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. [\ref{Zak}]). By computing endomorphism rings of these tilting complexes we get graded algebras.
We also compute ${\rm Out}^K(A_Γ)$, the group of outer automorphisms that fix 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.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Departamento de Matemáticas y Estadística, Universidad del Norte（北大学数学与统计系）
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
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.
In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. 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 Broué's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Broué's conjecture are consistent with both the mathematics and philo…
We introduce group graded basic Morita equivalences between algebras deter- mined by blocks of normal subgroups, and by using the extended Brauer quotient, we show that they induce graded basic Morita equivalences at local levels.
We show that any Brauer tree algebra has precisely $\binom{2n}{n}$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.
We introduce some graded versions of the walled Brauer algebra, working over a field of characteristic zero. This allows us to prove that the walled Brauer algebra is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov's arc algebra, as suggested by recent work of Cox and De Visscher. We deduce that the walled Brauer algebra is Koszul whenever its defining parameter is non-zero.
Let $\Bbbk$ be a field, $G$ a group, and $(Q, I)$ a bound quiver. A map $W\colon Q_1 \to G$ is called a $G$-weight on $Q$, which defines a $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$, and $W$ is called homogeneous if $I$ is a homogeneous ideal of the $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$. Then we have a $G$-graded $\Bbbk$-category $\Bbbk(Q, I, W):= \Bbbk(Q, W)/I$. We can then form a smash product $\Bbbk(Q, I, W)\# G$ of $\Bbbk(Q, I, W)$ and $G$, which canonically defines a Galois covering $\Bbbk(Q, I, W)\# G \to \Bbbk(Q, I)$ with group $G$ (we will see that all such Galois coverings to $\Bbb…
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 whenever the defining characteristic is good. 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 trees (in particular, the decomposition matrix) of the principal $\ell$-block of $E_7(q)$ when $\el…
Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. 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 the ordinary Brauer tree algebra and a particular algebra called 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 tr…
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；IMJ-PRG（IMJ-PRG研究所）；Centre for Quantum Mathematics, Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机科学系量子数学中心）；Yau Mathematical Sciences Center（丘成桐数学科学中心）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
We define a degenerate affine version of the walled Brauer algebra, that has the same role plaid by the degenerate affine Hecke algebra for the symmetric group algebra. We use it to prove a higher level mixed Schur-Weyl duality for gl_N. We consider then families of cyclotomic quotients of level two which appear naturally in Lie theory and we prove that they inherit from there a natural grading and a graded cellular structure.
In this paper we introduce mutation of Brauer trees. We show that our mutation of Brauer trees explicitly describes the tilting mutation of Brauer tree algebras introduced by Okuyama and Rickard.
We construct a naturally $\mathbb Z$-graded algebra $\mathscr G_n(δ)$ over $R$ with KLR-like relations and give an explicit isomorphism between $\mathscr G_n(δ)$ and $\mathscr B_n(δ)$, the Brauer algebras over $R$, when $R$ is a field of characteristic 0. This isomorphism allows us to exhibit a non-trivial $\mathbb Z$-grading on the Brauer algebras over a field of characteristic 0. As a byproduct of the proof, we also construct an explicit homogeneous cellular basis for $\mathscr G_n(δ)$.
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 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). This result allows computations in the dg-stable category to be performed by reducing to 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…
Let $G$ be the finite simple Chevalley group of type $^2E_6(2)$. It has a Schur multiplier of type $C_2^2 \times C_3$. We determine the ordinary character tables of the central extensions $3.G$, $6.G$, $(2^2\times 3).G$ of $G$ and their extensions by an automorphism of order $2$, that is $3.G.2$, $6.G.2$ and $(2^2\times 3).G.2$. Furthermore we determine all Brauer trees of all groups of type $Z.G.A$ (where $Z$ is central in $Z.G \lhd Z.G.A$ and $A \cong Z.G.A/Z.G$) for which the ordinary character table is known.
We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively 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 S…
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. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.
