Related papers: On defect groups for generalized blocks of the sym…

We consider Donovan's conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N \lhd G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we…

This paper is an attempt to compute the decomposition numbers of the blocks of the symmetric group which have "small defect"; that is, blocks of weight smaller than the characteristic. We present various methods for computing such…

We study a weakened version of the Holm--Willems Local Conjecture. The problem is reduced to quasi-simple groups under the assumption that the defect group is abelian. Complete proofs are provided in the case \(p = 2\).

We show that each block of an alternating group over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent. This provides new evidence for a refined version of Brou\'{e}'s abelian defect…

We classify up to Morita equivalence all blocks whose defect groups are Suzuki $2$-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact…

We define a new invariant for a $p$-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete…

We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for…

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for…

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan…

Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main…

L. Puig defined inertial blocks. In this paper, we prove that 2-blocks with defect group $C_{2^{n_1}}\times C_{2^{n_2}}\times...\times C_{2^{n_t}}$ are inertial, where $n_i\geq 2$ for all $i$.

The objective of this research paper is to study the relationship between a block of a finite group and a defect group of such block. We define a new notion which is called a strongly $k(D)$-block and give a necessary and sufficient…

We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent…

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain…

The algebraic $L$-groups $L_*(\A,X)$ are defined for an additive category $\A$ with chain duality and a $\Delta$-set $X$, and identified with the generalized homology groups $H_*(X;\LL_{\bullet}(\A))$ of $X$ with coefficients in the…

The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and,…

We show that the 3-block of the sporadic simple Janko group J_4 with defect group C_3 x C_3, and the principal 3-block of the alternating group A_8 are Puig equivalent, answering a question posed in earlier work of Koshitani-Kunugi-Waki. To…

The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class…

We study blocks of the double covers of symmetric and alternating groups. The main result is a `local' description, up to Morita equivalence, of arbitrary defect RoCK blocks of these groups in terms of generalized Schur superalgebras…

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and…