论文标题
怪物的计算机友好型结构
A computer-friendly construction of the monster
论文作者
论文摘要
令$ \ mathbb {m} $为最大的零星有限简单组的怪物组，并首先是由格里斯（Griess）于1982年建造的。 1985年，康威（Conway）在$ \ mathbb {z}中构建了$ \ mathbb {m} $ $ \ mathbb {m} $的196884维代表$ρ$ρ$ of mathbb {m} $。因此，这些矩阵可以减少任何（不一定是prime）奇数$ p $，从而导致奇数特征中的$ \ m athbb {m} $的表示形式。表示$ρ$基于两个最大子组的表示形式$ g_ {x0} $和$ n_0 $的$ \ \ mthbb {m} $。在ATLAS符号中，$ g_ {x0} $具有结构$ 2 _+^{1+24}。\ mbox {co} _1 $和$ n_0 $有结构$ 2^{2+11+22}。（m_ {24} \ times s_3）$。 Conway构建了一组$ n_0 $的显式发电机，但不为$ g_ {x0} $。 本文实质上是通过明确构造$ g_ {x0} \ setMinus n_0 $的明确结构来增强Conway构造的重写。这为我们提供了一组$ \ mathbb {m} $的发电机集。事实证明，$ \ mathbb {m} $的所有发电机的矩阵由单个块和块组成，这些块基本上是hadamard矩阵，由两个负功率缩放。如果Modulus $ P $为Shape $ 2^K-1 $，则可以非常有效地编程使用这种发电机的乘法。 因此，本文可以被视为程序员的参考，用于Conway构建Monster Group $ \ Mathbb {M} $。我们已经实施了$ \ mathbb {m} $ modulo 3、7、15、31、127和255的表示形式。
Let $\mathbb{M}$ be the monster group which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985, Conway has constructed a 196884-dimensional representation $ρ$ of $\mathbb{M}$ with matrix coefficients in $\mathbb{Z}[\frac{1}{2}]$. So these matrices may be reduced modulo any (not necessarily prime) odd number $p$, leading to representations of $\mathbb{M}$ in odd characteristic. The representation $ρ$ is based on representations of two maximal subgroups $G_{x0}$ and $N_0$ of $\mathbb{M}$. In ATLAS notation, $G_{x0}$ has structure $2_+^{1+24}.\mbox{Co}_1$ and $N_0$ has structure $2^{2+11+22}.( M_{24} \times S_3)$. Conway has constructed an explicit set of generators of $N_0$, but not of $G_{x0}$. This paper is essentially a rewrite of Conway's construction augmented by an explicit construction of an element of $G_{x0} \setminus N_0$. This gives us a complete set of generators of $\mathbb{M}$. It turns out that the matrices of all generators of $\mathbb{M}$ consist of monomial blocks, and of blocks which are essentially Hadamard matrices scaled by a negative power of two. Multiplication with such a generator can be programmed very efficiently if the modulus $p$ is of shape $2^k-1$. So this paper may be considered a as programmer's reference for Conway's construction of the monster group $\mathbb{M}$. We have implemented representations of $\mathbb{M}$ modulo 3, 7, 15, 31, 127, and 255.