We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, $\mathfrak{g}$, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the te…
We show that the Brauer algebra over the complex numbers for an integral parameter delta can be equipped with a grading, in the case of delta being non-zero turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter N. The parameter delta appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in pa…
We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic Sylow $p$-subgroup, we have obtained a necessary condition when the Brauer tree of the principal $p$-block of $G$ is a star.
Let $k$ be an arbitrary field, $Λ$ be a $k$-algebra and $V$ be a $Λ$-module. When it exists, the universal deformation ring $R(Λ,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up to $R\otimes_k Λ$, 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 bi…
Let $K$ be a normal subgroup of the finite group $H$. To a block of a $K$-interior $H$-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to a block given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results "Block extensions" Section 12.
We introduce the notion of reflections for selfinjective algebras from the point of view of torsion theories induced by two-term tilting complexes. As an application, we determine the transformations of Brauer trees associated with reflections. In particular, we provide a way to transform every Brauer tree into a Brauer line.
We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends 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 …
We study the space of stability conditions attached to the derived category of $A_{n}$-mod for $A_{n}$ the Brauer tree algebra of the line with $n$ edges. These algebras arise in the study of cyclic defect blocks of group algebras, and they are also related to the zig-zag algebras introduced by Huerfano and Khovanov. We show that for the Brauer tree algebra $A_{3}$, the connected component of the natural heart of the space of stability conditions is simply connected. However, unlike certains examples arising in geometry, the Bridgeland homomorphism is not a covering map.
We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors $F_{μ,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^θ$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^θ$. Furthermore thes…
The purpose of this paper is to discuss the validity of the assumptions (W) and (S) stated in a previous work, about the torsion in the modular l-adic cohomology of Deligne-Lusztig varieties associated to Coxeter elements. We prove that both (W) and (S) hold except for groups of type E_7 or E_8.
Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be isomorphic to each other. Under some restrictions, these loop modules are completely reducible (as ungraded modules), and some of their invariants --- inertia group, graded Brauer invariant and Schur index --- which were previously defined for simple modules over graded finite-dimensional semisimple Lie algebras over an algebraically closed field of charact…
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(F_q) 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) and the derived equivalence predicted by the geometric version of Broué's conjecture.
We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the walled Brauer algebra. For the first two algebras, these lists coincide with the list of all finite traces of the infinite symmetric group. For the walled Brauer algebra, the list of finite traces coincide with the list of finite traces of the square of the latter group. We introduce the operation which corresponds to the graph another graph which called "Pascliz…
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. In particular, we classify the Auslander-Reiten (=AR) quivers of CM $A$ for those $A$ in terms of $(-d)$-Calabi-Yau (=CY) configurations, which are Riedtmann's configuration for the case $d=1$. For any…
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
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 $\mathbb{Z}_{p}$ is of star type.
Gendo-symmetric algebras were introduced by Fang and Koenig 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 by Hu and Xi, between representation-finite gendo-sy…
For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of $G$-graded simple $\mathcal{L}$-modules, as well as necessary and sufficient conditions for an $\mathcal{L}$-module to admit a $G$-grading compatible with the given $G$-grading on $\mathcal{L}$.
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 $H_*ΩBG^{^\wedge}_p$. 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.
In our proceeding 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.
The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R from finite families of primitive elements. Furthermore, we classify these comodules by restricting the possible families of primitive elements, and taking the quotient by the action of certain groups. In the next section, we give a formula about primitive elements of the subalgebra of ladders, and construct a projection operator on the space of primitive elements. It allows us to obtain a basis of the pri…
In representation theory of graded Iwanaga-Gorenstein algebras, tilting theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of graded Cohen-Macaulay modules plays a prominent role. In this paper we study the following two central problems of tilting theory of $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ in the case where $A$ is finite dimensional: (1) Does $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have a tilting object? (2) Does the endomorphism algebras of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have finite global dimension? To the problem (2) we give the com…
