DUAL-EXTREME LATTICES among PERFECT LATTICES in DIMENSIONS n <= 8
Put on line: Feb. 11th, 2017.
Recall that a lattice L is dual-extreme if
\gamma'(L):=(\min L \times \min L^*)^1/2 achieves a local maximum on L,
and that this is equivalent to L to be both
(1) dual-perfect and (2) dual-eutactic, i.e. :
(1) The orthogonal projections p_x onto (\mathbb R)x, x\in S(L)\cup S(L^*)
span E;
(2) There exists a relation
\sum_{x\in S(L)} a_x p_x = \sum_{y\in S(L^*)} b_y p_y
with stricly positive coefficients a_x, b_y.
Here (1) is a consequence of perfection.
n < = 5: all perfect lattices are extreme and have (strongly) eutactic duals,
hence are dual-extreme (1+1+1+2+3=8 lattices).
Next: test of s(L^*)>=n, then of rank (S^*)>=n.
n=6. Survivors P6[i], i=1,2,3,5,7; all have a (strongly) eutactic dual,
hence are dual-extreme (5 lattices).
n=7. Survivors P7[i], 1 2 4 15 20 27 33 ; 1,2,4,33: E_7,E_7^*,D_7,A_7.
i=15. 4 orbits on S, 1 on S^*. Dual-eutaxy needs negative coefficients.
i=20. 4 orbits on S, 1 on S^*. Dual-eutaxy needs negative coefficients.
i=27. 2 orbits on S, 1 on S^*. Unique relation: (4,23,216). Dual-extreme.
Dual-extreme: P7[i], i=1,2,4,27,33 (5 lattices).
n=8. Survivors. p8d7[i], i=1,4,8,245,274,1175; p8d3[i], i=3296.
i=1,245,1175: E_8,D_8,A_8.
i=4. L and L^* strongly eutatic. Dual-extreme.
i=8. 3 orbits on S, 2 on S^*. L eutactic e.g., (1,4,2;4);
L^* strongly semi-eutactic. Dual-extreme.
i=274. L and L^* strongly eutatic. Dual-extreme.
i=3296. 5 orbits on S, 1 on S^*. M=x1*M1+...+x5*M5-y*Md;
M[1,1]=0 <===> y=52*x1+45*x2+42*x3+4*x4+10*x5;
Giving y this value makes M=0. Dual-extreme,
with L eutactic, L^* NOT eutactic (7 lattices).
Remark. All perfect, dual-extreme lattices up to n = 8 are extreme; moreover
their duals have a single orbit of minimal vectors except L=p8d7[8],
the dual of which is strongly semi-eutactic with two orbits.