Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2002.10921 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915318545252352 |
|---|---|
| author | Seysen, Martin |
| author_facet | Seysen, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_10921 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A computer-friendly construction of the monster Seysen, Martin Group Theory 20C34 (Primary) 20D08, 20C11 (Secondary) 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. |
| title | A computer-friendly construction of the monster |
| topic | Group Theory 20C34 (Primary) 20D08, 20C11 (Secondary) |
| url | https://arxiv.org/abs/2002.10921 |