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:
- Why do these $K_{n}$ have their respective structures instead of just $S_{n}$? What do they have in common?
- 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
Best Answer
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.