\\ MINIMAL CLASSES in DIMENSIONS 2 to 4 \\ \\ NOTE. We only consider WELL-ROUNDED minimal classes. \\ 1. The classes. \\ 2. Generic matrices. \\ 3. Eutactic matrices. \\ 4. Strict automorphisms. \\ 1. THE CLASSES \\ Minimal classes are denoted by S{n}x{r} where n is the dimension, \\ r is the perfection rank, and the letter x (=a,b,c, or d) \\ characterizes the class. The notation for dimension 4 \\ is that of "Perfect Lattices in Euclidean Spaces"; \\ for dimensions 2 and 3, it follows the notation above. \\ "Streut" means "strongly eutactic". \\ Together with each class we display the order \\ (to be read under emacs, or vi, or ...) \\ of its automorphism group, namely that of the barycenter matrix \\ (positive, definite) a*a, for a =S2a2, S2a3, ..., S4b10. \\ IN ALL CASES, Aut^+ = Aut. \\ n = 2. Streut: both classes. S2a2=[1,0;0,1]; \\ |Aut|=8=2^3 S2a3=[1,0,1;0,1,1]; \\ |Aut|=24=2^3.3 \\ n=3. Streut: S3a3, S3a4, S3a6. S3a3=[1,0,0;0,1,0;0,0,1]; \\ |Aut|=48=2^4.3 S3a4=[1,0,0,1;0,1,0,1;0,0,1,1]; \\ |Aut|=48=2^4.3 S3b4=[1,0,0,1;0,1,0,1;0,0,1,0]; \\ |Aut|=24=2^3.3 S3a5=[1,0,0,1,1;0,1,0,1,0;0,0,1,0,1]; \\ |Aut|=16=2^4 S3a6=[1,0,0,1,1,0;0,1,0,1,0,1;0,0,1,0,1,-1]; \\ |Aut|=48=2^4.3 \\ n=4. Streut: S4a4, S4a5, S4d6, S4a9, S4a10, S4b10; \\ S4b7 is strongly semi-eutactic with one zero eutaxy coefficient; \\ S4a61 ; [1,0],[0,1]: D_4; [1,1]: A_4 {M4b8=[2,-1,-1,-1;-1,2,1,x1;-1,1,2,x2;-1,x1,x2,2];} \\ e_5=e_1+e_2,e_6=e_1+e_3,e_7=e_1+e_4,e_8=e_2-e_3 \\ 0 Aut_0=Aut holds on the 4 perfect classes (S2a3,S3a6,S4a10,S4b10) \\ and on the class S4b9. \\ ---> |Aut_0|=2 holds on S3a3,S3b4, S4a4,S4a5,S4b5,S4c5,S4b6,S4d6,S4c7,S4d7. \\ ---> Aut_0 is of type (2,2) on S2a2,S4a6,S4a7,S4b7,S4b8, \\ ---> of type (2,2,2) on S3a4,S3a5,S4a8, \\ ---> and of order 72 on S4a9. \\ [a_9: Aut_0=(D_6XD_6)/{+-1} and Aut=2.Aut_0=Aut(A_2 tensor A_2).] \\ \\ ****************************** \\ ** END of FILE ** \\ ******************************