off echo, nat$
out "rcontrolrootsE7.r"$

%1) positive roots of E7.

dim:=63$

           %(Fulton-Harris, page 333)

x(1):=(e1-e2-e3-e4-e5-e6+sqrt(2)*e7)/2$
x(2):=e1+e2$
x(3):=e2-e1$
x(4):=e3-e2$
x(5):=e4-e3$
x(6):=e5-e4$
x(7):=e6-e5$

x(8):=e1+e3$
x(9):=e1+e4$
x(10):=e1+e5$
x(11):=e1+e6$
x(12):=e2+e3$
x(13):=e2+e4$
x(14):=e2+e5$
x(15):=e2+e6$
x(16):=e3+e4$
x(17):=e3+e5$
x(18):=e3+e6$
x(19):=e4+e5$
x(20):=e4+e6$
x(21):=e5+e6$

x(22):=e3-e1$
x(23):=e4-e1$
x(24):=e5-e1$
x(25):=e6-e1$

x(26):=e4-e2$
x(27):=e5-e2$
x(28):=e6-e2$

x(29):=e5-e3$
x(30):=e6-e3$

x(31):=e6-e4$

x(32):= sqrt(2)*e7$

%1 signe -

x(33):=(-e1+e2+e3+e4+e5+e6+sqrt(2)*e7)/2$ % - + + + + +
x(34):=(+e1-e2+e3+e4+e5+e6+sqrt(2)*e7)/2$ % + - + + + +
x(35):=(+e1+e2-e3+e4+e5+e6+sqrt(2)*e7)/2$ % + + - + + +
x(36):=(+e1+e2+e3-e4+e5+e6+sqrt(2)*e7)/2$ % + + + - + +
x(37):=(+e1+e2+e3+e4-e5+e6+sqrt(2)*e7)/2$ % + + + + - +
x(38):=(+e1+e2+e3+e4+e5-e6+sqrt(2)*e7)/2$ % + + + + + -


%3 signe -
x(39):=(-e1-e2-e3+e4+e5+e6+sqrt(2)*e7)/2$ % - - - + + +
x(40):=(-e1-e2+e3-e4+e5+e6+sqrt(2)*e7)/2$ % - - + - + +
x(41):=(-e1-e2+e3+e4-e5+e6+sqrt(2)*e7)/2$ % - - + + - +
x(42):=(-e1-e2+e3+e4+e5-e6+sqrt(2)*e7)/2$ % - - + + + -
x(43):=(-e1+e2-e3-e4+e5+e6+sqrt(2)*e7)/2$ % - + - - + +
x(44):=(-e1+e2-e3+e4-e5+e6+sqrt(2)*e7)/2$ % - + - + - +
x(45):=(-e1+e2-e3+e4+e5-e6+sqrt(2)*e7)/2$ % - + - + + -
x(46):=(-e1+e2+e3-e4-e5+e6+sqrt(2)*e7)/2$ % - + + - - +
x(47):=(-e1+e2+e3-e4+e5-e6+sqrt(2)*e7)/2$ % - + + - + -
x(48):=(-e1+e2+e3+e4-e5-e6+sqrt(2)*e7)/2$ % - + + + - -

x(49):=(+e1-e2-e3-e4+e5+e6+sqrt(2)*e7)/2$ % + - - - + +
x(50):=(+e1-e2-e3+e4-e5+e6+sqrt(2)*e7)/2$ % + - - + - +
x(51):=(+e1-e2-e3+e4+e5-e6+sqrt(2)*e7)/2$ % + - - + + -
x(52):=(+e1-e2+e3-e4-e5+e6+sqrt(2)*e7)/2$ % + - + - - +
x(53):=(+e1-e2+e3-e4+e5-e6+sqrt(2)*e7)/2$ % + - + - + -
x(54):=(+e1-e2+e3+e4-e5-e6+sqrt(2)*e7)/2$ % + - + + - -

x(55):=(+e1+e2-e3-e4-e5+e6+sqrt(2)*e7)/2$ % + + - - - +
x(56):=(+e1+e2-e3-e4+e5-e6+sqrt(2)*e7)/2$ % + + - - + -
x(57):=(+e1+e2-e3+e4-e5-e6+sqrt(2)*e7)/2$ % + + - + - -
x(58):=(+e1+e2+e3-e4-e5-e6+sqrt(2)*e7)/2$ % + + + - - -

%5 signe -  i.e. 1 signe +
                                          % + - - - - -    x(1)
x(59):=(-e1+e2-e3-e4-e5-e6+sqrt(2)*e7)/2$ % - + - - - -
x(60):=(-e1-e2+e3-e4-e5-e6+sqrt(2)*e7)/2$ % - - + - - -
x(61):=(-e1-e2-e3+e4-e5-e6+sqrt(2)*e7)/2$ % - - - + - -
x(62):=(-e1-e2-e3-e4+e5-e6+sqrt(2)*e7)/2$ % - - - - + -
x(63):=(-e1-e2-e3-e4-e5+e6+sqrt(2)*e7)/2$ % - - - - - +

%2) sum_of_positive_roots_list.
write "The following list collects {i,x(i)} where x(i) is the i-th positive root."$
laliste:=for i:=1:DIM join {{i,x(i)}};
lalisteprime :=for i:=1:DIM collect x(i);


write "The following list sum_of_positive_roots_list collects"$
write "{{i,j},k} such that x(i)+x(j) =x(k)."$


on time$
sum_of_positive_roots_list:=
for i:=1:DIM-1 join for j:=i+1:DIM join
 if x(i)+x(j) member lalisteprime then
           for k:=1:DIM join    if x(k)=x(i)+x(j) then {{{i,j},k}}
                                else {}
 else {}$


write "sum_of_positive_roots_list:=",sum_of_positive_roots_list$



%3) verification of Property (P).
listesverif:= for i:=1:DIM-1 join for j:=i+1:DIM join
                                        for k:=i+1:DIM join
    if   x(i)+x(j) member lalisteprime
    and  x(i)+x(k) member lalisteprime
    and  x(j)+x(k) member lalisteprime
 then

           for l:=1:DIM join
                   if x(l)=x(j)+x(k) then {{{i,j,k},x(j)+x(k), l}}
                   else {}
 else {}
 $




write "listesverif:=",ws$
 IF listesverif={} then  DO <<write "Property (P) OK">>
 ELSE  DO <<write "Property (P) not verified for",listesverif >>$

bye$