥1} 1_{A_{i}}C^{n}1_{A_{j}} of C^{n}.

Lemma 13. The algebra C_{1}^{n} is ℤ-graded, concentrated in degrees 0,1, and 2. In descending vertical order, its components in degrees 0, 1 and 2 are,

⨁_{1≤i≤n} A_{i}

⨁_{1≤i≤n−1}(_{i}T_{i+1} ⊕_{i}T_{i+1}^{∗})

⨁_{1≤i≤n−1} A_{i}^{∗}.

Lemma 14. Suppose that T ≅ T^{∗}, as A-A-bimodules. Then the infinite dihedral group D_{∞} acts as automorphisms of C. The space

T_{1}^{n} = ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{i}}C1_{A_{j}}

has the structure of a self-dual C_{1}^{n}-C_{1}^{n}-bimodule.

Proof.

We define an action of D_{∞} on C as follows: The involution σ sends A_{i} to A_{−i} via the identity, A_{i}^{∗} to A_{−i}^{∗} via the identity, _{i−1}T_{i} to _{−(i−1)}T_{−i}^{∗} via the isomorphism T ≅ T^{∗}, and analogously _{i+1}T_{i}^{∗} to _{−(i+1)}T_{−i}. Thanks to the assumption that AT_{A}^{∗} ≅ _{A}T_{A}, we see that this is indeed an algebra isomorphism. The translation τ in D_{∞} just maps the components C1_{A_{i}} to the analogous components of C1_{A_{i+1}}, which is also clearly an isomorphism.

Of course, C itself is a C-C-bimodule, but what about the truncation T_{1}^{n} ? The idempotents 1_{A_{i}}, for i > n act as zero on C1_{A_{j}}, for j < n. Therefore, C_{1}^{n} acts naturally on the left of T_{1}^{n}. After twisting the right action of C on itself by the automorphism σ ∘ τ^{−n}, we can similarly observe a right action of C_{1}^{n} on T_{1}^{n}.

Now

( ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{i}}C1_{A_{j}})^{∗} ≅ ( ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{j}}C1_{A_{i}})

by self-duality of C. □

Definition 15. Let

C_{n} ⟳ {algebras with a self-dual bimodule}

be the map which takes the pair (A, T) to the pair (C_{1}^{n}, T_{1}^{n}).

When employing the above definition, we sometimes forget the self-dual bimodules, and write simply C_{n}(A) for the algebra C_{1}^{n}.

Assume now that A is a quasi-hereditary algebra with poset Λ of weights. Let Λ_{B}^{1} = ⨿_{i∈ℤ}Λ[i] of weights, with the same ordering as in Λ within each Λ[i], and λ[i] < μ[j] for i ≠ j if and only if i > j ∈ ℤ. Let Λ_{B}^{2} = ⨿_{i∈ℤ}Λ[i] of weights, with the same ordering as in Λ within each Λ[i], and λ[i] < μ[j] for i ≠ j if and only if i < j ∈ ℤ.

The partially ordered sets Λ_{B}^{1}, Λ_{B}^{2} index the irreducible B_{0}-modules. Indeed, B_{0} is obviously locally finite-dimensional and quasi-hereditary with respect to the posets Λ_{B}^{1}, Λ_{B}^{2}.

For the remainder of this chapter, we assume that A is Ringel self-dual, and that T is a tilting bimodule for A, such that T_{A} ≅ (_{A}T)^{∗}. Thus, T_{A} ∈ F(Δ^{r}) ∩ F(∇^{r}).

Theorem 16. B is quasi-hereditary with respect to the poset Λ_{B}^{1}, with standard and costandard modules

Δ_{B}^{1}(λ[i]) = Δ_{B_{0}}(λ[i])  and  ∇_{B}^{1}(λ[i]) = Hom_{B_{0}}(B, ∇_{B_{0}}(λ[i])).

B is quasi-hereditary with respect to Λ_{B}^{2}, with standard and costandard modules

Δ_{B}^{2}(λ[i]) = B ⊗_{B_{0}} Δ_{B_{0}}(λ[i])  and  ∇_{B}^{2}(λ[i]) = ∇_{B_{0}}(λ[i]).

B is Ringel self-dual and Ringel duality exchanges the two quasi-hereditary structures on B.

Proof.

First observe that indeed B is locally finite-dimensional and Λ_{B} indexes simple modules since B_{1} forms a nilpotent ideal in B.

(1) Δ_{B}^{1}(λ[i]) has a simple top and the radical consists of composition factors with smaller indices.

Obvious from B_{0}.

(2) B -proj ⊂ F(Δ_{B}^{1}) with order relations as required.

We show that _{B}B1_{A_{i}} ∈ F(Δ_{B}^{1}) for all i ∈ ℤ. But _{B}B1_{A_{i}} has a filtration with a submodule _{i−1}T_{i} as submodule and A_{i} as quotient. As left B-module, _{i−1}T_{i} is filtered by Δ_{B_{0}}(λ[i − 1]) and A_{i} is filtered by Δ_{B_{0}}(λ[i]) with λ ∈ Λ. Since for A_{i} the filtration by Δ_{B_{0}}’s is in the right order (on every direct summand) and Δ_{B_{0}}(λ[i − 1]) > Δ_{B_{0}}(μ[i]) for all λ, μ ∈ Λ, the filtration respects the necessary inequalities on labels.

(3) Δ_{B}^{2}(λ[i]) has a simple top and the radical consists of composition factors with smaller indices.

Δ_{B}^{2}(λ[i]) has a submodule isomorphic to B_{1} ⊗_{B_{0}} Δ_{B_{0}}(λ[i]) ≅ _{i−1}T_{i} ⊗_{A_{i}} Δ_{A_{i}}(λ), the quotient by which is isomorphic to B_{0} ⊗_{B_{0}} Δ_{B_{0}}(λ[i]) ≅ Δ_{B_{0}}(λ[i]). The latter has simple head, and all other compositions factors have smaller indices by the quasihereditary structure of B_{0}. The former has composition factors with labels in Λ[i − 1] which, since in this ordering i − 1 < i, are smaller as desired. Furthermore B_{1} is a nilpotent ideal in B, thus the above submodule does not contribute to the head of the module.

(4) B -proj ⊂ F(Δ_{B}^{2}) with order relations as required.

B_{B_{0}} ≅ (B_{0})_{B_{0}} ⊕ (B_{1})_{B_{0}} and (B_{0})_{B_{0}} is projective, hence flat. We claim that (B_{1})_{B_{0}} ⊗_{B_{0}} − is exact on F(Δ_{B_{0}}). To prove this, it suffices to check that _{i−1}T_{i} ⊗_{A_{i}} − is exact on F(Δ_{A_{i}}). So let M ∈ F(Δ_{A_{i}}) and consider _{i−1}T_{i} ⊗_{A_{i}} M. This being in F(Δ_{A_{i−1}}), is equivalent to (_{i−1}T_{i} ⊗_{A_{i}} M)^{∗} being in F(∇_{A_{i−1}}^{r}). Now (_{i−1}T_{i} ⊗_{A_{i}} M)^{∗} =

Hom_{F} (_{i−1}T_{i} ⊗_{A_{i}} M, F) ≅ Hom_{mod- A_{i}}(T_{i}, M^{∗}). But M ∈ F(Δ_{A_{i}}) implies M^{∗} ∈

F(∇_{A_{i}}^{r}) and, by the assumption that T_{A} ≅ (_{A}T)^{∗}, T_{A} is also a tilting module for mod- A. Therefore, Hom_{mod- A_{i}}(T_{i}, −) is exact on F(∇_{A_{i}}^{r}) by [7], A4 (1), and thus

(_{i−1}T_{i} ⊗_{A_{i}} M)^{∗} ∈ F(∇_{A_{i−1}}^{r}). So B ⊗_{B_{0}} − is exact on F(Δ_{B_{0}}), and _{B}B ∈ F(Δ_{B}^{2}). The

required ordering conditions follow immediately from those for B_{0}.

This finishes the proof of B having two quasihereditary structures.

Similarly, we find that for the right module categories, with respect to Λ_{B}^{1}, we have

Δ_{B}^{1,r}(λ[i]) = Δ_{B_{0}}^{r}(λ[i]) ⊗_{B_{0}} B

and with respect to Λ_{B}^{2},

Δ_{B}^{2,r}(λ[i]) = Δ_{B_{0}}^{r}(λ[i])

By duality, we now see that

∇_{B}^{1}(λ[i]) = (Δ^{1,r}(λ[i]))^{∗} = Hom_{F} (Δ_{B_{0}}^{r}(λ[i]) ⊗_{B_{0}} B, F)
≅ Hom_{B_{0}}(B, (Δ_{B_{0}}^{r}(λ[i]))^{∗}) ≅ Hom_{B_{0}}(B, ∇_{B_{0}}(λ[i])),
∇^{1,r}(λ[i]) = (Δ^{1}(λ[i]))^{∗} = (Δ_{B_{0}}(λ[i]))^{∗} = ∇_{B_{0}}^{r}(λ[i]),
∇_{B}^{2}(λ[i]) = (Δ^{2,r}(λ[i]))^{∗} = (Δ_{B_{0}}^{r}(λ[i]))^{∗} = ∇_{B_{0}}(λ[i]),

and

∇^{2,r}(λ[i]) = (Δ^{2}(λ[i]))^{∗} = Hom_{F} (B ⊗_{B_{0}} Δ_{B_{0}}(λ[i]), F)
≅ Hom_{mod- B_{0}}(B, (Δ_{B_{0}}(λ[i]))^{∗}) ≅ Hom_{B_{0}}(B, ∇_{B_{0}}^{r}(λ[i])).

