Braid groups as self-equivalences of derived categories

Alexander Zimmermann^{∗}

Faculté de Mathématiques, Université de Picardie Jules Verne, 33 rue St Leu, F-80039 Amiens, FRANCE

alexander.zimmermann@u-picardie.fr

1. Introduction

Derived categories of algebras become more and more the appropriate working place of representation theorists. In the representation theory of groups this interest arose from various places notably from Broué’s conjecture which states in its simplest form that for a big enough field k of characteristic p the derived categories of the principal block of the group ring kG of the finite group G and that of the principal block of the group ring kN_{G}(P) of the normalizer N_{G}(P) of an abelian p-Sylow subgroup P of G are equivalent as triangulated categories. Though this is our main motivation at the moment, there are more situations which we would like to recall. There is the development of character sheaves which was pushed very far by Lusztig, Shoji and others to describe generically the representation theory of algebraic groups. Moreover, many relationships between commutative or non commutative geometries to the representation theory of algebras has been discovered. This relationship is made precise and can be formulated by an equivalence between various corresponding derived categories. In these notes we shall be concerned with consequences of the established or conjectured equivalences. In particular we work on symmetries of the representation theory which are implied by the existence of the derived category in the background. To be more precise for R a commutative ring we shall study the group of self-equivalences of the derived category of an R-algebra which is projective as an R-module. This group often is quite rich as we established in joint work with Raphaël Rouquier in (RZ-96). In fact, braid groups seems to play an important rôle in the structure of these groups. Parallel developments by Khovanov, Lenzing, Meltzer, Miyachi, Polishchuk, Seidel, Thomas, Yekutieli and others also indicate in parts that for nice situations braid group symmetries arise. In the present note we shall review some of these developments. Furthermore we shall compute the case of the principal block of the group ring of a group with Klein Four Sylow subgroup. There, a generalized braid group of type Ã_{2} occurs. Moreover, we explain how these self-equivalences of the derived category of a principal block of a group ring RG act on the cohomology ring H^{∗}(G, R) in a functorial way. Furthermore, we study this action for the case of the Klein Four Sylow subgroup, and in particular for the alternating group 𝔄_{4}.

2. Generalities on equivalences between derived categories

We recall some facts about equivalences between derived categories as is developed in Rickard (Ric-89; Ric-91; Ric-96) and Keller (Ke-93). For the notations we refer to (KZ-98).

2.1. The algebraic setup

Jeremy Rickard proved in his fundamental work a necessary and sufficient criterion for two derived categories of rings to be equivalent. In the situation we study here this criterion is as follows.

THEOREM 1 (Jeremy Rickard (Ric-91), Bernhard Keller (Ke-93)). Let S be a commutative ring and let A be a finite dimensional S-algebra which is projective as S-module. Then, in case B is another

∗ These notes are an extended version of the talk I gave at the Euroconference ”Rings, Modules and Representations“ in Constanţa in August 2000. I acknowledge support by the CCCI.

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.afourkluwer.tex; 8/11/2000; 9:41; p.1

S-algebra and D^{b}(A) ≃ D^{b}(B) as triangulated categories, there is a complex X ∈ D^{b}(B ⊗_{S} A^{op}) so that F_{X} := X ⊗_{A}^{𝕃} − is a functor inducing an equivalence.

We call functors standard if they are of the form F_{X} and if they induce an equivalence D^{b}(A) ≃ D^{b}(B). In case F_{X} is a standard equivalence, we call X a twosided tilting complex.

(2.1) Recall that an S-algebra A for an integral domain S with field of fractions K is an S-order if first A is finitely generated projective as S-module and second K ⊗_{S} A is a semisimple K-algebra. An S-algebra A is symmetric provided Hom_{S}(A, S) ≃ A as A ⊗_{S} A^{op}-modules. Suppose now S is a field and A an arbitrary S-algebra or S is a Dedekind domain and A is a symmetric S-order. Then if D^{b}(A) ≃ D^{b}(B), also B is projective as S-module (Z-99a) and a quasi-inverse of F_{X} is standard as well.

(2.2) We keep the assumption that A is an S-projective S-algebra. We define (RZ-96)

T rP ic_{S}(A) := { isomorphism classes [X] | X ∈ D^{b}(A ⊗_{S} A^{op}) is a twosided tilting complex}.

Then, T rP ic_{S}(A) is a group under the group law [X] · [Y] := [X ⊗_{A}^{𝕃} Y]. This group contains the classical Picard group of A (cf (Ba-68)), with it also the group of automorphisms of A modulo inner automorphisms of A, and an infinite cyclic central subgroup generated by shift in degrees.

