Notes on Loewy series of centers of $p$-blocks

Explicit calculations in an infinitesimal singular block of $SL_N$

Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$ there exists a singular block in the category of $G_1$-modules which contains precisely $n+1$ irreducible modules. We are interested in the lift of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also …

On centers of blocks with non-cyclic defect groups

In this short note we study the center ZB of a block B of a finite group over an algebraically closed field of prime characteristic through its Loewy length llZB. A result of Okuyama in 1981 gave an upper bound for llZB in terms of defect group of B. The purpose of this note is to improve this bound for non-cyclic defect groups.

A note on the Grothendieck ring of the symmetric group

Let $p$ be a prime number and let $n$ be a non-zero natural number. We compute the descending Loewy series of the algebra $R\_n/pR\_n$, where $R\_n$ denotes the ring of virtual ordinary characters of the symmetric group $S\_n$.

Centres of blocks of finite groups with trivial intersection Sylow $p$-subgroups

For finite groups $G$ with non-abelian, trivial intersection Sylow $p$-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of $G$ and the centre of the Brauer correspondent. This was already known for the Suzuki groups; the result will be generalised to cover more groups with trivial intersection Sylow $p$-subgroups.

Rigidity of tilting modules in category O

In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. These results are well-known by several specialists, but seem difficult to find in the existing literature.

关于Morita型稳定等价与幂零块

On stable equivalences of Morita type and nilpotent blocks

本文用模理论方法重新证明了Puig的结果：与幂零块成Morita型稳定等价的块也是幂零块。

Morita Equivalent Blocks of Symmetric Groups

A well-known result of Scopes states that there are only finitely many Morita equivalence classes of $p$-blocks of symmetric groups with a given weight (or defect). In this note we investigate a lower bound on the number of those Morita equivalence classes.

A note on the Navarro conjecture for alternating groups with abelian defect

In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro conjecture is verified for the p-singular characters of the principal block of the alternating groups in the case of abelian defect when p is odd.

A note on Humphreys' conjecture on blocks

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_χ({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic $p>0$ and $χ\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result for ${\mathfrak g}$ of type $G_2$ in characteristic 3, a case in which the previousl…

关于有限群块源代数中基猜想的简化定理

Reduction theorems for a conjecture on basis in source algebras of blocks of finite groups

本文研究了有限群块源代数中稳定基的猜想，提出了存在稳定单位基的简化定理，并探讨了有限简单群块的相关问题。

等价于主块上一个条件的群论条件

A group-theoretic condition equivalent to a condition on principal blocks

本文提出一个群论条件，等价于有限群在指定素数集的主块中仅含平凡复不可约特征的性质。

On blocks with cyclic defect group and their head orders

It is shown that Section 8 of Plesken's 1983 lecture notes describes blocks of cyclic defect group 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.

On Picard groups of blocks with normal defect groups

Let $b$ be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of $b$ has linear source. We note that this improves upon results of Zhou and also Boltje, Kessar and Linckelmann.

A note on perfect isometries between finite general linear and unitary groups at unitary primes

Let $q$ be a power of a prime, $l$ a prime not dividing $q$, $d$ a positive integer coprime to both $l$ and the multiplicative order of $q\mod l$ and $n$ a positive integer. A. Watanabe proved that there is a perfect isometry between the principal $l-$blocks of $GL_n(q)$ and $GL_n(q^d)$ where the correspondence of characters is give by Shintani descent. In the same paper Watanabe also prove that if $l$ and $q$ are odd and $l$ does not divide $GL_n(q^2)|/|U_n(q)|$ then there is a perfect isometry between the principal $l-$blocks of $U_n(q)$ and $GL_n(q^2)$ with the correspondence of characters …

The strong Frobenius numbers for cyclic defect blocks are equal to one

The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to one. This answers a question of Farrell and Kessar.

On diagonal entries of Cartan matrices of p-blocks

In this short note, we show some inequalities on Cartan matrices, centers and socles of blocks of group algebras. Our main theorems are generalizations of the facts on dimension of Reynolds ideals.

关于块代数的Hochschild上同调非零性与Happel性质的一个约化定理

A reduction theorem for non-vanishing of Hochschild cohomology of block algebras and Happel's property

本文证明了有限群块代数一阶Hochschild上同调非零性的约化定理，并研究了有限单群代数的块，同时构造了满足Happel性质的范畴代数例子。

Block Basis for Coinvariants of Modular Pseudo-reflection Groups

As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ GL_n(q) $ which generalizes the Weyl groups of restricted Cartan type Lie algebra.

Classifying blocks with abelian defect groups of rank $3$ for the prime $2$

In this paper we classify all blocks with defect group $C_{2^n}\times C_2\times C_2$ up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of $2$-blocks with abelian defect groups of rank at most $3$. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index $2$, so requires nove…

代数几何中的共形块

Conformal blocks in algebraic geometry

本文从代数几何视角总结了共形块的构造及其性质，探讨了共形块如何定义曲线模空间上的向量包，并通过共形块解释主丛模上线包及其截面的结构。

完备离散赋值环上对称群双覆盖的 RoCK 块

RoCK blocks for double covers of symmetric groups over a complete discrete valuation ring

本文证明奇特征域上对称群双覆盖的 RoCK 块可提升至离散赋值环，并与其 Brauer 对应 splendid 导出等价，证明仅依赖有限群表示论。