To prove the Ringel self-duality of B, we need the following lemma.

Lemma 17. Δ_{B}^{2}(λ[i]) ≅ ∇_{B}^{1}(λ′[i − 1]).

Proof of the lemma. We know that (∇_{B}^{1}(λ′[i−1]))^{∗} ≅ Δ_{B}^{1,r}(λ′[i−1])), so it suffices to show that there exists a non-degenerate bilinear form ⟨ , ⟩ : Δ_{B}^{1,r}(λ′[i − 1])) × Δ_{B}^{2}(λ[i]) ⟶ F with the property that ⟨x, by⟩ = ⟨xb, y⟩ for x ∈ Δ_{B}^{1,r}(λ′[i − 1])), y ∈ Δ_{B}^{2}(λ[i]), b ∈ B. This is equivalent to having a linear map Δ_{B}^{1,r}(λ′[i − 1]))_{F} ⊗ Δ_{B}^{2}(λ[i]) ⟶ F which factors over

Δ_{B}^{1,r}(λ′[i − 1])) ⊗_{B} Δ_{B}^{2}(λ[i]) = Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} B ⊗_{B} B ⊗_{B_{0}} Δ_{B_{0}}(λ[i])

≅ Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} B ⊗_{B_{0}} Δ_{B_{0}}(λ[i])

≅ Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} B_{0} ⊗_{B_{0}} Δ_{B_{0}}(λ[i])

⊕ Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} B_{1} ⊗_{B_{0}} Δ_{B_{0}}(λ[i]).

But Δ_{B_{0}}^{r}(λ′[i−1]) ⊗_{B_{0}}B_{0} ⊗_{B_{0}}Δ_{B_{0}}(λ[i]) ≅ Δ_{B_{0}}^{r}(λ′[i−1])1_{A_{i−1}} ⊗_{B_{0}}1_{A_{i}}Δ_{B_{0}}(λ[i]) ≅ 0 and

we claim that Δ_{B_{0}}^{r}(λ′[i−1]) ⊗_{B_{0}}B_{1} ⊗_{B_{0}}Δ_{B_{0}}(λ[i]) is isomorphic to ∇_{B_{0}}^{r}(λ[i]) ⊗_{B_{0}}Δ_{B_{0}}(λ[i]) on the one hand and to Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} ∇_{B_{0}}(λ′[i − 1]) on the other hand. Indeed,

(_{i−1}T_{i} ⊗_{A_{i}} Δ_{A_{i}}(λ))^{∗} = Hom_{F} (_{i−1}T_{i} ⊗_{A_{i}} Δ_{A_{i}}(λ), F) ≅ Hom_{mod- A_{i}}(_{i−1}T_{i}, (Δ_{A_{i}}(λ))^{∗})
≅ Hom_{mod- A_{i}}(_{i−1}T_{i}, ∇_{A_{i}}^{r}(λ)) ≅ Δ_{A_{i−1}}^{r}(λ′),

whence_{i−1}T_{i} ⊗_{B_{0}}Δ_{B_{0}}(λ[i]) = _{i−1}T_{i} ⊗_{A_{i}}Δ_{A_{i}}(λ) ≅ ∇_{A_{i}}(λ′) = ∇_{B_{0}}(λ′[i−1]). The second

isomorphism, Δ_{B_{0}}^{r}(λ′[i − 1]) ⊗_{B_{0}} B_{1} ≅ ∇_{B_{0}}^{r}(λ[i]) follows by the same arguments for right modules. But ∇_{B_{0}}^{r}(λ[i]) and Δ_{B_{0}}(λ[i]) are dual to one another as are Δ_{B_{0}}^{r}(λ′[i−1]) and ∇_{B_{0}}(λ′[i−1]). Thus we have unique (up to scalar) non-degenerate bilinear forms on both pairs. Defining our bilinear form as the sum of both gives us a B-equivariant nondegenerate bilinear form on Δ_{B}^{1,r}(λ′[i − 1])) × Δ_{B}^{2}(λ[i]). (It is nondegenerate because, as B_{0}-modules, Δ_{B}^{1,r}(λ′[i−1])) ≅ Δ_{B_{0}}^{r}(λ′[i−1])⊕∇_{B_{0}}^{r}(λ[i]) and Δ_{B}^{2}(λ[i]) ≅ Δ_{B_{0}}(λ[i]) ⊕ ∇_{B_{0}}(λ′[i − 1]) and it is nondegenerate on both dual pairs.) This completes the proof of the lemma. □

Proof of theorem, continued. By the lemma B -proj ⊂ F(∇_{B}^{1}) = F(Δ_{B}^{2}), but we also have B -proj ⊂ F(Δ_{B}^{1}), hence projective modules are tilting modules in the first highest weight structure on B -mod. But clearly

⨁_{λ[i],μ[j]∈Λ_{B}} Hom_{B}(P (λ[i]), P (μ[j])) ≅ B

so B is indeed Ringel self-dual. Denoting the new standard modules by Δ̃_{B} we obtain

Δ̃_{B}^{1}(λ[i]) = Hom_{B}(B, ∇_{B}^{1}(λ[i])) ≅ ∇_{B}^{1}(λ[i]) ≅ Δ_{B}^{2}(λ′[i + 1]).

By the right analogue of the lemma we see that (right) projectives are tilting modules for the second highest weight structure on mod- B, and by the same computation as above, we obtain

Δ̃_{B}^{2,r}(λ[i]) = Hom_{B}(B, ∇_{B}^{2,r}(λ[i])) ≅ ∇_{B}^{2,r}(λ[i]) ≅ Δ_{B}^{1,r}(λ′[i − 1]).

Dualizing we see that

˜∇_{B}^{1,r}(λ[i]) ≅ (Δ̃_{B}^{1}(λ[i]))^{∗} ≅ (Δ_{B}^{2}(λ′[i + 1]))^{∗} ≅ ∇_{B}^{2,r}(λ′[i + 1])

and

˜∇_{B}^{2}(λ[i]) ≅ (Δ̃_{B}^{2,r}(λ[i]))^{∗} ≅ (Δ_{B}^{1,r}(λ′[i − 1]))^{∗} ≅ ∇_{B}^{1}(λ′[i − 1]).

Since Δ’s and ∇’s determine each other, this completes the proof of the theorem. □

Let Λ_{C}^{1} = Λ_{B}^{1}, and Λ_{C}^{2} = Λ_{B}^{2}.

Theorem 18. C is quasi-hereditary with poset Λ_{C}^{1}, as well as with poset Λ_{C}^{2}. We have

∇_{C}^{1}(λ[i]) = ∇_{B}^{1}(λ[i]), Δ_{C}^{2}(λ[i]) = Δ_{B}^{2}(λ[i]).

Furthermore, C is Ringel self-dual, and Ringel duality exchanges the two highest weight structures on C.

Proof. The equality of the indexing sets for simple modules follows from the nilpotency of B^{∗} in C. Now, C1_{A_{i}} has a filtration with submodule B^{∗}1_{A_{i}} ≅ (1_{A_{i}}B)^{∗} and quotient B1_{A_{i}}. The latter has a filtration by Δ_{B}^{2}(λ[i]), where λ ∈ Λ, with the necessary properties by Theorem 16. The former has a filtration by (Δ_{B}^{1,op}(λ[i]))^{∗} ≅ ∇_{B}^{1}(λ[i]) ≅ Δ_{B}^{2}(λ′[i + 1]). So, since i + 1 > i we have a filtration respecting the necessary inequalities on labels.

The fact that C is symmetric follows from a general statement the the trivial extension of an algebra by its dual is symmetric.

Ringel self-duality follows immediately from symmetry, since projectives have a Δ-filtration, but since they are the same as injectives, also a ∇-filtration, thus projectives are tilting modules, implying Ringel self-duality. □

Set J_{n} := ⋃_{j≤n}Λ[j] and I_{k} := ⋃_{i≥k}Λ[i] and adopt the notational convention C^{n} :=

C^{J_{n}}, C_{k} := C_{I_{k}}, and C_{k}^{n} := C_{I_{k}}^{J_{n}}.