2.2. The geometric setup

There is a geometric analogue to this construction. Let X and Y be algebraic varieties of the same dimension and let D^{b}(X) be the derived category of bounded complexes of coherent sheaves on X and likewise D^{b}(Y) be the derived category of bounded complexes of coherent sheaves on Y.

(2.3) Let P be a bounded complex of coherent sheaves on X × Y. Denote by p_{X} the projection of X × Y on X and by p_{Y} the projection of X × Y on Y. Then, in case

F_{P} (−) := (p_{X})_{∗}(P ⊗_{O_{X×Y}}^{𝕃} (p_{Y})^{∗}−) : D^{b}(Y) ⟶ D^{b}(X)

is an equivalence of categories, this equivalence is called ((M-81; Brid-99; P-95)) Fourier-Mukai transformation. If X is a point, then a Fourier-Mukai transformation is a standard equivalence.

(2.4) Tom Bridgeland gave in (Brid-99) some partial analogue to Rickard’s theorem. For this suppose that X and Y are smooth projective varieties of the same dimension and that P is a vector bundle on X × Y. The bundle P is called strongly simple over Y if for each y ∈ Y the bundle P_{y} on X is simple and if Ext_{X}^{i}(P_{y_{1}}, P_{y_{2}}) = 0 for any (y_{1}, y_{2}) ∈ Y × Y and any i.

THEOREM 2 (Tom Bridgeland (Brid-99)). The functor F_{P} is fully faithful if and only if P is strongly simple over Y. The functor F_{P} is an equivalence precisely when one also has P_{y} ≃ P_{y} ⊗ ω_{X} for all y ∈ Y.

Theorem 2 is some sort of analogue to the theorem of Rickard and Keller in case the twosided tilting complex is just a module. As far as I know there is no general theory like Rickard’s for the geometric situation.

3. Some results on braid groups

We recall briefly some facts about braid groups.

(3.1) Let W be a real reflection group generated by orthogonal reflections {s_{1}, …, s_{n}} =: S on some hyperplanes {H_{1}, … H_{n}} in some real euclidean space E. We associate to (W, S) a graph Γ_{(W,S)}. The vertices of Γ_{(W,S)} are the elements of S. There is an edge with weight m_{i,j} between s_{i} and s_{j} if and only if s_{i} and s_{j} do not commute and the order of s_{i}s_{j} is m_{i,j} + 2. It is well known that W is finite if and only if Γ_{(W,S)} is a Dynkin diagram.

(3.2) Now, W acts on the arrangement of hyperplanes and also on the complement. It is a result of Brieskorn (Brie-71) that π_{1}((ℂ ⊗_{ℝ} E \ ⋃_{i=1}^{n} ℂ ⊗_{ℝ} H_{i})/W) =: Braid(Γ_{(W,S)}) is a group with

presentation Braid(Γ_{(W,S)}) =< t_{1}, …, t_{n}| \underbrace{t_{i}t_{j}t_{i} …}_{m_{i,j}+2 factors} = \underbrace{t_{j}t_{i}t_{j} …}_{m_{i,j}+2 factors} > . The case of Dynkin

diagram Γ_{(W,S)} = A_{e} is due to Arnold while the other cases are due to Brieskorn.

(3.3) This item was communicated to me by Claudio Procesi. We shall be concerned in particular with the braid group associated to the Dynkin diagram A_{2} (which is just a graph with one edge and 2 vertices) and the affine Dynkin diagram Ã_{2} (which is the unique graph with three edges and three vertices, each vertex being adjacent to two edges). Then Br(A_{2}) =< s, t| sts = tst > and Br(Ã_{2}) =< s_{1}, s_{2}, s_{3}| s_{i}s_{j}s_{i} = s_{j}s_{i}s_{j}∀ i, j ∈ {1, 2, 3} > . The mappings Br(A_{2}) →^{α} Br(Ã_{2}) defined by α(s) = s_{1} and α(t) = s_{2} as well as the mapping Br(Ã_{2}) ⟶^{β} Br(A_{2}) given by β(s_{1}) = s, β(s_{2}) = t and β(s_{3}) = t are group homomorphisms. We see immediately β ∘ α = id_{Br(A_{2})}. Hence, Br(Ã_{2}) = kerβ ×Br(A_{2}) .

(3.4) Independently of the existence of a Weyl group we define for any weighted graph Γ a generalized braid group Braid(Γ). The case of complex reflection groups (W, S) is studied by Broué, Malle, Rouquier (BMR-98) and even more involved types of diagrams and associated braid groups have been introduced.

4. Some results on self-equivalences of derived categories

4.1. The abelian situation

The group T rP ic_{S}(A) was first studied in (RZ-96) (see also (Z-96)). As one of the main results we got the following description.

