Orders of the automorphism groups of the Niemeier lattices

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Numbers

$R(L)$

$|O(L)|$

$\varnothing$ (Leech lattice):

8315553613086720000

comment: The Leech lattice $\Lambda_{24}$ has no roots. Its full automorphism group is $\mathrm{Co}_0=2.\mathrm{Co}_1$.

$A_{1}^{24}$:

4107449023856640

$A_{2}^{12}$:

413762786426880

$A_{3}^{8}$:

295882444505088

$A_{4}^{6}$:

716636160000000

$A_{5}^{4}\,D_{4}$:

2476694568960000

$D_{4}^{6}$:

108208436847575040

$A_{6}^{4}$:

15485790781440000

$A_{7}^{2}\,D_{5}^{2}$:

47943914618880000

$A_{8}^{3}$:

573416710078464000

$A_{9}^{2}\,D_{6}$:

1213580338790400000

$D_{6}^{4}$:

6763027302973440000

$A_{11}\,D_{7}\,E_{6}$:

16019260472033280000

$E_{6}^{4}$:

346657985428193280000

$A_{12}^{2}$:

155103152174530560000

$D_{8}^{3}$:

824788751971516416000

$A_{15}\,D_{9}$:

3887340541213409280000

$A_{17}\,E_{7}$:

37172693925353226240000

$D_{10}\,E_{7}^{2}$:

31316197926418513920000

$D_{12}^{2}$:

1924703466207817236480000

$A_{24}$:

31022420086661971968000000

$D_{16}\,E_{8}$:

477676405704303732326400000

$E_{8}^{3}$:

2029289625631919702016000000

comment: The orthogonal direct sum $E_8\oplus E_8\oplus E_8$.

$D_{24}$:

5204698426366666226930810880000

Definition

For each Niemeier lattice $L$, a positive definite even unimodular lattice of rank $24$, the table gives the order $|O(L)|$ of its full linear isometry group $O(L)=\{g\in O(\mathbb{R}^{24}):gL=L\}$ [1].

Parameters

$R(L)$

— root system (the root system $R(L)=\{v\in L:(v,v)=2\}$ of a positive definite even unimodular lattice of rank $24$, including the empty root system of the Leech lattice)

Formulas

(1)

$|O(L)|=|G(L)|\prod_i |W(R_i)|^{m_i}$ when $R(L)=\bigoplus_i R_i^{m_i}$, where the $R_i$ are distinct irreducible root systems, $m_i$ their multiplicities, and $G(L)=O(L)/W(R(L))$ [1].

(2)

$|W(A_n)|=(n+1)!$ for $n\geq1$, $|W(D_n)|=2^{n-1}n!$ for $n\geq4$, and $|W(E_6)|=51840$, $|W(E_7)|=2903040$, $|W(E_8)|=696729600$. These are the values at $t=1$ of the corresponding Coxeter Poincaré polynomials.

(3)

$|O(\Lambda_{24})|=2|\mathrm{Co}_1|$, $|O(N(A_1^{24}))|=2^{24}|\mathrm{M}_{24}|$, and $|O(N(A_2^{12}))|=2\cdot6^{12}|\mathrm{M}_{12}|$, where $N(R)$ is the Niemeier lattice with root system $R$ [1]. The simple-group orders are $|\mathrm{Co}_1|$, $|\mathrm{M}_{24}|$ and $|\mathrm{M}_{12}|$.

(4)

$\displaystyle\sum_{[L]}\frac{1}{|O(L)|}=\frac{|B_{12}|}{24}\prod_{j=1}^{11}\frac{|B_{2j}|}{4j}$, where the sum runs over the $24$ Niemeier isometry classes and $B_k$ are the Bernoulli numbers. This is the rank-$24$ Minkowski-Siegel mass formula [2], Section 2.

Comments

(5)

The root system determines the isometry class of $L$: there are $23$ nonempty possibilities and the rootless Leech lattice $\Lambda_{24}$ [1], Section 5.3. A power in a root-system label denotes the number of identical irreducible components. Rows follow increasing common Coxeter number, with the Leech lattice first, in the order of [3].

(6)

All lattices have Gram determinant $1$. A root has squared norm $2$, and the Leech lattice has minimum squared norm $4$, consistent with the classical-lattice conventions. The group includes $-I$ and all component permutations that preserve the lattice; it contains no translations.

(7)

For a lattice with roots, $W(R)$ is the group generated by reflections in the roots, and $G(L)=O(L)/W(R)$ is the finite quotient that records the remaining symmetries. The quotient must preserve the glue defining $L$, so a root system with repeated components need not admit every permutation of those components [1], Section 5.3.

(8)

The Leech lattice is the same $\Lambda_{24}$ whose automorphism order occurs in the classical-lattice table.

References

[1]

G. Chenevier, The characteristic masses of Niemeier lattices, Journal de théorie des nombres de Bordeaux 32 (2020), 545–583. Section 5.3, Example 1.3 and Table C.5. (arXiv) (doi)

[2]

O. D. King, A mass formula for unimodular lattices with no roots, Mathematics of Computation 72 (2003), 839–863. (arXiv) (doi)

Links

[3]

R. E. Borcherds, the 24 norm-zero vectors corresponding to the Niemeier lattices

Similar tables

Orders of the automorphism groups of the classical lattices — full automorphism orders for classical lattices, including the same Leech lattice

Poincaré polynomials of the finite Coxeter groups — evaluation at 1 supplies the Weyl-group factors for the root components

Orders of sporadic finite simple groups — Mathieu and Conway orders occur in the automorphism groups

Packing densities and Hermite numbers of the classical lattices — packing invariants of classical lattices in the same normalisations

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: yes (it holds all 24 isometry classes of positive definite even unimodular lattices of rank 24, including the Leech lattice)

Repeats values from: Orders of the automorphism groups of the classical lattices

How they were obtained:

The generator multiplies exact Weyl-group orders by the quotient orders in [3]. The Leech order is read directly because its reflection group is trivial. All quotient orders were compared with the group identifications in [1], Section 5.3; its Table C.5 gives the reciprocal Leech order as the mass of the identity characteristic polynomial.

The independent check computes Weyl orders from Sage positive-root heights using $\prod_{\alpha>0}(\operatorname{ht}(\alpha)+1)/\operatorname{ht}(\alpha)$ and uses separately transcribed quotient factors. It verifies all 24 orders and the exact rank-24 mass formula, with rank-8 and rank-16 mass controls. Increasing one order by 1 and omitting the Leech row both fail the mass check. A separate enumeration of rank-24 ADE root systems with equal Coxeter numbers gives exactly the 23 nonempty labels. Comparisons also use the 18 applicable stored Coxeter Poincaré polynomials, the three sporadic-group relations, and the classical-lattice Leech entry. These checks are repeated on the values read back from the draft.