Let us now assume that _{A}T_{A}^{∗} ≅ _{A}T_{A} as a bimodule. Recall that in these circumstances, D_{∞} =< σ, τ > acts on C. Note that in the Ringel duality in theorem 18, C′ = τ^{−1}(C), since the projective P (λ[i]) has a submodule Δ_{C}(λ′[i + 1]), implying

P (λ[i]) ≅ T_{C}(λ[i + 1]) and P_{C′}(λ[i]) = Hom_{C}( ⨁_{\begin{matrix}j∈ℤ \\ λ∈Λ \end{matrix}}P (λ[j]), P (λ[i − 1])).

Theorem 19. C_{1}^{n} for n ≥ 1 is Ringel self-dual, and the tilting bimodule T_{C_{1}^{n}} is a self-dual bimodule.

Proof. By Proposition 12, (C_{1}^{n})′ ≅ (C′)_{J′_{n}}^{I′_{1}} with the ordering i > i + 1 on ℤ. Therefore

(C_{1}^{n})′ ≅ (τ^{−1}C)_{J′_{n}}^{I′_{1}} ≅ C_{J′_{n−1}}^{I′_{0}} ≅^{σ} C_{−(n−1)}^{0} ≅^{τ^{n}} C_{1}^{n}.

The tilting module T_{C_{1}^{n}} satisfies

T_{C_{1}^{n}} = ⨁_{\begin{matrix}j≤n \\ λ∈Λ \end{matrix}} Hom_{C}(⨁_{i≥1}C1_{A_{i}}, T_{C}(λ[j]))

≅ ⨁_{\begin{matrix}j≤n \\ λ∈Λ \end{matrix}} Hom_{C}(⨁_{i≥1}C1_{A_{i}}, P_{C}(λ[j − 1]))

≅ ⨁_{\begin{matrix}j≤n−1 \\ λ∈Λ \end{matrix}} Hom_{C}(⨁_{i≥1}C1_{A_{i}}, P_{C}(λ[j]))

= ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}} Hom_{C}(C1_{A_{i}}, C1_{A_{j}})

≅ ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{i}}C1_{A_{j}}.

The first equality comes from the fact that factoring out a heredity ideal doesn’t change the tilting modules for the remaining labels and that the tilting module for a heredity subalgebra is the tilting module multiplied by the idempotent.The fourth equality takes into account that we only have nonzero maps from C1_{A_{i}} to itself or to C1_{A_{i±1}}. Now

(_{C_{1}^{n}}( ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{i}}C1_{A_{j}})_{(C_{1}^{n})′})^{∗} = _{(C_{1}^{n})′}( ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{j}}C1_{A_{i}})_{C_{1}^{n}}

(by self-duality of C), but to view this as a (C_{1}^{n}, (C_{1}^{n})′)-bimodule we have to twist with σ ∘ τ^{−n} on the left and its inverse on the right which yields ⨁_{\begin{matrix}0≤j≤n−1 \\ 1≤i≤n \end{matrix}}1_{A_{i}}C1_{A_{j}}

as desired. □

Corollary 20. The map C_{n} restricts to a map

C_{n} ⟳ {quasi-hereditary algebras with a self-dual tilting bimodule}. □

4. Schur algebras

Let M denote the algebra of n × n matrices over F. Recall the Schur algebra S(n, r) is defined to be the subalgebra (M^{⊗r})^{Σ_{r}} of fixed points under the action of the symmetric group Σ_{r} on M^{⊗r}. The category of representations of S(n, r) can be identified with the category of polynomial representations of GL_{n}(F), of degree r [9].

Let Λ^{+}(n, r), the set of partitions of r with n parts or fewer, given the dominance ordering. The algebra S(n, r) is quasi-hereditary with respect to the poset Λ^{+}(n, r). We write ξ_{λ} ∈ S(n, r) for Green’s idempotents in S(n, r), for λ ∈ Λ(n, r).

In this paper, we are only concerned with S(2, r), but it will be useful to recall some facts about Ringel duality which hold for general n.

Lemma 21. (S. Donkin, [7], 4.1) Let n ≥ r. Then ⋀^{r}(M) is a tilting S(n, r)- S(n, r)-bimodule. □

When n ≥ r, let S′(n, r) = S(n, r). When n < r, let

S′(n, r) ≅ S(r, r)/ ∑_{λ∉Λ(n,r)} S(r, r)ξ_{λ}S(r, r).

The algebras S(n, r), S′(n, r) are Ringel dual.

The Schur algebra possesses a natural anti-automorphism inherited from the transpose operator on M. We also call this antiautomorphism the transpose operator, and denote by s^{T} the twist of an element s by the transpose operator. Since ξ_{λ}^{T} = ξ_{λ} for all λ, the transpose operator descends to an antiautomorphism of S′(n, r)

If A is an algebra, endowed with an antiautomorphism x, then given any left/right A-module M, we define the right/left A-module M^{op} to be that obtained by twisting the action of A on M by x. If A_{1}, A_{2} are algebras, endowed with antiautomorphisms x_{1}, x_{2}, then given an A_{1}-A_{2}-bimodule M, we define the A_{2}-A_{1}-bimodule M^{op} to be that obtained by twisting the actions of A_{1}, A_{2} on M by x_{1}, x_{2}.

Lemma 22. Let _{S(n,r)}T_{S′(n,r)} be a tilting bimodule. Then T^{op} ≅ T^{∗}, as S′(n, r)- S(n, r)-bimodules, where T^{op} is obtained after twisting T by the transpose antiautomorphisms of S′(n, r), S(n, r).

Proof. In case n ≥ r, we have T ≅ ⋀^{r}(M). However, it is well known that ⋀^{r}(M) is self-dual, which is to say ⋀^{r}(M)^{op} ≅ ⋀^{r}(M)^{∗}.

The case n < r follows by truncation from the case n = r. Indeed, in this case, we have T = (∑_{λ∈Λ(n,r)} ξ_{λ}) ⋀^{r}(M). Since ξ_{λ}^{T} = ξ_{λ}, this bimodule is also self-dual. □

We now restrict our study to the case n = 2. Suppose F is a field of characteristic p > 0. Let S = S(2, r) be the Schur algebra over F, where r = ap^{k}−2 or r = ap^{k}−3 for some k ≥ 1 and 2 ≤ a ≤ p. Along with the cases r < p^{2} and r = ap^{k} − 1, these are exactly the Schur algebras which are Ringel self-dual ([8], Theorem 27). Furthermore, S(2, ap^{k} − 1) is Morita equivalent to S(2, ap^{k} − 3) ⊕ F ([8], Corollary 2).

If r is odd, our index set Λ for the quasi-hereditary structure of S consists of all odd natural numbers up to r; if r is even, it consists of all even natural numbers up to r, including 0.

The following definitions assume p odd. If r is odd, let A = S(2, p^{k} − 2) and if r is even, let A = S(2, p^{k} − 3). We define subsets I_{j}, for 1 ≤ j ≤ a, of Λ as follows:

RATIONAL REPRESENTATIONS OF GL_{2}. 13

r odd
j odd I_{j} = {λ ∈ Λ | (j − 1)p^{k} + 1 ≤ λ ≤ jp^{k} − 2}
j even I_{j} = {λ ∈ Λ | (j − 1)p^{k} ≤ λ ≤ jp^{k} − 3}
r even
j odd  I_{j} = {λ ∈ Λ | (j − 1)p^{k} ≤ λ ≤ jp^{k} − 3}
j even I_{j} = {λ ∈ Λ | (j − 1)p^{k} + 1 ≤ λ ≤ jp^{k} − 2}

In case p = 2 (and thus necessarily a = 2), let A = S(2, 2^{k} − 3) if r is odd, and A = S(2, 2^{k} − 2) if r is even. We define subsets I_{j}, for j = 1, 2, of Λ as follows:

r odd

j = 1  I_{j} = {1, 3, …, 2^{k−1} − 3}
j = 2 I_{j} = { 2^{k−1} + 1, 2^{k−1} + 3, …, 2^{k} − 3}
r even
j = 1  I_{j} = {0, 2, …, 2^{k−1} − 2}
j = 2  I_{j} = {2^{k−1}, 2^{k−1} + 2, …, 2^{k} − 2}

Let us define I_{0} := Λ \ ( ⋃_{1≤j≤a}I_{j}).

For 1 ≤ j ≤ a, set b_{j} := min{I_{j}}, r_{j} := max{I_{j}}

We choose orthogonal idempotents {e_{λ}}_{λ∈Λ} in S, such that S ≅ ⨁_{λ,μ∈Λ} e_{λ}Se_{μ}, and S/J(S) = ⨁_{λ∈Λ} M_{λ} is a direct sum of matrix rings M_{λ} over F, where e_{λ} is the unit of M_{λ}.

Let f_{j} := ∑_{λ∈I_{j}} e_{λ}, where e_{λ} ∈ S is the primitive idempotent corresponding to the projective P (λ). Let ε_{j} = ∑_{i≥j} f_{i}.

