I have been doing research on the Niemeier lattices with root systems of type, $A\_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups seemed random at first, and they appeared to have a connection to certain projective linear groups over Galois fields with a prime number of elements. These root systems correspond to the divisors of $24$. I looked into the structures of the groups that permute the separate root systems and I was intrigued:


We will use the notation $K\_{n}$ for the group that induces permutations on the corresponding Niemeier lattice $A\_{k}^n$ (where $n·k = 24$).


Cases $K\_{1}$ and $K\_{2}$
---------------------------


$K\_{1}$ can only be the trivial group as there is only $1$ vertex to act on and the only group that acts faithfully over any singleton set is trivial.


$K\_{2}$ acts faithfully on $2$ vertices, corresponding to the exchange between the two root systems.


Case $K\_{3}$
-------------


This group is isomorphic to the symmetric group of degree $3$ (order $3!$ or $6$). It also happens(?) to be isomorphic to the projective special linear group, $L\_{2}(2)$


Case $K\_{4}$
-------------


This group is isomorphic to the alternating group of degree $4$ (order $4!/2$ or $12$). This one is also isomorphic to another linear group, namely $L\_{2}(3)$.


Case $K\_{6}$
-------------


This group is isomorphic to the symmetric group of degree $5$ (order $5!$ or $120$), but the striking thing about this is that it’s acting primitively on $6$ vertices, not $5$ vertices. This group is related to the projective special linear group $L\_{2}(5)$. It has one representation that acts primitively on the $6$ vertices of $PG\_{1}(5)$, and another representation on $5$ vertices, (the alternating group, $A\_{5}$), and both are contained within the symmetric group over $6$ vertices, $S\_{6}$.


Case $K\_{8}$
-------------


This group is isomorphic to $AGL\_{3}(2)$ an affine group of rank $3$, over the Galois field of $2$ elements (order $8·7·6·4$ or $1344$). like all the other $K\_{n}$ This group acts primitively on $8$ vertices. This one contains two distinct representations of the projective special linear group, $L\_{2}(7)$ one of which is the action of the group $L\_{2}(7)$ primitive over the $8$ vertices of $PG\_{1}(7)$ the other is a group isomorphic to the first, another linear group $L\_{3}(2)$. this one acts primitively over the $7$ vertices of $PG\_{2}(2)$


Case $K\_{12}$
--------------


This group is isomorphic to one of the sporadic groups, namely $M\_{12}$ the Mathieu group of degree $12$ (order $8·9·10·11·12$ or $95040$). this group contains two separate representations of the linear group $L\_{2}(11)$, but unlike the previous two cases, only one of these groups is maximal within $K\_{12}$ (namely the group $L\_{2}(11)$ and its primitive action over the $12$ vertices of $PG\_{1}(11)$). The other one is more subtle, as it’s not maximal in $M\_{12}$, however it *is* maximal in another sporadic group $M\_{11}$ (the stabiliser of either a point or a total within $M\_{12}$). This representation of $L\_{2}(11)$ acts primitively over $11$ vertices.


Case $K\_{24}$
--------------


This group is the most notable one, as it contains all the groups mentioned previously. It is isomorphic to a very interesting sporadic group called $M\_{24}$ the Mathieu group of degree $24$ (order $3·16·20·21·22·23·24$ or $244823040$). It contains a linear subgroup of type $L\_{2}(23)$, but unlike the previous cases there is only a single instance of this group, and there is no additional representation over $n - 1$ vertices. There is a lot of structure happening here, and frankly I haven’t enough space for that.


A recapitulation of the groups $K\_{n}$
---------------------------------------


**Orders:**


$|K\_{1}| = 1 $


$|K\_{2}| = 2 $


$|K\_{3}| = 6 $


$|K\_{4}| = 12 $


$|K\_{6}| = 120 $


$|K\_{8}| = 1344 $


$|K\_{12}| = 95040 $


$|K\_{24}| = 244823040 $


**Other facts:**


$K\_{3} \cong S\_{3} \cong L\_{2}(2) $


$K\_{4} \cong A\_{4} \cong L\_{2}(3) $


$K\_{6} \cong S\_{5} \supset L\_{2}(5) \cong A\_{5} $


$K\_{8} \cong AGL\_{3}(2) \supset (L\_{2}(7) \cong L\_{3}(2)) $


$K\_{12} \cong M\_{12} \supset M\_{11} \supset L\_{2}(11) $


$K\_{12} \cong M\_{12} \supset L\_{2}(11) $


$K\_{24} \cong M\_{24} \supset L\_{2}(23) $


$M\_{24} \supset \text{ (all previous ones)} $


I have only two questions:


1. Why do these $K\_{n}$ have their respective structures instead of just $S\_{n}$? What do they have in common?
2. What sort of structure must these $K\_{n}$ maintain under their respective group actions?