THEOREM 3 (Raphaël Rouquier and Alexander Zimmermann (RZ-96)). Let k be a prime field and let B_{e} be an indecomposable Brauer tree algebra with no exceptional vertex for a Brauer tree with e edges. Then, there is a group homomorphism φ_{e} : Braid(A_{e}) ⟶ T rP ic_{k}(B_{e}) . Moreover, φ_{2} is injective and the image of φ_{2} is normal of index 8. For any e > 1 the image of a standard generator is infinite cyclic.

(4.1) Injectivity is proven by identifying Braid(A_{2})/Z(Braid(A_{2})) ≃ P SL_{2}(ℤ) and one then identifies the images of projective indecomposable modules with the matrix coefficients of elements in P SL_{2}(ℤ). Actually, let P_{+} and P_{−} be the two indecomposable projective B_{2}-modules. For [X] ∈ T rP ic_{k}(B_{2}) let T_{+} := X ⊗_{B_{2}} P_{+} and T_{−} := X ⊗_{B_{2}} P_{−}. Replace T_{+} by the representative in the isomorphism class of T_{+} with the smallest k-dimension and likewise for T_{−}. Then, T_{+} = P_{+}^{n_{+}} ⊕ P_{−}^{n_{−}} and T_{−} = P_{+}^{m_{+}} ⊕ P_{−}^{m_{−}} as modules, forgetting the differentials. Then one proves

that the image of X in P SL_{2}(ℤ) is the matrix ( \begin{matrix}±n_{+} & ±m_{+} \\ n_{−} & m_{−} \end{matrix} ). This description immediately gives

the statement. Surjectivity is proven by using that there are only two simple modules up to isomorphism and any complex has two ends. For more ample details the reader should consult (RZ-96).

(4.2) There is some analogue for orders over a complete discrete valuation domain. There, so-called Green orders take the rôle of Brauer tree algebras and one gets the same situation but replacing B_{e} by the corresponding order Λ_{e}. The interest is that tensoring with the field of fractions gives a very different situation to tensoring obtained by tensoring with the residue field. Some nice arithmetic structure comes into the play. For more ample details the reader might consult (Z-00a).

(4.3) Later, independently, Mike Khovanov and Paul Seidel discovered this homomorphism φ_{e}.

THEOREM 4 (M. Khovanov and P. Seidel (KS-00; ST-00; HK-00)). φ_{e} is injective for all e > 2.

A more general class of algebras, so-called zig-zag algebras, is covered. The proof is very involved and uses sophisticated methods related to Mirror symmetry. In particular one of the key steps is to establish an equivalence of categories between the derived category of the algebra under consideration and some object coming from symplectic geometry. Using Floer cohomology the authors are able to prove injectivity.

(4.4) In the geometrical setting we get some braid group picture as well.

THEOREM 5 (Alexander Polishchuk (P-95)). Let A be a connected abelian variety of dimension g over a field k endowed with a principal symmetric polarisation. Then, Braid(A_{2}) acts on D^{b}(A) by Fourier-Mukai transformations.

The symmetric polarisation is used to identify the variety with its dual. The construction goes along the following lines. One finds some sort of involution and a complex inducing a Fourier-Mukai transformation. Conjugating the complex by the involution one gets a second complex and these two complexes are the braid group generators.

There is one more example in this direction using Fourier-Mukai transformation actions on derived categories.

THEOREM 6 (Raphaël Rouquier (Rou-98)). Let G be a semisimple complex algebraic group with Weyl group W and let B be the variety of Borel subgroups of G. Then, the braid group B_{W} associated to W acts on D^{b}(B) by Fourier-Mukai transformations.

4.2. Non abelian statements

(4.5) Amnon Yekutieli is interested in non communicative geometry and was interested for this reason in algebras with finite global dimension.

THEOREM 7 (Jun-Ichi Miyachi and Amnon Yekutieli (MY-99)). Let A be a hereditary basic finite dimensional k-algebra of finite representation type with quiver Δ. Then, there is a group isomorphism T rP ic_{k}(A) ≃ Aut_{k}(\overrightarrow{ℤ}Δ)^{τ}.

Here, τ = A^{∗}[−1] ⊗_{A}^{𝕃} − and \overrightarrow{ℤ}Δ is a quiver obtained by repetition of Δ (Ried-80). By this, Miyachi and Yekutieli get explicitly the case of Δ being a Dynkin diagram. No braid group arises, though the groups coming up are all of a nice algebraic nature.

(4.6) Helmut Lenzing and Hagen Meltzer study all kinds of properties for a rather special class of algebras, so-called canonical algebras. These are algebras of finite global dimension and are defined by quivers and relations. Moreover, these algebras can serve as a model for a ’weighted projective line’. For details the reader may consult Geigle and Lenzing (GL-87). Recently these algebras came into play by the attempt of Happel and Reiten to classify all abelian categories which are derived equivalent to a hereditary one. Canonical algebras play an important rôle in the list of categories they expect to get. Here is one of the main results on the group of self-equivalences of the derived category of these algebras.