By work of A. Henke and S. Koenig, there are idempotents η_{j} ∈ S (denoted ξ_{l}^{o} in [11]), and explicit isomorphisms Φ_{j} : A → η_{j}Sη_{j}/η_{j}Sη_{j+1}Sη_{j}, for 1 ≤ j ≤ a ([11], Theorem 3.3).

We now assume that the idempotents e_{λ} are chosen in such a way that e_{λ}η_{j} = η_{j}e_{λ}, for λ ∈ Λ, 1 ≤ j ≤ a. It therefore follows that the idempotents η_{j} commute with f_{i}, ϵ_{i} as well, and we have ϵ_{k}η_{j} = η_{k}, for 1 ≤ j ≤ k ≤ a

Let α_{j} := f_{j}Sf_{j}/f_{j}Sf_{j+1}Sf_{j}, for 1 ≤ j ≤ a.

Lemma 23. The algebra α_{j} is Morita equivalent to A, for 1 ≤ j ≤ a. We have f_{j}Sf_{i} = 0 unless j − 1 ≤ i ≤ j + 1, and

\begin{aligned}& S = ⨁_{j=1}^{a} f_{j}Sf_{j} ⊕ ⨁_{j=1}^{a−1}(f_{j}Sf_{j+1} + f_{j+1}Sf_{j}) ⊕ ⨁_{λ∈I_{0}} e_{λ}Se_{λ}. \\ (1) \end{aligned}

Proof: From the decomposition matrix of S, we see that f_{j}Sf_{i} = 0 unless j − 1 ≤ i ≤ j + 1 and that for λ ∈ I_{0}, e_{λ}Se_{μ} = e_{μ}Se_{λ} = 0 unless μ = λ when it is isomorphic to F. Hence

α_{j} = f_{j}Sf_{j}/f_{j}Sf_{j+1}Sf_{j} ≅ ε_{j}Sε_{j}/ε_{j}Sε_{j+1}Sε_{j},

This algebra is Morita equivalent to η_{j}Sη_{j}/η_{j}Sη_{j+1}Sη_{j}, which is isomorphic to A. This completes the proof of the lemma. □

Remark 24 It will be important to us that the Henke-Koenig isomorphism Φ_{j} between A and η_{j}Sη_{j}/η_{j}Sη_{j+1}Sη_{j} is compatible with the transpose operators on S, A. To be more explicit, η_{j}^{T} = η_{j}, and Φ_{j}(a^{T}) = Φ_{j}(a)^{T}, for a ∈ A.

Lemma 25. We have f_{a}Sf_{a−1}Sf_{a} = 0.

Proof. This is a reformulation of [8], Proposition 25. Indeed, according to this proposition, Sf_{a−1}Sf_{a} is the submodule of Sf_{a} consisting of all composition factors of the form L(λ), λ ∈ I_{a−1}, implying

f_{a}Sf_{a−1}Sf_{a} ≅ Hom_{S}(Sf_{a}, Sf_{a−1}Sf_{a}) = 0. □

Lemma 26. X_{j} = f_{j}Sf_{j+1} is an α_{j}-α_{j+1}-tilting bimodule.

Proof. By lemmas 23 and 25, the f_{j}Sf_{j}-f_{j+1}Sf_{j+1}-bimodule X_{j} is in fact an α_{j}-α_{j+1}-bimodule. It remains to show that _{α_{j}}X_{j} is a full tilting module, and End_{α_{j}}(X_{j}) = α_{j+1}.

By the same argument as in lemma 23 we can reduce to the case where a = 2 by considering all modules for the subalgebra ε_{j}Sε_{j}/ε_{j}Sε_{j+2}Sε_{j}. So let S = S(2, r) where r ∈ {2p^{k} −2, 2p^{k} −3} and use the notation from above. We need to show that f_{1}Sf_{2} ∈ S/Sε_{2}S -mod is a tilting module. But by [8], Proposition 25, Sf_{1}Sf_{2} ⊆ Sf_{2} is the submodule consisting of all composition factors of the form L(λ) for λ ∈ I_{1} and is isomorphic to the full tilting module for S(2, max{I_{1}}). But by the first of these facts the action factors over α_{1} = S/Sε_{2}S ≅ S(2, r_{1}), so it is a full tilting module for this algebra.

Now we have a canonical map from α_{2} = f_{2}Sf_{2} to End_{α_{1}}(f_{1}Sf_{2}). Given the fact that A is Ringel self-dual, we know that α_{2}, A, and End_{α_{1}}(f_{1}Sf_{2}) are Morita equivalent, thus α_{2} and End_{α_{1}}(f_{1}Sf_{2}) are isomorphic. It therefore suffices to prove injectivity of this map. So, suppose it has a nontrivial kernel. This is equivalent to the existence of an endomorphism ϕ of Sf_{2}, annihilating all composition factors of the form L(λ) for λ ∈ I_{1} (namely Sf_{1}Sf_{2}). But all composition factors of the socle of Sf_{2} are of the form L(λ) for λ ∈ I_{1}, by [8], Lemma 3, and thus im ϕ∩soc Sf_{2} = 0 forcing ϕ to be zero. □

Remark 27 Note that it follows from the proof of the lemma that f_{j}Sf_{j+1}Sf_{j} is the annihilator of f_{j}Sf_{j+1} in f_{j}Sf_{j}. Since by Remark 23 f_{j−1}Sf_{j}Sf_{j+1} ⊆ f_{j−1}Sf_{j+1} = 0, it follows that f_{j}Sf_{j−1}Sf_{j} ⊆ f_{j}Sf_{j+1}Sf_{j}.

Let X̄_{j} = f_{j+1}Sf_{j}. By lemmas 23 and 25, X̄_{j} is an α_{j+1}-α_{j}-bimodule.

Let X_{j}^{op} be the α_{j+1}-α_{j}-bimodule obtained by passing _{α_{j}}X_{jα_{j+1}} via the established Morita equivalences to the category of A-A-bimodules, twisting on both sides by the transpose automorphism of A, and then passing via Morita equivalence to the category of α_{j+1}-α_{j}-bimodules.

Lemma 28. There is an isomorphism of α_{j+1}-α_{j}-bimodules, X̄_{j} ≅ X_{j}^{op}.

Proof.

We have

X_{j} = f_{j}Sf_{j+1} ≅ ϵ_{j}§ϵ_{j+1}/ϵ_{j+1}Sϵ_{j+1}.

This passes, via Morita equivalence, to the A-A-bimodule

η_{j}Sf_{j} ⊗_{f_{j}Sf_{j}} f_{j}Sf_{j+1} ⊗_{f_{j+1}Sf_{j+1}} f_{j+1}Sη_{j+1} ≅

η_{j}f_{j}Sf_{j+1}η_{j+1} ≅ η_{j}Sη_{j+1}/η_{j}ϵ_{j+1}Sη_{j+1}.

Since twisting by the transpose operator exchanges the irreducible modules L(λ), L^{r}(λ), the projective S-modules Sf_{j} and Sf_{j}^{T} are isomorphic. We therefore have

X̄_{j} = f_{j+1}Sf_{j} = f_{j+1}Sf_{j+1}^{T}Sf_{j}^{T} Sf_{j} ≅

f_{j+1}Sf_{j+1}^{T} ⊗_{ϵ_{j+1}^{T}Sϵ_{j+1}^{T}} (ϵ_{j+1}^{T}Sϵ_{j}^{T} /ϵ_{j+1}^{T}Sϵ_{j+1}^{T}) ⊗_{ϵ_{j}^{T} Sϵ_{j}^{T}} f_{j}^{T} Sf_{j}.

This passes, via Morita equivalence, to the A-A-bimodule

η_{j+1}Sf_{j+1}^{T} ⊗_{f_{j+1}^{T}Sf_{j+1}^{T}} f_{j+1}^{T}Sf_{j}^{T} ⊗_{f_{j}^{T} Sf_{j}^{T}} f_{j}^{T} Sη_{j+1} ≅

η_{j+1}f_{j+1}^{T}Sf_{j}^{T} η_{j} ≅ η_{j+1}Sη_{j}/η_{j+1}Sϵ_{j+1}^{T}η_{j}.

Applying the transpose anti-automorphism to S, we exchange the bimodules η_{j}Sη_{j+1}/η_{j}ϵ_{j+1}Sη_{j+1} and η_{j+1}Sη_{j}/η_{j+1}Sϵ_{j+1}^{T}η_{j}, the left and right actions being twisted by the transpose operator. However, the transpose operator is compatible with the Henke-Koenig isomorphisms, and therefore an equivalent statement is that passing to the opposite exchanges X_{j} and X̄_{j}. We therefore have X̄_{j} ≅ X_{j}^{op}, as required. □

Let us define

N := ∑_{j=1}^{a−1}(f_{j}Sf_{j+1} + f_{j+1}Sf_{j} + f_{j}Sf_{j+1}Sf_{j}),

N_{2} := ∑_{j=1}^{a−1} f_{j}Sf_{j+1}Sf_{j}.

