arXiv 2201.117152022-01-27math.RT

具有阿贝尔$p'$惯性商的正缺陷块的结构

Structure of blocks with normal defect and abelian $p'$ inertial quotient

David Benson, Radha Kessar, Markus Linckelmann

备注：: 21 pages

摘要：

设$k$是素特征$p$的代数闭域.设$kGe$是有限群$G$的群代数的块，它具有正规亏群$P$和交换$p'$惯性商$L$ .证明了$kGe$是$P$的半直积与$L$的某个子群的群代数的量子化形式上的矩阵代数.要做到这一点，我们首先检查相关的分次代数，使用詹宁斯-奎伦式定理。作为例子，我们计算了指数为$p$、阶为$p^3$的超特殊$p$ -群$P$与中心作用平凡的八阶四元数群的半直积情况下非主块的基本代数的伴随阶数.在$p=3$的情况下，我们给出了明确的生成元和关系的基本代数作为一个量化版本的$kP$。作为第二个例子，我们给出了明确的发电机和关系的情况下，一组形状$2^{1+4}:3^{1+2}$的特征2。

英文摘要：

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of $P$ with a certain subgroup of $L$. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extraspecial $p$-group $P$ of exponent $p$ and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case $p=3$ we give explicit generators and relations for the basic algebra as a quantised version of $kP$. As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

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