THEOREM 8 (Meltzer and Lenzing (LM-00)). Let X be a weighted projective line of tubular type. Then the group of isomorphism classes of self-equivalences of D^{b}(X) is isomorphic to the group Braid(A_{2}) ×(P ic_{0}(X) ×AutX).

For details on the definitions we refer to (GL-87). It is interesting to see that also in this case the generators for the braid group part are analogous to the case studied in (RZ-96) for Brauer tree algebras with two simple modules.

(4.7) It is surprising to get the braid group also in this non abelian situation. It seems that this happens since the assumption of the weighted projective line to be tubular has some regularizing effect, and so the situation becomes “almost abelian”. In fact for general types the braid groups do not arise and actually the derived category in this case has only relatively few symmetries.

5. Action on the cohomology ring

Let G be a finite group and let R and be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p. In fact in order to establish the theory that follows the assumptions are much too strict on R but the hypotheses on R as above simplify a lot the presentation. Let S be any commutative noetherian ring.

(5.1) In order to study the group T rP ic_{R}(B_{0}(RG)) we shall look for natural modules. Since for any two SG-modules M and N we have

Ext_{SG}^{n}(N, M) ≃ Hom_{D^{b}(SG)}(N, M[n])

it is natural to consider M = N and to look at the subgroup of T rP ic_{S}(SG) that fixes the module M. Then, morally at least, this group should act on Ext_{SG}^{∗}(M, M) in a multiplicative way. Since H^{∗}(G, S) = Ext^{∗}(S, S) is the cohomology ring of the group G with values in S, we are particularly interested in the case M = S. Denote

HD_{S}(G) := {[X] ∈ T rP ic_{S}(SG) | X ⊗_{SG} S ≃ S}

It is reasonable to restrict in case of S = R to the derived category of the principal block. We shall do this without mentioning and hope that this will not produce any confusion. To get an action of HD_{S}(G) on H^{∗}(G, S) there are many obstacles on the way. E.g. there are many isomorphisms we choose on the way, the functors are equivalences and not isomorphisms of categories to cite only a few. But, we get the following first result.

THEOREM 9 ((Z-99b)). Keep the assumptions of this section. Then, the cohomology ring H^{∗}(G, S) is an S HD_{S}(G)-module. This action is compatible with the ring structure of H^{∗}(G, S). If S_{1} ⟶ S_{2} is a ring homomorphism, then this ring homomorphism induces a group homomorphism HD_{S_{1}}(G) ⟶ HD_{S_{2}}(G) and this way the ring homomorphism H^{∗}(G, S_{1}) ⟶ H^{∗}(G, S_{2}) is S_{1} HD_{S_{1}}(G)-linear.

The proof is not only a checking of being well defined, but an actual construction of a bigger group acting properly, and then seeing that the extension that had to be considered actually enlarges the group by something in the kernel of the action.

(5.2) What do we know with respect to functoriality of this structure in the first variable? Here we have to face the problem that derived equivalences behave very badly with respect to subrings, quotient rings, extension rings or similar constructions. There is one exception, the so-called splendid tilting complexes introduced by Jeremy Rickard (Ric-96).

DEFINITION 5.1. (Jeremy Rickard (Ric-96)) Let G and H be finite groups with a common p-Sylow subgroup P and let R be a complete discrete valuation domain of characteristic 0 with residue field of characteristic p or a field of characteristic p. Denote by ΔP a fixed diagonal embedding of P in G × H. Let X be a bounded complex of B_{0}(RG) ⊗_{R} B_{0}(RH)^{op} modules so that

− Each homogeneous component of X is projective as RG-module and projective as RHmodule,

− Each homogeneous component of X is a ΔP -projective p-permutation module (that is a direct factor of a permutation module), − Hom_{RG}(X, X) is homotopic to B_{0}(RH) as complex of bimodules and Hom_{RH}(X, X) is homotopic to B_{0}(RG) as complex of bimodules.

(5.3) I do not know any twosided tilting complex between two principal blocks of group rings which cannot be modified by a Morita equivalence so that the modified complex is splendid.

(5.4) For splendid tilting complexes Rickard proves that the Brauer construction induces equivalences on the local levels. We shall explain this statement below. Let G be a finite group and let k be a field of characteristic p. Then, for any kG-module M and a p-subgroup U of G define the kN_{G}(U)-module M(U) := M^{U}/ ∑_{V <U} T r_{V}^{U}(M^{V}). This construction, the Brauer construction, is functorial and carries over to the homotopy category. It is the right environment to restrict splendid derived equivalences to centralizers of subgroups.