Proposition 29. We have a filtration by of S by ideals,

\begin{aligned}& S ⊃ N ⊃ N^{2} ⊃ 0. \\ (2) \end{aligned}

Furthermore N^{2} = N_{2}, and N^{3} = 0. We have isomorphisms of S-S-bimodules,

S/N ≅ ⨁_{1≤j≤a}α_{j} ⊕ ⨁_{λ∈I_{0}} e_{λ}Se_{λ},

N /N^{2} ≅ ⨁_{1≤j≤a−1} (X_{j} ⊕ X_{j}^{∗}) ,

N^{2} ≅ ⨁_{1≤j≤a−1}α_{j}^{∗}.

Proof. The first statement as well as N^{2} = N_{2} and N^{3} = 0, are easily verified using Lemma 23, Lemma 25 and Remark 27. From (1) we see that

S/N ≅ ⨁_{1≤j≤a}f_{j}Sf_{j}/(f_{j}Sf_{j+1}Sf_{j}) ≅

⨁_{1≤j≤a}α_{j} ⊕ ⨁_{λ∈I_{0}} e_{λ}Se_{λ},

and by lemmas 22, 26 and 28,

N /N^{2} ≅ ⨁_{1≤j≤a−1}(f_{j}Sf_{j+1} + f_{j+1}Sf_{j}) ≅

⨁_{1≤j≤a−1}(X_{j} + X_{j}^{op}) ≅ ⨁_{1≤j≤a−1}(X_{j} + X_{j}^{∗}).

Now all that is left to show is that f_{j}Sf_{j+1}Sf_{j} ≅ α_{j}^{∗}. To see this, note that by repeatedly applying Remark 23

\begin{aligned}f_{j}Sf_{j+1}Sf_{j} & = f_{j}Sε_{r_{j+1}}Sf_{j} ≅ f_{j}Sε_{r_{j+1}} ⊗_{ε_{r_{j+1}} Sε_{r_{j+1}}} ε_{r_{j+1}}Sf_{j} \\ & ≅ f_{j}Sf_{j+1} ⊗_{α_{j+1}} f_{j+1}Sf_{j} \end{aligned}

But

Hom_{F} (f_{j}Sf_{j+1} ⊗_{α_{j+1}} f_{j+1}Sf_{j}, F) ≅ Hom_{mod- α_{j+1}}(X_{j}, X_{j}^{op∗})
≅ End_{mod- α_{j+1}}(X_{j}) ≅ α_{j},

thus f_{j}Sf_{j+1}Sf_{j} ≅ α_{j}^{∗} as claimed. □

Let C_{1}^{a} = C_{1}^{a}(A) be the algebra obtained by applying the construction C_{1}^{a} of the previous chapter to the algebra A, and its self-dual bimodule T.

Theorem 30. The graded algebra S_{gr} associated to the filtration S ⊃ N ⊃ N^{2} ⊃ 0 is Morita equivalent to C_{1}^{a} ⊕ F^{⊕I_{0}}.

Proof. By Proposition 29, we know that S_{gr} is Morita eqivalent to C̃_{1}^{a} ⊕ F^{⊕I_{0}}, where C̃_{1}^{a} is ℤ-graded, concentrated in degrees 0,1, and 2. In descending vertical order, the components of C̃_{1}^{a} in degrees 0, 1 and 2 are,

⨁_{1≤i≤a} Ã_{i}

⨁_{1≤i≤n−1}(_{i}T̃_{i+1} ⊕_{i}T̃_{i+1}^{∗})

⨁_{1≤i≤n−1} Ã_{i}^{∗},

where Ã_{i} is isomorphic to A, and where _{i}T̃_{i+1} is a tilting Ã_{i}-Ã_{i+1}-bimodule. Twisting the isomorphisms Ã_{i} ≅ A by automorphisms of A if necessary, we may assume that _{i}T̃_{i+1} ≅ _{A}T_{A}. We proceed to piece together an algebra isomorphism between C̃_{1}^{a} and C_{1}^{a} itself.

We know from the proof of the previous proposition that multiplication f_{j}Sf_{j+1}⊗_{F} f_{j+1}Sf_{j} ↠ α_{j}^{∗} is surjective, for 1 ≤ j ≤ a − 1. Therefore, multiplication _{j}T̃_{j+1} ⊗_{F} jT̃_{j+1}^{∗} ↠ Ã_{j}^{∗} is also surjective.

Since we have a canonical isomorphism _{j}T_{j+1} ⊗_{A_{j+1}} _{j}T_{j+1}^{∗} ≅ A_{j}^{∗}, we consequently

obtain an isomorphism Ã_{j}^{∗} ≅ A^{∗} of A-A-bimodules.

We now claim that multiplication _{j}T̃_{j+1}^{∗} ⊗_{F} _{j}T̃_{j+1} ↠ Ã_{j+1}^{∗} is also surjective, for 1 ≤ j ≤ a−2. Equivalently, we claim that multiplication f_{j+1}Sf_{j} ⊗_{F}f_{j}Sf_{j+1} → α_{j+1}^{∗} is surjective. Indeed, this multiplication is inherited from the left module structure on the maximal submodule M of Sf_{j+1} whose composition factors L(λ) respect λ ∈ I_{j+1}. The submodule M has a filtration with submodule α_{j+1}^{∗} and quotient f_{j}Sf_{j+1}. Note that Sf_{j} is a tilting module ([8], Corollary 21, Lemma 24) and therefore self-dual. Therefore M^{op∗} is the maximal quotient of Sf_{j+1} all of whose composition factors L(λ) respect λ ≤ I_{j+1}. M^{op∗} has a filtration with submodule f_{j}Sf_{j+1} and quotient isomorphic to α_{j+1}. However, we know the structure of this module precisely. For instance, the product f_{j}Sf_{j+1} ⊗_{F}α_{j+1} → f_{j}Sf_{j+1} corresponds to the right action of T ⊗A → T. Since the product on M is dual to that on M, the map f_{j+1}Sf_{j} ⊗_{F} f_{j}Sf_{j+1} ↠ Ã_{j+1}^{∗} is dual to the map A ↪ T ⊗ T^{∗}, and is therefore surjective, as required.

We have now proven that f_{i}Sf_{i−1}Sf_{i} = f_{i}Sf_{i+1}Sf_{i}, for 2 ≤ i ≤ a − 1. We therefore have isomorphisms

Ã_{i}^{∗} ≅ _{i}T̃_{i−1} ⊗_{Ã_{i−1}} _{i−1}T̃_{i} ≅ _{i}T̃_{i+1} ⊗_{Ã_{i+1}} _{i+1}T̃_{i} ≅ Ã_{i}^{∗},

of Ã_{i}-Ã_{i}-bimodules. Let us denote this chain of isomorphisms ϕ_{i}. We have

Hom_{A⊗A^{op}}(A^{∗}, A^{∗}) ≅ Hom_{A⊗A^{op}}(A, A) ≅ Z(A),

and thus ϕ_{i} is multiplication by a central element in Ã_{i}. Multiplying the bimodules iT̃_{i+1} by these central elements if necessary, we can assume that in fact ϕ_{i} = 1, for 1 ≤ i ≤ a − 1.

It is now clear that the sum of our bimodule isomorphisms

Ã_{i} ≅ A_{i}, _{i}T̃_{i+1} ≅ _{i}T_{i+1}, _{i}T̃_{i+1}^{∗} ≅ _{i}T_{i+1}^{∗}, Ã_{i}^{∗} ≅ A_{i}^{∗}

defines an algebra isomorphism from C̃_{1}^{a} to C_{1}^{a}, as required. □

5. GL_{2}

In this chapter, we give precise statements of Theorem 1 and Conjecture 2, together with a justification of theorem 1.

The determinant representation of GL_{n}(F) is a polynomial representation of degree n. Therefore, tensoring with the determinant representation defines an exact functor from the category of polynomial GL_{n}(F) representations of degree r to the category of polynomial GL_{n}-representations of degree r+n, carrying simple modules to simple modules. Correspondingly, the Schur algebra S(n, r) can be realised as a quotient of S(n, r + n) by an idempotent ideal S(n, r + n)iS(n, r + n). We denote by S(n, \underline{r}) the inverse limit of the sequence of algebra epimorphisms

S(n, r) ↞ S(n, r + n) ↞ S(n, r + 2n) ↞ ...

The centre Z of GL_{n}(F) is isomorphic to F^{×}, and its group of rational characters is therefore isomorphic to ℤ. The category of rational representations of GL_{n}(F) on which Z acts by the character r ∈ ℤ is naturally equivalent to S(n, \underline{r}) -mod. The category of rational representations of GL_{n}(F) is therefore isomorphic to the module category of ⨁_{r∈ℤ} S(n, \underline{r}).

