# Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

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

• The symbol $\subset$ means 'subset' or 'subgroup'. But $M_{12}$ is a supergroup of $M_{11}$. So most (or all?) symbols $\subset$ should be replaced by $\supset$. Oct 18, 2021 at 21:18
• I just noticed that the statement about $K_{6}$ isn’t entirely true. I was actually thinking about $S_{6}$ and I confused it with $K_{6}$ in my mind. Oct 19, 2021 at 11:36
• The group $M_{24}$ preserves the binary Golay code. For background, see [1]. A natural structure preserved by the automorphism group of a Niemeier lattice is the glue code as described in [1], Chapter 16. * [1] Conway, Sloane, Sphere packings, Lattices and Groups. Oct 20, 2021 at 9:47
• Luckily for me, I happen to own that book, Martin. but, I haven’t been able to find an explanation for this phenomenon involving certain "glue codes". Oct 21, 2021 at 10:07
• To clarify what I meant, by "phenomenon". I was talking about how the automorphism groups of these "glue codes" were identified, i.e. the reason for their established structures. The case with $D_{k}^n$ for $k \geq 4$ and $k · n = 24$ is a little simpler than this case, as the automorphism group of the former has order $n!$ Oct 21, 2021 at 19:06

## 1 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.

• Do these glue codes have anything to do with Umbral Moonshine theory? Nov 8, 2021 at 2:08