THEOREM 10 (Jeremy Rickard (Ric-96)). Let X be a splendid tilting complex in D^{b}(B_{0}(kG) ⊗_{k} B_{0}(kH)). Then, for any p-subgroup Q of G the complex X(ΔQ) is a splendid tilting complex in D^{b}(B_{0}(kC_{G}(Q)) ⊗_{k} B_{0}(kC_{H}(Q))).

Moreover, Rickard proves that for any splendid tilting complex \overline{X} in D^{b}(B_{0}(kG) ⊗_{k} B_{0}(kH)) there is an, up to isomorphism, unique splendid tilting complex X in D^{b}(B_{0}(RG) ⊗_{R} B_{0}(RH)) with X ⊗_{R} k ≃ \overline{X}.

(5.5) Denote by [X] the isomorphism class of a complex X in the derived category and denote by (X) the isomorphism class of a complex X in the homotopy category. Note that being homotopy equivalent is stronger than being quasi-isomorphic.

(5.6) It is natural to define (Z-01)

SplenP ic_{R}(G) := {(X) | X is a splendid tilting complex in D^{b}(B_{0}(RG) ⊗_{R} B_{0}(RG))}

and similarly for k as coefficient domain. It is readily verified that this in fact is a group under tensor product. It is clear that we get a group homomorphism SplenP ic_{R}(G) ⟶ T rP ic_{R}(B_{0}(RG)) by setting ν( (X) ) := [X]. The same notation is used for k as coefficient domain. In view of the action on the cohomology ring we define

HSplen_{R}(RG) := ν^{−1}(HD_{R}(G)) ∩ SplenP ic_{R}(RG)

and similarly for k instead of R.

(5.7) The action of HSplen_{R}(G) on H^{∗}(G, R) is via φ though. So, working up to homotopy equivalence only or working up to quasi-isomorphism does not make any difference for the action on the cohomology ring of the group.

(5.8) Why do we like to work up to homotopy equivalence only instead of up to quasi-isomorphism? The reason is the following result.

PROPOSITION 5.1 ((Z-01)). Let k be a field of characteristic p, let G be a finite group and let Q be a p-subgroup of G. Then, the Brauer construction provides a group homomorphism SplenP ic_{k}(G) ⟶ SplenP ic_{k}(C_{G}(Q))

If one would work over isomorphism classes in the derived category the Brauer construction is not necessarily well defined. Actually there is no obvious reason why [B_{0}(kG)] = [X] implies [B_{0}(kC_{G}(Q)))] = [X(ΔQ)].

(5.9) Let us study the obstruction. The equation [B_{0}(kG)] = [X] implies that there is a quasiisomorphism B_{0}(kG) ⟶^{s} X (and not necessarily in the other direction) since the homology of X is concentrated in degree 0 and is isomorphic to B_{0}(kG). The mapping cone of s is therefore acyclic (i.e. with homology 0). Form the mapping cone by the standard construction

C(s) := (B_{0}(kG)[1] ⊕ X, ( \begin{matrix}d_{B_{0}(kG)} & s \\ 0 & d_{X} \end{matrix} )) .

Then, it is easily verified that C(s(ΔQ)) = (C(s))(ΔQ) . In order to define the mapping in Proposition 5.1 in the derived category we have to prove that the Brauer functor applied to an acyclic complex is acyclic. Actually this is slightly stronger than what is needed since not every acyclic complex is the mapping cone of a quasi-isomorphism as above. Nevertheless, we shall give an example showing that it is not correct that the Brauer construction applied to an acyclic complex of p-permutation modules is acyclic.

(5.10) Let k be the field with 2 elements. The sequence 0 ⟶ k ⟶ kC_{2} ⟶ k ⟶ 0 coming from the augmentation of the cyclic group of order 2 is exact. Moreover, this exact sequence can be seen as acyclic complex of kC_{2}-permutation modules with three non zero terms. Applying the Brauer functor −(C_{2}) to this complex gives (… ⟶ 0 ⟶ k ⟶ 0 ⟶ k ⟶ 0 ⟶ …) since kC_{2}(C_{2}) = 0 and k(C_{2}) = k. Obviously, this complex is not acyclic.