For any finite dimensional algebra A, the algebra C_{n}(A) has an ideal

⨁_{1≤i≤n−1}(A_{i+1} ⊕_{i}T_{i+1} ⊕_{i}T_{i+1}^{∗} ⊕ A_{i}^{∗}),

the quotient by which is A_{1} ≅ A. In this way, we obtain a sequence of algebra epimorphisms,

A ↞ C_{n}(A) ↞ C_{n}(C_{n}(A)) ↞ ...

We denote by \underleftarrow{C}_{n}(A) the inverse limit of this sequence of maps. The statement of Conjecture 2 is now completely precise:

Conjecture 2. Every block of rational representations of GL_{2}(F) is equivalent to \underleftarrow{C}_{p}(F) -mod.

An equivalent statement is that every block of S(2, \underline{r}) is Morita eqivalent to \underleftarrow{C}_{p}(F). Another is that S ≅ S_{gr}, in the notation of the last chapter.

We now give some corollaries of our work in chapter 4. Let S, N , A, T be as defined there, and let U be an S-S-tilting bimodule.

Lemma 31. We have N U = UN, and N S^{∗} = S^{∗}N.

Proof:

A tilting bimodule for S is given by U = (⨁_{i=1}^{p−1} Sf_{i}) ⊕ T. Thus,

N U = ⨁_{1≤i,j≤p−1} f_{j}Sf_{i} = UN .

We have S^{∗} ≅ (⨁_{i=1}^{p−1} Sf_{i}) ⊕ Sf_{p}^{∗}. Making this identification, we find

N S^{∗} ≅ ( ⨁_{1≤i,j≤p−1} f_{j}Sf_{i}) ⊕ A_{p}^{∗} ≅ S^{∗}N . □

Corollary 32. The space

U_{gr} = ⨁_{i=0,1,2} N^{i}U/N^{i+1}U

is a S-S-tilting bimodule. The space

(S^{∗})_{gr} = ⨁_{i=0,1,2} N^{i}S^{∗}/N^{i+1}S^{∗}

is a S-S-bimodule, isomorphic to (S_{gr})^{∗}. □

By Theorem 30, S_{gr} is Morita equivalent to C_{p}(A) ⊕ F^{⊕I_{0}}, where A is another Ringel self-dual Schur algebra, by induction we obtain the following:

Corollary 33. Then there is a filtration of S by ideals, refining the radical filtration, whose associated graded ring G is Morita equivalent to a direct sum of algebras of the form \underleftarrow{C}_{p}^{d}(F), for d ∈ ℤ_{+}. □

Given r ∈ ℤ_{+}, we choose d ≥ r, such that S = S(2, d) is Ringel self-dual, and d = r (mod 2). We have S(2, r) ≅ S/SjS, for some idempotent j. We define G(2, r) to be G/GjG, where G is the graded ring associated to S by Corollary 33. The algebra G(2, r) is independent of choice of d, and we have algebra epimorphisms

G(2, r) ↞ G(2, r + 2) ↞ G(2, r + 4) ↞ ...

between graded rings G(2, r) = G(S(2, r)) of Schur algebras.

The statement of Theorem 1 is now completely precise. Its truth is clear from Corollary 33.

Theorem 1. Every block of G(2, \underline{r}) is Morita equivalent to \underleftarrow{C}_{p}(F). □

6. Stable equivalence

A deep conjecture of M. Broué predicts that a block of a finite group abelian of abelian defect is equivalent to its Brauer correspondent [3]. R. Rouquier has proved that the snag to an inductive proof of this conjecture is the lifting of a stable equivalence to a derived equivalence; he has also observed an analogy between this and a basic problem in algebraic geometry: proving that birational Calabi-Yau varieties have equivalent derived categories [20], [2]. In this chapter, we prove that the ability to overcome such difficulties would also facilitate a proof of Conjecture 2.

We define here a pair of infinite dimensional self-injective quasi-hereditary algebras L_{1} and L_{2}. We define a stable equivalence between L_{1} and L_{2}, sending simple modules to simple modules. If we could lift this stable equivalence to a Morita equivalence, we would have a proof of Conjecture 2.

For background on triangulated categories, we refer to Neeman’s book [15]. For a concrete approach, and a proof that the stable module category of a self-injective algebra is triangulated, the reader may consult the book of Happel [10].

Let S, A denote the Schur algebras defined in chapter four. Let Q denote the algebra fSf, where f = ∑_{i=1}^{p−1} f_{i}.

Let L_{1} = C = C(A). We now define an algebra L_{2}, by removing a copy of C_{1}^{n} from L_{1}, and gluing a copy of Q in its place.

Let _{0}T_{1} be a tilting A-α_{1}-bimodule, and _{p−1}T_{p} a tilting α_{p−1}-A-bimodule. We have canonical bimodule isomorphisms,

0T_{1}^{∗} ⊗_{A}_{0} T_{1} ≅ α_{1}^{∗}, _{0}T_{1} ⊗_{α_{1}} _{0}T_{1}^{∗} ≅ A^{∗},

p−1T_{p}^{∗} ⊗_{α_{p−1}} _{p−1}T_{p} ≅ A^{∗}, _{p−1}T_{p} ⊗_{A} _{p−1}T_{p}^{∗} ≅ α_{p−1}^{∗}.

We define an algebra L_{2} in the following way: It consists of three subalgebras C_{J′_{0}}, Q and C_{p} glued together with the bimodules _{0}T_{1}, _{0}T_{1}^{∗}, _{p−1}T_{p}, _{p}T_{p−1}^{∗},

L_{2} := ( \begin{matrix}C_{J′_{0}} & _{0}T_{1} & 0 \\ _{0}T_{1}^{∗} & Q & _{p−1}T_{p} \\ 0 & _{p−1}T_{p}^{∗} & C_{p} \end{matrix} )

where the multiplications

C_{J′_{0}} ⊗_{0}T_{1} → _{0}T_{1} _{0}T_{1}^{∗} ⊗ C_{J′_{0}} → _{0}T_{1}^{∗}
0T_{1} ⊗ Q → _{0}T_{1} Q ⊗_{0}T_{1}^{∗} → _{0}T_{1}^{∗}
C_{p} ⊗_{p−1}T_{p}^{∗} → _{p−1}T_{p}^{∗} _{p−1}T_{p} ⊗ C_{p} → _{p−1}T_{p}
p−1T_{p}^{∗} ⊗ Q → _{p−1}T_{p}^{∗} Q ⊗_{p−1}T_{p} → _{p−1}T_{p}

are given by the action of the corresponding quotient A_{i} of the involved algebra R ∈ {C_{J′_{0}}, Q, C_{p}} on the tilting module, the kernel of the surjection R ↠ A_{i} acting as zero. The multiplications between the tilting modules

0T_{1}⊗_{T} _{1}^{∗} → A_{0}^{∗} ⊂ C_{J′_{0}},
0T_{1}^{∗}⊗_{0}T_{1} → α_{1}^{∗} ⊂ Q,
p−1T_{p}⊗_{p−1}T_{p}^{∗} → α_{p−1}^{∗} ⊂ Q,
p−1T_{p}^{∗}⊗_{p−1}T_{p} → A_{p}^{∗} ⊂ C_{p},

are given by the canonical isomorphisms above. All other products between elements of the bimodules are zero. Similarly multiplying elements of two different subalgebras out of the three yields zero.

Let L be either L_{1} or L_{2}. We prove statements concerning both these algebras.

When L = L_{2}, we let f_{i} denote 1_{A_{i}}, for i ∈ ℤ.

When L = L_{2}, we let A_{i} denote the component A_{i} of C_{J′_{0}} for i ≤ 0, the component α_{i} for 1 ≤ i ≤ p − 1, and the component A_{i} of C_{p} for i ≥ p. We also let _{i}T_{i+1} denote the component _{i}T_{i+1} of C_{J′_{0}} for i < 0, the component X_{i} for 1 ≤ i ≤ p − 2, and the component T_{i} of C_{p} for i ≥ p. We let f_{i} denote the idempotent 1_{A_{i}} of C_{J′_{0}} for i ≤ 0, the idempotent f_{i} ∈ S for 1 ≤ i ≤ p − 1, and the idemponent 1_{A_{i}} of C_{p} for i ≥ p.

In either case L = L_{1}, L_{2}, we let B denote that quotient, obtained be factoring out the ideal ⊕_{i∈ℤ}(_{i}T_{i+1}^{∗} ⊕ A_{i}^{∗}). We let B^{t} denote that quotient, obtained be factoring out the ideal ⊕_{i∈ℤ}(_{i}T_{i+1} ⊕ A_{i}^{∗}). The algebras B, B^{t} are both isomorphic to the algebra B = B(A) defined in chapter 3.

Let Λ_{L}^{1} = Λ_{B}^{1}, and Λ_{L}^{2} = Λ_{B}^{2}.

