Brauer Tree Research Articles

Cohomology and Ext for blocks whose Brauer trees are lines or stars

We compute the dimensions of Ext G n ( V , W ) for all irreducible modules V, W lying in r-blocks of cyclic defect whose Brauer tree is either a star or a line (open polygon), which in the line case extends a result of Dudas. We also make use of this information to aid in completing the same computation for all cross characteristic r-blocks of cyclic defect in rank one groups of Lie type.

Broué's abelian defect group conjecture for blocks with cyclic hyperfocal subgroups

AbstractIn this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

Decomposition matrices and Brauer trees of spin characters for S_25 , p=17

In this paper, we will find the Brauer trees and the decomposition matrices of the symmetric groups , when n=25 for p=17 by using -inducing for calculates modular characters from the ordinary characters of the symmetric group. Also, we find all the blocks of defect 0 and 1 and we will depend on the results of the decomposition matrices of the symmetric groups when n=24 for p=17

The Ext-algebra of the Brauer tree algebra associated to a line

The Ext-algebra of the Brauer tree algebra associated to a line

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

On the source algebra equivalence class of blocks with cyclic defect groups, I

We investigate the source algebra class of a p-block with cyclic defect groups of the group algebra of a finite group. By the work of Linckelmann this class is parametrized by the Brauer tree of the block together with a sign function on its vertices and an endo-permutation module of a defect group. We prove that this endo-permutation module can be read off from the character table of the group. We also prove that this module is trivial for all cyclic p-blocks of quasisimple groups with a simple quotient which is a sporadic group, an alternating group, a group of Lie type in defining characteristic, or a group of Lie type in cross-characteristic for which the prime p is large enough in a certain sense.

The Singularity and Cosingularity Categories of C∗BG for Groups with Cyclic Sylow p-subgroups

We construct a differential graded algebra (DGA) modelling certain A_{infty } algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H_{*}{Omega } BG{^{^{wedge }}_p}. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.

Blocks of profinite groups with cyclic defect group

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group Z p $\mathbb {Z}_{p}$ is of star type.

The differential graded stable category of a self-injective algebra

Let A be a finite-dimensional, self-injective algebra, graded in non-positive degree. We define A -dgstab, the differential graded stable category of A , to be the Verdier quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab / Ω ( 1 ) , reducing computations in the dg-stable category to those in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

Trivial source characters in blocks with cyclic defect groups

We describe the ordinary characters of trivial source modules lying in blocks with cyclic defect groups relying on their recent classification in terms of paths on the Brauer tree by G. Hiss and the second author. In particular, we show how to recover the exceptional constituents of such characters using the source algebra of the block.

Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations

In this paper, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. We also study representation-finite d-self-injective dg algebras A in detail for some positive integer d. In particular, we classify the Auslander-Reiten (=AR) quivers of CMA for a large class of d-self-injective dg algebras A in terms of (−d)-Calabi-Yau (=CY) configurations, which are Riedtmann's configurations for the case d=1. For any given (−d)-CY configuration C, we show there exists a d-self-injective dg algebra A, such that the AR quiver of CMA is given by C. For type An, by using a bijection between (−d)-CY configurations and certain purely combinatorial objects which we call maximal d-Brauer relations given by Coelho Simões, we construct such A through a Brauer tree dg algebra.

Mutations and Pointing for Brauer Tree Algebras

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Brauer trees of unipotent blocks

In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author in [19]. As a consequence, the Brauer trees of principal \ell -blocks of finite groups are known for \ell > 71 .

The Classification of the Trivial Source Modules in Blocks with Cyclic Defect Groups

Relying on the classification of the indecomposable liftable modules in arbitrary blocks with non-trivial cyclic defect groups by the first author and Naehrig we give a complete classification of the trivial source modules lying in such blocks, describing in particular their associated path on the Brauer tree of the block in the sense of Janusz’ classification of the indecomposable modules of such blocks. Furthermore, the appendix contains a minor correction to the statement of the classification theorem of the indecomposable liftable modules, as well as a description of the minimal distance from an arbitrary indecomposable liftable module to the boundary of the stable Auslander-Reiten quiver of the block.

Simplicial complexes and tilting theory for Brauer tree algebras

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f-vector of the simplicial complexes of Brauer tree algebras only depends on the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support τ-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.

Spherical objects of the multiplicity free Brauer tree algebra with two edges

We explicitly describe all spherical objects, as well as tilting objects, of the multiplicity free Brauer tree algebra with two edges.

Smash products of group weighted bound quivers and Brauer graphs

Let be a field, G a group, and (Q, I) a bound quiver. A map is called a G-weight on Q, which defines a G-graded -category , and W is called homogeneous if I is a homogeneous ideal of the G-graded -category . Then we have a G-graded -category . We can then form a smash product of and G, which canonically defines a Galois covering with group G [we will see that all such Galois coverings to have this form for some W]. First we give a quiver presentation of the smash product . Next if (Q, I, W) is defined by a Brauer graph with an admissible weight, then the smash product is again defined by a Brauer graph, which will be computed explicitly. The computation is simplified by introducing a concept of Brauer permutations as an intermediate one between Brauer graphs and Brauer bound quivers. This extends and simplifies the result by Green–Schroll–Snashall on the computation of coverings of Brauer graphs, which dealt with the case that G is a finite abelian group, while in our case G is an arbitrary group. In particular, it enables us to delete all cycles in Brauer graphs to transform it to an infinite Brauer tree.

FOLDINGS AND TWO-SIDED TILTING COMPLEXES FOR BRAUER TREE ALGEBRAS

In this note, for a Brauer tree algebra $A$ and a star-shaped Brauer tree algebra $B$ which is derived equivalent to $A$, we give operations on the two-sided tilting complex $D_T$ of $A\otimes B^{op}$-modules constructed in [3] which is isomorphic to the Rickard tree-to-star complex $T$ constructed in [5] in $D^b(A)$, and we show that the operations on $D_T$ correspond to operations called $foldings$ on the Rickard tree-to-star complex $T$ given in [7].

Two-sided tilting complexes for Brauer tree algebras

In this note, we construct two-sided tilting complexes corresponding to one-sided tilting complexes for Brauer tree algebras.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.