THEOREM 11 ((Z-01; Z-00b)). Let k be a field of characteristic p, let G be a finite group, P a p-Sylow subgroup and let Q be a p-subgroup of G. Then, for any [X] ∈ HSplen_{k}(G) so that X(ΔQ) ∈ HSplen_{k}(C_{G}(Q)), we get HSplen_{R}(G) ≃ HSplen_{k}(G) and the diagrams

\begin{matrix}H^{∗}(G, k) & ⟶^{X⊗−} & H^{∗}(G, k) \\ res_{C_{G}(Q)}^{G} ↓ & & res_{C_{G}(Q)}^{G} ↓ \\ H^{∗}(C_{G}(Q), k) & ⟶^{X(ΔQ)⊗−} & H^{∗}(C_{G}(Q), k) \end{matrix} and \begin{matrix}H^{∗}(G, k) & ⟶^{X⊗−} & H^{∗}(G, k) \\ ↑ trans_{C_{G}(Q)}^{G} & & ↑ trans_{C_{G}(Q)}^{G} \\ H^{∗}(C_{G}(Q), k) & ⟶^{X(ΔQ)⊗−} & H^{∗}(C_{G}(Q), k) \end{matrix}

are commutative. In case P is abelian, an X ∈ HSplen_{k}(G) ∩ (−(ΔP))^{−1}(HSplen_{k}(C_{G}(P))) acts on H^{∗}(G, k) via automorphisms of P.

(5.11) It should be noted that the first point is not a trivial consequence of Rickard’s result that SplenP ic_{k}(G) ≃ SplenP ic_{R}(G). It has to be shown that the unique lift fixes the trivial module as well. The second relation is essentially proved by first showing that X(ΔQ) ⊗_{C_{G}(Q)} k ≃ X^{G}(Q). The third item uses the second and then as essential ingredient the theorem of Roggenkamp and Scott on the isomorphism problem for integral group rings of p-groups (RS-87; Rog-92).

6. The alternating group of degree 4

We shall give the group ring R𝔄_{4} for R being a complete discrete valuation ring of characteristic 0 and residue field of characteristic 2, so that R has a third root of unity. For this one may as well use the description in (N-98) of R SL_{2}(4) by Gabriele Nebe using that SL_{2}(4) ≃ 𝔄_{5} and the explicit tilting complex inducing a splendid equivalence between the principal block of R SL_{2}(4) and R𝔄_{4} given by Jeremy Rickard in (Ric-96). Nevertheless, a more direct approach is possible as is shown below. Let k be a field of characteristic 2 containing a third root of unity and let R be a complete discrete valuation ring of characteristic 0 containing a third root of unity and with residue field k of characteristic 2. Since Ẑ_{2}[ζ_{3}] is a splitting field for 𝔄_{4} and all of its subgroups, it is enough to compute R𝔄_{4} for R being the (unique) unramified extension of ℤ̂_{2} of degree 2. Let K be the field of fractions of R. Since 𝔄_{4} ≃ (C_{2} × C_{2}) ×C_{3} K is a splitting field for 𝔄_{4}. We use a representation A_{4} ≃< a, b, c : a^{2}, b^{2}, c^{3}, (a, b), a^{c} = ab, b^{c} = a >. Now, ℚ𝔄_{4} is a direct product of a copy of ℚ, a copy of ℚ[ζ_{3}] and a 3 × 3 matrix ring over ℚ. Moreover, K ⊗_{ℚ} ℚ[ζ_{3}] ≃ K × K. We are interested, how R𝔄_{4} is embedded into K𝔄_{4}. Over R, however, we get 3 indecomposable projective modules, over ℤ̂_{2} we get 2 projective indecomposable modules. Corresponding to the irreducible characters χ_{i}, i = 1, 2, 3 of C_{3}, we get the idempotents e_{i} := \frac{1}{3} ∑_{j=1}^{3} χ_{i}(c^{j})c^{j} Over ℤ̂_{2} we get the two indecomposable idempotents χ_{1} and χ_{2} + χ_{3}. We know, that for i ≠ j e_{i}R𝔄_{4}e_{j} is a non zero fractional R–ideal in K and for i = j we know that e_{i}RA_{4}e_{i} is an R–order in K × K.

The central idempotent δ := \frac{1}{4}(1 + a + ab + b) in K𝔄_{4} corresponds to the natural projection onto KC_{3}. This induces a pullback diagram

\begin{matrix}R𝔄_{4} & ⟶ & RC_{3} \\ ↓ & & ↓ \\ RA_{4} · (1 − δ) & ⟶ & (R/4R)C_{3} \end{matrix} .

Λ := R𝔄_{4} · (1 − δ) is an R-order in ( \begin{matrix}K & K & K \\ K & K & K \\ K & K & K \end{matrix} ) . We use the induced representation from a one

dimensional character of the Klein Four group

a ⟶ ( \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , b ⟶ ( \begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and c ⟶ ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) .