Proposition 34. The algebra L is self-injective, and quasi-hereditary with respect to both Λ_{L}^{1}, and Λ_{L}^{2}.

Proof. Note that for any i, L_{2}f_{i} has the same filtration as Cf_{i} by a submodule B^{∗}f_{i} and a quotient Bf_{i}. Using the same definition of standard modules as for C (which were the same as for B) we see that by the same arguments as for C, L_{2} is quasi-hereditary. The other quasihereditary structure comes from the filtration of C1_{A_{i}} with quotient B^{t}f_{i} and submodule B^{t∗}f_{i}.

To show that L_{2} is selfinjective, consider _{L_{2}}L_{2}^{∗} which has components

( \begin{matrix}C_{J′_{0}}^{∗} & _{0}T_{1} & 0 \\ _{0}T_{1}^{∗} & Q^{∗} & _{p−1}T_{p} \\ 0 & _{p−1}T_{p}^{∗} & C_{p}^{∗} \end{matrix} ) ,

where C_{J′_{0}}^{∗} = ⨁_{i≤0} Hom(C_{J′_{0}}f_{i}, F), and C_{p}^{∗} = ⨁_{i≥p} Hom(C_{p}f_{i}, F). Taking into account the self-injectivity of Q, which holds since projective modules Qf_{i}, 1 ≤ i ≤ p − 1 are tilting, the selfinjectivity of C_{J′_{0}} and C_{p}, which hold since every projective for C is selfdual and this isn’t changed by cutting to a heredity ideal, and the fact that T ≅ T^{∗} for a tilting A-A-bimodule T, we see that this is isomorphic to L_{2} as a left L_{2}-module, proving the claim. □

Lemma 35. L -\underline{mod} is generated by add A_{p}, add A_{p+1}.

Proof. Let T be the smallest triangulated subcategory of L -\underline{mod} containing the subcategories add A_{p}, add A_{p+1}. To prove the lemma, we show by induction on n that all A_{i}-modules are in T , for p + 1 − n ≤ i ≤ p + n. Since A has finite global dimension, we can form a finite projective resolution of length m in A_{i} -mod for every A_{i}-module (i = p, p + 1) M. Then Ω^{m}(M) is in add A_{i}, and by considering triangles stemming from exact sequences in A_{i} -mod we see that Ω^{m−1}M, Ω^{m−2}M, …, M ∈ T . Thus the statement holds in case n = 1. Suppose the statement is true for n = N. There is a triangle of L-modules

A_{i}^{∗} → Ω(A_{i}) → _{i−1}T_{i} ⊕_{i}T_{i+1}^{∗} ⇝

in T for p + 1 − N ≤ i ≤ p + N. Putting i = p + 1 − N, we conclude that p−NT_{p+1−N} ∈ T . Putting i = p + N, we conclude that _{p+N}T_{p+N+1}^{∗} ∈ T . Taking direct summands, we find that add_{p−N}T_{p+1−N}, add_{p+N}T_{p+N+1}^{∗} ⊂ T . Since there is a finite resolution A ↪ U_{1} → ⋯ ↠ U_{r} with U_{j} ∈ add T for all j, we find that the inductive hypothesis is true for n = N + 1, as required. □

Let _{A}T̂_{A} = ⋯ → T̂^{(2)} → T̂^{(1)} → T̂^{(0)} be a projective resolution of the bimodule AT_{A}^{∗} of minimal length. This is finite since A ⊗ A^{op} is finite-dimensional and quasihereditary, and therefore of finite global dimension. Then the total complex of AT̂_{A} ⊗_{A}⋯⊗_{A}_{A}T̂_{A} (k factors) is a projective resolution of _{A}T_{A}^{∗} ⊗_{A}⋯⊗_{A}_{A}T_{A}^{∗} (k factors). Also, for any primitive idempotent e of A, _{A}T̂e is a projective resolution of the indecomposable summand T^{∗}e of the tilting module.

Let q : L ↠ B be the algebra epimorphism of L onto B, which factors out the ideal ⨁_{i∈ℤ}(_{i}T_{i+1}^{∗} ⊕ A_{i}^{∗}). The endomorphism ring of Bf_{i} is A_{i}. Let K = ker(q). We have an isomorphism Kf_{i} ≅ Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}^{∗} of (L, A_{i})-bimodules, coming from the right multiplication with _{i}T_{i+1}.

Given a complex of A_{i}-modules C, we define Lf_{i} ⊙_{A_{i}} C to be some complex of projective L-modules such that A_{i} ⊗_{L} Lf_{i} ⊙_{A_{i}} C ≅ C. Such a complex exists, by the usual lifting argument, but is not necessarily unique. However, Lf_{i} ⊙_{A_{i}} C contains a canonical subcomplex isomorphic to Kf_{i} ⊗_{A_{i}} C, the quotient by which is isomorphic

to Bf_{i} ⊗_{A_{i}} C.

Lemma 36. Every direct summand of _{L}Bf_{i} has infinite projective dimension.

Proof Let e be any idempotent which is a summand of f_{i}. We manufacture an infinite projective resolution of _{L}Be, whose components in degree >> 0 are components of Lf_{j}, for j >> 0. This allows us to prove that given any m >> 0, there exists a simple A_{j}-module L, for some j >> i, such that Ext^{m}(Be, L) ≠ 0.

We have an isomorphism Kf_{i} ≅ Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}^{∗}. Right multiplication by e gives an isomorphism of L-modules Ke ≅ Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}^{∗}e. So we have short exact sequences

(3)  Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}^{∗} → Lf_{i} → Bf_{i}

and

(4)  Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}^{∗}e → Le → Be.

We thus obtain a natural map of complexes of L-modules,

ϕ_{i} : Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i} → Lf_{i}

with direct summand

ϕ_{e} : Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e → Le.

Since all modules occurring in the complexes _{i+1}T̂_{i} are projective as left A_{i+1}- modules, and hence Δ-filtered, and

Bf_{i+1} ⊗_{A_{i+1}} − ≅ (A_{i+1} ⊕_{i}T_{i+1}) ⊗_{A_{i+1}} −

is exact on F(Δ), we find that Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i} is quasi-isomorphic to Bf_{i+1} ⊗_{A_{i+1}}

i+1T_{i}, and Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e is quasi-isomorphic to Bf_{i+1} ⊗_{A_{i+1}} _{i}T_{i+1}e . Therefore, the cone of ϕ_{i} is quasi-isomorphic to Bf_{i} and the cone of ϕ_{e} is quasi-isomorphic to Be.

We have an exact sequence of complexes of L-modules

Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i} → Lf_{i+1} ⊙_{A_{i+1}} _{i+1}T̂_{i} → Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i},

again with direct summand

Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e → Lf_{i+1} ⊙_{A_{i+1}} _{i+1}T̂_{i}e → Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e.

These are exact in every degree, since the sequence in a given degree is obtained by tensoring the short exact sequence (3) with the corresponding entry of _{i+1}T̂_{i} (resp. _{i+1}T̂_{i}e), which is flat as a left A_{i+1}-module. Therefore Bf_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e is quasi-isomorphic to the total complex of

Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i} → Lf_{i+1} ⊙_{A_{i+1}} _{i+1}T̂_{i}e.

We now claim that Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e is quasi-isomorphic to the total

complex of Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T̂_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e. Indeed, Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T_{i+1} is quasi-

isomorphic to Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T̂_{i+1} by the above, and since every module occurring in

i+1T̂_{i}e is projective as a left A_{i+1}-module and every module occurring in Bf_{i+2} ⊗_{A_{i+2}}

i+2T̂_{i+1} is projective as a right A_{i+1}-module, the rows and columns in the double complex are exact, proving the claim.

We now know that Be is quasi-isomorphic to the total complex of

Bf_{i+2} ⊗_{A_{i+2}} _{i+2}T̂_{i+1} ⊗_{A_{i+1}} _{i+1}T̂_{i}e → Lf_{i+1} ⊙_{A_{i+1}} _{i+1}T̂_{i}e → Le.

Iterating this procedure, we obtain a projective resolution P_{.}(Be) of _{L}Be, with a filtration whose sections are isomorphic to

Lf_{j} ⊙_{A_{j}} _{j}T̂_{j−1} ⊗_{A_{j−1}} _{j−1}T̂_{j−2} ⊗_{A_{j−2}} ⋯ ⊗_{A_{i+1}} _{i+1}T̂_{i}e[i − j],

as L-modules for j ≥ i.