I will also remind us of what we defined the groups $K\_{n}$ to be: the group of permutations of the separate root systems of the Niemeier lattices of type $A\_{k}^n$ (with $n·k = 24$)


Edit: I forgot to mention: As of writing this, I have no proper experience in Niemeier lattices. I’d prefer it if the explanation were limited to group theory and a basic understanding of lattice theory.


Update: I had just found something interesting. These $8$ groups seem closely related to certain Umbral Groups (particularly, the ones of lambdacy $2$, $3$, $4$, $5$, $7$, $9$, $13$, and $25$) [7th November 2021, 21:05]


Proof of the Umbral Moonshine Conjecture:
<https://arxiv.org/pdf/1503.01472.pdf>


Umbral Moonshine and the Neiemeir Lattices:
<https://arxiv.org/pdf/1307.5793.pdf>



Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$
The information required for answering this question is contained in [1]. There it is scattered over many chapters; so I will give a summary here.


Let $N$ be a Niemeier lattice, $R$ be the sublattice of $N$ generated by
its roots and and $R^\*$ be the dual lattice of $R$. Then $G = R^\*/R$ is
an Abelian group, the *glue group*. The elements of $G$
are called the *glue vectors*
of $G$. Since $N$ is unimodular, the subgroup $H = N/R$ of $G$ has
order $|G|^{1/2}$ and also index $|G|^{1/2}$ in $G$. See [1],
Chapter 4.3 for an introduction to gluing theory.


If $R = R\_1 \oplus \ldots \oplus R\_n$ then $G$ has structure
$G= G\_1 \oplus \ldots \oplus G\_n$ with $G\_i = R\_i^\* / R\_i$.


For each glue vector $v \in G$ let $l(v)$ be the norm of its shortest
representative in $R^\*$. Then we have a decomposition
$v = v\_1 + \ldots + v\_n$ with $v\_i \in G\_i$; and we have
$l(v) = l(v\_1) + \ldots l(v\_n)$, where $l(v\_i)$ is the norm of the
shortest representative of $v\_i$ in $R\_i^\*$.


The glue groups of the root lattices and their length functions $l$
are known; for the case $A\_n$ see [1], Chapter 4.6. The glue group
of $A\_n$ is cyclic of order $n+1$. If $g$ is a generator of that
group then we have $l(k\cdot g) = \frac{k \cdot (n+1-k)}{n+1}$
for $0 \leq k \leq n$.


Since $N$ is an even lattice, $l(v)$ must be an even integer for
all $v \in H$. All roots of $N$ lie in $G$; so $l(v) > 2$ must hold
for all nonzero glue vectors $v\in H$.


The conditions listed above impose some rather severe restrictions
on the subgroup $H$ of $G$, which are discussed in [1], Chapter 18.4.
Obviously, a permutation of the root lattices preserving a Niemeier
lattice must also preserve the subgroup $H$ and the norm function $l$
on $H$.


Thus the subgroup $H$ of $G = R^\* / R$ and the norm function $l$ on
$H$ is the structure preserved by the permutation groups $K\_n$ of
the Niemeier lattices $A\_k^{24/k}$.


An answer to the question why a certain group $K\_n$ has a certain
structure is to some extent opinion based. For the *large* cases
$K\_{24/(n-1)}, n = 2,3,$ the Abelian group $G$ happens to be a vector space
over $\mbox{GF}\_n$, and the norm function turns out to be proportional to
the Hamming weight of a vector. So we have to look for linear codes over
$\mbox{GF}\_n^k$, where $(n-1)\cdot k = 24$, with Hamming distance
at least 8 in case $n=2$ and 6 in case $n=3$.
Such codes are known as Golay codes and also discussed in [1].


For the *small* cases $K\_k, k \leq 8$ we have to look for
permutation groups on $k$ letters. Here the linear groups
$\mbox{SL}\_2(k-1)$ have such a permutation representation.


[1] Conway, Sloane, Sphere packings, Lattices and Groups.