Then, we conjugate by ( \begin{matrix}1 & 1 & 1 \\ 1 & ζ_{3}^{2} & −ζ_{3} \\ 1 & −ζ_{3} & ζ_{3}^{2} \end{matrix} ) to get Λ = ( \begin{matrix}R & 2R & 2R \\ 2R & R & 2R \\ 2R & 2R & R \end{matrix} ) . After some computation we

get R𝔄_{4} ≃ {(a_{1}, a_{2}, a_{3}, (b_{i,j})_{1≤i,j≤3}) ∈ R × R × R × Λ | b_{i,i} − a_{i} ∈ 4R ∀1 ≤ i ≤ 3}

7. The braid group generators

We keep the hypotheses of the beginning of section 6. The group ring R𝔄_{4} has three projective indecomposable modules P_{1}, P_{2}, P_{3} up to isomorphism. Denote by P_{1} the projective cover of the trivial module. Note that if R does not have a third root of unity, then there are only two projective indecomposable modules. The discussion in that case differs considerably of what we shall develop here. Denote in order not to overload the notation by A the group ring R𝔄_{4}. We denote by S_{i} the complex

… ⟶ 0 ⟶ P_{i} ⊗ P_{i}^{∗} ⟶ A ⟶ 0 ⟶ …

and the functors F_{i} := S_{i} ⊗_{A} − for any i ∈ {1, 2, 3}. The following result was known to Joe Chuang 1 for k as coefficient domain as action on the simple modules. We shall include a proof for R and this implies at once the result for k.

PROPOSITION 7.1. S_{i} ⊗_{A} S_{j} ⊗_{A} S_{i} ≃ S_{j} ⊗_{A} S_{i} ⊗_{A} S_{j} for any pair (i, j) ∈ {1, 2, 3} × {1, 2, 3}.

Proof. It is obvious by the symmetry of R𝔄_{4} that it is enough to prove the statement for i = 1 and j = 2. We compute S_{1} ⊗P_{1} ≃ P_{1} ⟶ 0, S_{1} ⊗P_{2} ≃ P_{1} ⟶ P_{2} and S_{1} ⊗P_{3} ≃ P_{1} ⟶ P_{3} where the right most position for the complexes on the right of the isomorphism is in degree 0. Analogous results hold for S_{2} instead of S_{1}. Then, S_{2} ⊗_{A} S_{1} ⊗_{A} P_{1} ≃ S_{2} ⊗_{A} P_{1}[1] ≃ P_{2} ⟶ P_{1} ⟶ 0 again with the convention that the right hand side of the complex on the right is meant to be in degree 0.

S_{2} ⊗_{A} S_{1} ⊗_{A} P_{2} ≃ F_{2}(cone(P_{1} ⟶ P_{2})) ≃ cone(F_{2}(P_{1}) ⟶ F_{2}(P_{2})) ≃ P_{1}[1] .

Likewise S_{2} ⊗_{A} S_{1} ⊗_{A} P_{3} ≃ F_{2}(cone(P_{1} ⟶ P_{3})) ≃ P_{2} ⟶ P_{1} ⊕ P_{2} ⟶ P_{3} . Furthermore, F_{1}(P_{2} ⟶ P_{1} ⟶ 0) ≃ P_{2}[2] and F_{1}(P_{1}[1]) ≃ P_{1}[2]. Compute the image of P_{3}:

F_{1}(F_{2}(F_{1}(P_{3}))) ≃ F_{1}(cone((P_{2} → P_{1}) → (P_{2} → P_{3})))
≃ cone(F_{1}(P_{2} → P_{1}) → F_{1}(P_{2} → P_{3}))
≃ cone(P_{2}[1] → (P_{2} ⟶ P_{1} ⊕ P_{2} → P_{3}))
≃  P_{1} ⟶ P_{2} P_{2} ⟶ P_{1} ↘ ↗ P^{3} ⟶ 0 ⟶ . . . . . . → 0 ↗ ↘

1 In September 1999 in a private communication

where in the final complex the differential in degree 2 maps the projective P_{1} in degree 2 to the projective P_{2} in degree 1 and the projective P_{2} in degree 2 to the projective P_{1} in degree 1. In any case the result is invariant under changing indices 1 and 2. By (RZ-96) this proves that F_{1}F_{2}F_{1} differ from F_{2}F_{1}F_{2} only by an automorphism of R𝔄_{4}. As is easily seen from the structure of R𝔄_{4} in section 6, apart from the identity there is no R-linear automorphism which fixes each projective indecomposable up to isomorphism. We proved the statement.

COROLLARY 7.1. Let Br(Ã_{2}) be the braid group associated to the affine Dynkin diagram Ã_{2}. Then, there is a group homomorphism Br(Ã_{2}) ⟶ SplenP ic_{R}(𝔄_{4}).

It is clear that the group generated by F_{2} and F_{3} is the image under the above homomorphism of an Artin braid group on three strings. Moreover, these two self-equivalences stabilize the trivial module.

