Brauer Tree Research Articles

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ,V) and we describe R(Λ,V) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.

On Representation-Finite Gendo-Symmetric Biserial Algebras

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

Representation type for block algebras of Hecke algebras of classical type

We find representation type of the cyclotomic quiver Hecke algebras RΛ(β) of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type in good characteristic. We conjecture that this statement should hold for Hecke algebras of exceptional type.

Derived Picard groups of selfinjective Nakayama algebras

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

Two-Term Tilting Complexes over Brauer Tree Algebras

In this paper, all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described, using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.

Two-Term Partial Tilting Complexes Over Brauer Tree Algebras

In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are computed.

Two-term Tilting Complexes and Simple-minded Systems of Self-injective Nakayama Algebras

We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular polygon.

Coxeter orbits and Brauer trees III

This article is the final one of a series of articles on certain blocks of modular representations of finite groups of Lie type and the associated geometry. We prove the conjecture of Broué on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements. We also prove a conjecture of Hiß, Lübeck, and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer tree (in particular, the decomposition matrix) of the principal ℓ \ell -block of E 7 ( q ) E_7(q) when ℓ ∣ Φ 18 ( q ) \ell \mid \Phi _{18}(q) and E 8 ( q ) E_8(q) when ℓ ∣ Φ 18 ( q ) \ell \mid \Phi _{18}(q) or ℓ ∣ Φ 30 ( q ) \ell \mid \Phi _{30}(q) .

On Loewy lengths of blocks

AbstractWe give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

Cohomology of Deligne–Lusztig varieties for short-length regular elements in exceptional groups

We determine the cohomology of Deligne–Lusztig varieties associated with some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Φ14-block of E7 and the principal Φ24-block of E8.

The Brauer trees of non-crystallographic groups of Lie type

In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the ‘groups’ I2(n,q), H3(q) and H4(q). The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne–Lusztig theory and the geometric version of Brouéʼs conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Brouéʼs conjecture are consistent with both the mathematics and philosophy of blocks with cyclic defect groups.

The classification of the indecomposable liftable modules in blocks with cyclic defect groups

Let G be a finite group, k be an algebraically closed field of characteristic p and B a block of kG with cyclic defect group. We classify the indecomposable B-modules which are liftable with respect to a splitting p-modular system. The indecomposable non-projective modules in B are constructed from certain paths in the Brauer tree of B (see [Ja69]). We determine those paths that give rise to liftable modules. We also find the characters of the lifts of these modules.

Coxeter orbits and Brauer trees

We study the cohomology with modular coefficients of Deligne–Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal ℓ-block of a finite reductive group G(Fq) when the order of q modulo ℓ is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, Lübeck and Malle (1995) in [25]) and the derived equivalence predicted by the geometric version of Brouéʼs conjecture (Broué and Malle, 1993, [7]).

Scott modules in finite groups with cyclic Sylow p-subgroups

We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and the values of ordinary characters at non-identity p-elements. Previously, determining Scott modules often involved long calculations. The method is applicable to the case where finite groups have cyclic Sylow p-subgroups. The structure of a Scott module is closely related to the length of the shortest path from the vertex labeled by the trivial ordinary character to the exceptional vertex in the Brauer tree. We classify the structures of the Scott modules and the ordinary characters they afford, by the length of the associated path in the Brauer tree.

Cuspidal [formula omitted]-modules and deformations of certain Brauer tree algebras

We show that the algebras describing blocks of the category of cuspidal weight (resp. generalized weight) sl n -modules are one-parameter (resp. multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley–Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A Γ corresponding to an arbitrary Brauer tree Γ of type ( m , e ) . We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A S , whose tree is a star with the exceptional vertex in the middle, to A Γ . The grading on A S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras. We also compute Out K ( A Γ ) , the group of outer automorphisms that fix the isomorphism classes of simple A Γ -modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A Γ up to graded Morita equivalence and rescaling.

THE GRADED CENTER OF THE STABLE CATEGORY OF A BRAUER TREE ALGEBRA

We calculate the graded center of the stable category of a Brauer tree algebra. The canonical map from the Tate analogue of Hochschild cohomology to the graded center of the stable category is shown to induce an isomorphism module taking quotients by suitable nilpotent ideals. More precisely, we show that this map is surjective with nilpotent kernel in even degrees, while this map need not be surjective in odd degrees in general.

Representations of p′-valenced Schurian schemes

Let p be a prime number. We consider representations of p′- valenced Schurian schemes over a field of characteristic p, especially the case that the cardinality of the underlying set can be divided by p and not by p2. A typical example of such scheme is obtained by the following way. Let G be a finite group of order pq, where q is prime to p, and let H be a p′-subgroup of G. Define the scheme by the action of G on H\G. In this case, we will show that the adjacency algebra is a direct sum of some Brauer tree algebras and simple algebras, and hence it has finite representation type. Also we give some examples of the case that G is the symmetric group of degree p and H is its Young subgroup.

A construction of almost all Brauer trees

We know that almost all Brauer trees are shaped like a star. Given such a star and an odd prime l , we give an explicit method for constructing infinitely many groups with this star as the Brauer tree of some l -block. Furthermore we show that there is an infinite family of Brauer trees which cannot be realized in the principal block of any group.

ON BLOCKS WITH CYCLIC DEFECT GROUPS AND THEIR HEAD ORDERS

ABSTRACT It is shown that (Plesken, 1983, Theorem 8.5), describes blocks of cyclic defect groups up to Morita equivalence. In particular such a block is determined by its planar embedded Brauer tree. Applying the radical idealizer process, the head order of such blocks is calculated explicitly.