We now claim that for every m ≥ 0, there exists an irreducible L-module L such that Ext_{L}^{m}(Be, L) ≠ 0. From the projective resolution above we see that

P_{m}(Be) ≅

(Lf_{i+m} ⊙ (T̂^{(0)})^{⊗m}e) ⊕ ( ⨁_{\begin{matrix}k<m \\ ∑_{j=1}^{k} r_{j}=m−k \end{matrix}}Lf_{i+k} ⊙ T̂^{(r_{1})} ⊗ ⋯ ⊗ T̂^{(r_{k})}e) ,

where all tensors are taken over A_{i}, for some i. Now choose an irreducible A_{i+m}- module L in the head of (Lf_{i+m} ⊙ (T̂^{(0)})^{⊗m})e. Then Hom_{L}(P_{m}(Be), L) ≠ 0 but Hom_{L}(P_{k}(Be), L) = 0 for all k < m. Furthermore

Hom_{L}(P_{m+1}(Be), L) =

m−1 ; ⨁_{r=0} Hom_{L}(Lf_{i+m} ⊙ (T̂^{(0)})^{⊗r} ⊗ T̂^{(1)} ⊗ (T̂^{(0)})^{⊗m−r−1}e, L).

The map

m−1 ; ⨁_{r=0}Lf_{i+m} ⊙ (T̂^{(0)})^{⊗r} ⊗ T̂^{(1)} ⊗ (T̂^{(0)})^{⊗m−r−1}e → Lf_{i+m} ⊗ (T̂^{(0)})^{⊗m}e,

is not surjective as the cokernel at least has a quotient Lf_{i+m} ⊙ (T^{⊗m})e ≠ 0. Therefore we can choose L such that not every morphism in Hom_{L}(P_{m}(Be), L) comes from a morphism in Hom_{L}(P_{m+1}(Be), L). Hence Ext_{L}^{m}(Be, L) ≠ 0. □

Let L^{+} = L^{I′_{p}}. Thus, L^{+} is a quasi-hereditary quotient of L with poset Ip^{′}. By definition, there is an isomorphism L_{1}^{+} ≅ L_{2}^{+}.

By the theory of E. Cline, B. Parshall, and L. Scott ([5], Theorem 3.9), we have an embedding of derived categories

j : D^{b}(L^{+}) → D^{b}(L).

By a theorem of Rickard [17], we have a Verdier quotient of triangulated categories,

π : D^{b}(L) → L -\underline{mod},

whose kernel is the thick subcategory of perfect complexes. Note that even though his theorem only includes finite-dimensional self-injective algebras, the same proof goes through in the locally finite-dimensional case.

Proposition 37. The composition

D^{b}(L^{+}) →^{j} D^{b}(L) →^{π} L -\underline{mod}

has dense image, and kernel T , where T is the thick subcategory of D^{b}(L^{+}) generated by {L^{+}f_{i}, i > p}.

Proof. In the above composition of functors, M ∈ L^{+} -mod ⊂ D^{b}(L^{+}) maps to the isomorphism class of M in L -\underline{mod}. Considering add A_{p}, add A_{p+1} ⊂ L^{+} -mod, and applying Lemma 35, we see that the image is indeed dense.

It is obvious that T is contained in the kernel, since for i < 0, the projectives for L1_{A_{i}} and L^{+}1_{A_{i}} are the same, so under the inclusion j bounded complexes in projectives from L^{+}1_{A_{i}} (i < 0) map to bounded complexes in projectives for L, which become isomorphic to zero under π.

Suppose that E_{.} is a bounded complex of projective modules in K^{b}(L^{+}) of minimal length, such that E_{.} ∉ T . Therefore, some direct summand of L1_{A_{0}} occurs in E_{.}. By cutting in the “stupid way” and shifting in degree we may assume that E_{0} ≠ 0 is a direct sum of summands of L^{+}1_{A_{0}}, and E_{i} = 0, for i > 0. The image of E_{.} under j in D^{b}(L) ≅ K^{−,b}(L -proj) does not have a presentation as a finite complex of projective L-modules by Lemma 36, and therefore E_{.} is not contained in the kernel of π. This completes the proof of the proposition. □

Theorem 38. We have a stable equivalence,

L_{1} -\underline{mod} ≅ L_{2} -\underline{mod},

sending simple modules to simple modules.

Proof:

We have L_{1}^{+} ≅ L_{2}^{+}. Therefore, the stable equivalence is immediate from proposition 37. The fact that simple modules correspond to simple modules is obvious on the subcategory of L_{1}^{+}-modules. For the remaining simples, one proceeds by induction, exactly as in the proof of Lemma 35. □

7. Epilogue

We end with some remarks and open questions.

Remark 39 If there is a stable equivalence of Morita type between two finite dimensional algebras, sending simple modules to simple modules, then a theorem of M. Linckelmann states that the algebras are in fact Morita equivalent [14]. If there is a stable equivalence of Morita type between two finite dimensional algebras, one of which is graded, then a theorem of R. Rouquier states that there is a compatible grading on the other algebra [19]. However, we are unable to apply these results, since the stable equivalence of theorem 38 is not manifestly of Morita type.

Is it the case that any stable equivalence between locally finite dimensional algebras, one of which has a grading refining the radical filtration, and which sends simple modules to simple modules, must lift to a Morita equivalence ? If this were so, then Conjecture 2 would follow.

Remark 40 The problem of finding gradings on modular representation categories is rather a general one, related to the celebrated conjecture of G. Lusztig concerning irreducible characters of algebraic groups (see [12]). For example, one expects blocks of Schur algebras S(n, n) to have a grading refining the radical filtration, at least when the weight of the block is less than p.

We have conjectured that blocks of Schur algebras S(n, n) are all derived equivalent to certain subquotients of a symmetric quasi-hereditary algebra, the Schiver double D_{A_{∞}} (see [21], [22]). The most obvious barrier to a proof of this is the difficulty of finding a grading on the Rock blocks. Conjecture 2 can be thought of as a simple analogue of the Schiver double conjecture, the algebra C_{p}(A) playing a similar role in this paper, to that played by the algebra D_{A_{∞}} in the theory of Rock blocks. Indeed, the development of Conjecture 2 was made, with a view towards understanding the Schiver double conjecture better. We hope the method of defining stable equivalences introduced in this paper may prove useful, as a step towards a proof of the Schiver double conjecture.

Remark 41 It would be interesting if there were analogues of Conjecture 2 for algebraic groups other than GL_{2}. Let us speculate on what features such generalisations might possess.

Suppose that T is a tilting bimodule for A. Then we may think of the pair (A, T) as defining a collection of triangulated categories and exact functors,

... \xymatrix{D^{b}(A -mod) \ar[r]_{α_{1}} & D^{b}(A -mod) \ar[l]_{β_{1}} \ar[r]_{α_{2}} & D^{b}(A -mod) \ar[l]_{β_{2}} \ar[r]_{α_{3}} & D^{b}(A -mod) \ar[l]_{β_{3}}} ...

such that α_{i}β_{i} ≅ β_{i−1}α_{i−1}, for i ∈ ℤ. Here, α_{i} = β_{i} = − ⊗_{A}^{L} T. Passing to Grothendieck groups, we obtain a free representation of the preprojective algebra Π_{∞} on a quiver orienting an infinite line. In other words, the pair (A, T) defines a Π_{∞}-category, which we denote F(A, T) (cf. [1], [4]). The map C_{n} can therefore be thought of as a map

C_{n} ⟳ Π_{∞} -cat,

where Π_{∞} -cat denotes a collection of Π_{∞}-categories, taking F(A, T) to F(C_{n}(A, T)). Passing to Grothendieck groups, we see that

K(F(C_{n}(A, T))) ≅ K(F(A, T))^{⊕n}.

One way to think of the map C_{n} is therefore as a categorification of the functor −^{⊕n} on Π_{∞} -mod. To be more precise, one should define Π_{∞} -cat as a 2-category, and C_{n} as an endo-2-functor of Π_{∞} -cat.

Let q ∈ ℂ^{×} be a p^{th} root of unity. Any block of the quantum group q-GL_{2}(ℂ) is Morita equivalent to C_{1}(ℂ), the Brauer tree algebra on a semi-infinite line ([13], Corollary 7.3). Translation by p embeds a semi-infinite line in itself, and we have a corresponding algebra monomorphism from C_{1}(ℂ) to itelf, related to Steinberg’s tensor product theorem on q-GL_{2}(ℂ). By composition, we obtain a sequence of embeddings,

C_{1}(ℂ) ↪ C_{1}(ℂ) ↪ C_{1}(ℂ) ↪ ...,

whose direct limit is C(ℂ), the Brauer tree algebra on an infinite line. The preprojective algebra Π_{∞} is the Koszul dual of C(ℂ).

To explore the modular representation theory of blocks of a reductive algebraic group G(F), as we have done in this paper in case G = GL_{2}, one should perhaps first look for Koszul duals of direct limits of blocks of the corresponding quantum group q-G(ℂ), before looking to define categories over these Koszul duals, and categorifications of functors between their module categories.

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Vanessa Miemietz, Universität zu Köln, Mathematisches Institut, Weyertal 86-90, D-50931 Köln, Germany.

Will Turner, Department of Mathematics, University of Oxford, Oxford, England. Email: turnerw@maths.ox.ac.uk