8. Studying the action on the group cohomology

(8.1) The outer automorphism group of 𝔄_{4} is isomorphic to the cyclic group of order 2 since any automorphism of 𝔄_{4} induces an automorphism of 𝔖_{4} where each automorphism is inner. Moreover (A-M 94, Chapter III Theorem 1.3), H^{∗}(𝔄_{4}, 𝔽_{2}) = 𝔽_{2}[A, B](C)/(C^{3} + A^{2} + B^{2} + AB) where A = f_{1}^{3} + f_{2}^{3}, B = ζf_{1}^{3} + ζ^{2}f_{2}^{3} and C = f_{1}f_{2}. Here, ζ is a primitive third root of unity over 𝔽_{2}. The elements f_{1} and f_{2} are the two canonical generators of H^{∗}(C_{2}^{2}, 𝔽_{2}) in degree 1. The group of automorphisms of C_{2}^{2} is GL_{2}(2) ≃ 𝔖_{3}. The automorphisms of order 3 are inner in 𝔄_{4} and induce hence the identity on the subring H^{∗}(𝔄_{4}, 𝔽_{2}) of H^{∗}(C_{2}^{2}, 𝔽_{2}). It is an easy and straightforward calculation that the elements of GL_{2}(2) of order 2 act trivially on degree 2 cohomology and act on degree 3 cohomology as regular representation (A ⟶ A and B ⟶ −A − B).

(8.2) Carlson and Rouquier proved (CR-00) that for a group G = P ×Q where P is an abelian pgroup and Q is a cyclic p′-group, the group StP ic_{𝔽_{p}}(𝔽_{p}G) of self-equivalences of Morita type of the stable category is generated by the syzygy operator and the usual Picard group of 𝔽_{p}G. Hence, the action of HD_{𝔽_{p}}(G) on H^{∗}(G, 𝔽_{p}) is via automorphisms of 𝔽_{p}G. In case G = 𝔄_{4} and p = 2, the action described above is the only possible non trivial action of HSplen_{𝔽_{2}}(𝔄_{4}) on H^{∗}(𝔄_{4}, 𝔽_{2}).

(8.3) Recall that < F_{2}, F_{3} >, which is the homomorphic image of an Artin braid group, is in HSplen_{R}(R𝔄_{4}). By (Z-00b) we conclude in our case of abelian Sylow subgroups that HSplen_{R}(𝔄_{4}) acts by automorphisms of the Sylow subgroup on the cohomology. Since for a splendid tilting complex X one has X(ΔQ) ⊗_{C_{G}(Q)} R ≃ X^{G}(Q) (where G acts on the right and Q on the left), one sees easily that any element of < F_{2}(ΔC_{2}^{2}), F_{3}(ΔC_{2}^{2}) > fixes the trivial module as well. Moreover, it is easy to see that < F_{2}, F_{3} > acts trivially an H^{∗}(𝔄_{4}, 𝔽_{2}) since this action factors via the action of the group of stable self-equivalences (see (Z-99b)) and there S_{2} and S_{3} become just isomorphic to R𝔄_{4}. So, we proved that < F_{2}, F_{3} > does not contain Out(𝔄_{4}). Denoting by γ the element (of order 2) in Out(R𝔄_{4}) given by conjugation by an element in 𝔖_{4} \ 𝔄_{4}, we get < F_{2}, F_{3}, γ >=< F_{2}, F_{3} > ×C_{2} ≤ HD_{R}(𝔄_{4}) .

(8.4) We observe that F_{2}(P_{2}) ≃ P_{2}[1] and F_{2}(P_{3}) ≃ (P_{2} ⟶ P_{3}) with homology in degree 0 and 1, and likewise for F_{3} with the rôles of P_{2} and P_{3} interchanged. The very same proof as in (RZ-96, Theorem 4.6) shows that the morphism Br(A_{2}) ⟶< F_{2}, F_{3} >⊆ HSplen_{𝔽_{2}}(𝔄_{4}) is injective.

(8.5) Since the symmetric group of degree 3 acts naturally on the three generators of Br(Ã_{2}) and by the fact that the relations obey the same symmetry also on Br(Ã_{2}), we get

Br(Ã_{2}) × 𝔖_{3}  ≃ (kerβ ×Br(A_{2})) × (C_{3} ×C_{2}) ⟶ ⟶  Br(A_{2}) × C_{2}
↓  ↓^{∩}
< F_{1}, F_{2}, F_{3} > ×𝔖_{3}  < F_{2}, F_{3} > × < γ >
∩  ∩
T rP ic_{R}(R𝔄_{4})  HD_{R}(𝔄_{4})

(8.6) May it be that derived categories “in an abelian situation” have generalized braid group symmetries? I believe that the above observations are not just an accident.

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