相关论文: The Brauer trees of unipotent blocks

We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by R\'edei. If the defect group is also metacyclic, then the block invariants are known. In the remaining cases there are only two…

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological…

Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer…

We determine the decomposition matrices of unipotent $\ell$-blocks of defect $\Phi_4^2$ for exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first author, together with properties…

In the representation theory of finite groups, Brou\'e's abelian defect group conjecture says that for any prime p if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the…

In a paper of 2003, B. K\"ulshammer, J. B. Olsson and G. R. Robinson defined $\ell$-blocks for the symmetric groups, where $\ell >1$ is an arbitrary integer. In this paper, we give a definition for the defect group of the principal…

We put forward a conjecture for the leading constant in Malle's conjecture on number fields of bounded discriminant, guided by stacky versions of conjectures of Batyrev-Manin, Batyrev-Tschinkel, and Peyre on rational points of bounded…

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and…

In this article, we consider the finite exceptional groups of Lie type $E_6$ and $^2E_6$. We prove the inductive blockwise Alperin weight condition holds for unipotent $l$-blocks of $E_6^{\varepsilon}(q)$ if $2,3\nmid q$, $l\geq 5$

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…

We prove, for primes $p\ge5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero…

Let $G$ be a simple, simply connected algebraic group of exceptional type defined over $\mathbb{F}_q$ with Frobenius endomorphism $F: G \to G$. Let $\ell \nmid q$ be a good prime for $G$. We determine the number of irreducible Brauer…

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…

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper…

Let B be a p-block of a finite group G with abelian defect group D such that S\unlhd G, S'=S, G/Z(S)\le\Aut(S) and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Brou\'e.…

In this paper, we investigate the block that has an abelian defect group of rank $2$ and its Brauer correspondent has only one simple module. We will get an isotypy between the block and its Brauer correspondent. It will generalize the…

We compute the dimensions of $\operatorname{Ext}_G^n(V, W)$ for all irreducible $V$, $W$ lying in $r$-blocks of cyclic defect in the simple groups $\operatorname{Sz}(q)$, $\operatorname{PSU}_3(q)$ and $\operatorname{{}^2G}_2(q)$ in cross…

We introduce a correspondence between phylogenetic trees and Brauer diagrams, inspired by links between binary trees and matchings described by Diaconis and Holmes (1998). This correspondence gives rise to a range of semigroup structures on…

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the…

We obtain a presentation for the singular part of the Brauer monoid with respect to an irreducible system of generators, consisting of idempotents. As an application of this result we get a new construction of the symmetric group via…