MUTATIONS AND POINTED BRAUER TREES

概要

Brauer tree algebras are important objects in the modular representation theory because blocks with cyclic defect groups are Brauer tree algebras. Also it is known that for a block of G with cyclic defect group P, its Brauer correspondent with respect to NG(P) is a Brauer star algebra. In [7] Rickard showed that two Brauer tree algebras are derived equivalent if and only if their Brauer trees have the same numbers of edges and exceptional multiplicities. This implies that Broue´’s abelian defect conjecture holds in the case of cyclic defect groups. For the proof, for any Brauer tree algebra, Rickard constructed a tilting complex over the Brauer tree algebra which induces a derived equivalence from the Brauer tree algebra to the Brauer star algebra. Later, in [9], for any Brauer tree algebra by using pointing of the Brauer tree, Schaps-Zakay constructed a lot of tilting complexes over Brauer star algebra which in- duce equivalence opposite to the one constructed by Rickard, and the class of the complexes includes the tilting complex which induces the inverse equivalence to the one induced by Rickard tree-to-star complex. Also the class of Schaps-Zakay star-to-tree complexes is all of the two-restricted star-to-tree complexes (Definition 2.13). In [8] by using pointings of Brauer trees, Rickard-Schaps constructed tree-to-star complexes giving inverse equivalences to those induced by the Schaps-Zakay star-to-tree complexes.

On the other hand, nowadays silting mutations are studied by many people, in particu- lar, since mutations for tilting complexes over symmetric algebras produce various tilting complexes, we are interested in the mutations in case of symmetric algebras. As one of the results of the mutations, in [2] Aihara showed that if a symmetric algebra is of finite representation type, then the algebra is tilting-connected (in fact, tilting-discrete). Hence all tilting complexes over a Brauer tree algebra are controlled by mutations since Brauer tree algebras are of finite representation type. Also, for a Brauer tree algebra AG associ- ated to a Brauer tree G and for the Brauer star algebra B derived equivalent to AG, in [10] Schaps-Zvi gave a sequence of the irreducible mutations which converts the stalk complex B to the Schaps-Zakay star-to-tree complex corresponding to the reverse pointing or the left alternating pointing of the Brauer tree G.

Our aim is to generalize their result and show that for any Brauer tree G and for any pointing of G, we can find a sequence of irreducible mutations which converts the stalk complex of the Brauer star algebra to the Schaps-Zakay star-to-tree complex corresponding to the pointing. Since each Schaps-Zakay star-to-tree complex induces an inverse equivalence to the one induced by the Rickard-Schaps tree-to-star complex for the same pointing, giving a sequence of mutations which converts the stalk complex of the Brauer star algebra to the Schaps-Zakay star-to-tree complex is equivalent to giving one to convert the stalk complex of the Brauer tree algebra to the Rickard-Schaps tree-to-star complex. Therefore for any Brauer tree G and any pointing of G, we give, in Algorithm 3.2, a sequence of irreducible mutations which converts the stalk complex AG to the Rickard-Schaps tree-to-star complex T corresponding to the pointing (Theorem 4.1).

Also, when considering mutations of Brauer tree algebras, the Kauer move (Definition 2.7) is important. The Kauer move tells us how the Brauer tree mutates when we apply an irreducible mutation to Brauer tree algebra, that is, we can determine easily and explicitly the Brauer tree of the endomorphism algebra of the tilting complex obtained by the irreducible mutation. In that way, we want to determine easily and explicitly how to convert the Schaps- Zakay star-to-tree complex by mutations. There, we introduce the Kauer move for pointed Brauer trees, which is a local move for pointed Brauer tree G(p) converting it to another pointed Brauer tree μz(G(p)), where z ∈ {+, −}, satisfying the following property, which tells us how the two-restricted star-to-tree complex mutates by irreducible mutations (Theorem 4.3): Let G be a Brauer tree, G(p) a pointed Brauer tree of G and T (G(p)) a Schaps-Zakay star-to-tree complex, then μz(T (G(p)) T (μz(G(p)).

Throughout this paper, algebra means finite dimensional basic algebra. For an algebra Γ, Γ-modules means finitely generated left Γ-modules, and we denote by k an algebraically closed field, by A = AG a Brauer tree algebra associated to a Brauer tree G with e edges numbered as 1, 2,... , e and by B a Brauer star algebra derived equivalent to A. Moreover, for an algebra Γ, we denote by Db(Γ) the derived category of the complexes of finite generated Γ-modules.

We now describe the organization of this paper.

In Sect. 2, we recall some facts on Brauer tree algebras, mutations, and pointed Brauer trees.

In Sect. 3, we introduce a Kauer move for pointed Brauer trees, and give an algorithm which define a sequence of the irreducible mutations from the pointed Brauer tree.

In Sect. 4, we prove the sequence of irreducible mutations obtained in Sect. 3 converts the stalk complex to the Rickard-Schaps tree-to-star complex corresponding to the pointed Brauer tree.

In Sect. 5, for a concrete pointed Brauer tree, we explain how to get the sequence of irre- ducible mutations, and confirm that the sequence converts the stalk complex to the Rickard- Schaps tree-to-star complex.