On Triple Ratios: 5 of 6

S3 conjugations

          Define the “pivot” operator # on {1,2,3} this way:

                   a#b  =  c  if and only if

                   a=b=c=a     or a not= b not= c not= a

 

          That is, a#b is the same or the third.

The pivot a#b is “equally equal” to a and b. Here is its table:

#   1   2   3

1   1   3   2           

2   3   2   1           

3   2   1   3  

      

x#y pivots y around x in the triangle 1,2,3. Pivot generates the reflections symmetries of the triple; double reflections are rotations, so double pivot defines these rotations:

1+x    =   1#(2#x)  =  3#(1#x)  =  2#(3#x) ;

2+x    =   2#(1#x)  =  1#(3#x)  =  3#(2#x).

1+x moves x one step up the cycle 1 < 2 < 3 < 1.

2+x moves x one step down the cycle 1 < 2 < 3 < 1.

Here are function tables for the six permutations of the triple:

  x                        |         1       2       3

---------------------|--------------------------------

0 + x                    |         1       2       3

1 + x                    |         2       3       1

2 + x                    |         3       1       2 

1#x                      |         1       3       2  

2#x                      |         3       2       1  

3#x                      |         2       1       3  

This is the permutation group on three elements: S_3. It is the same as the symmetries of the triangle. It contains three rotations (a+x) and three reflections  (a#x). Here is S₃’s group table:

         

 a*b            |       0+     1+     2+     1#     2#     3#

 _________|_______________________________________

                   |

0+     |         0+     1+     2+     1#     2#     3#

1+     |         1+     2+     0+     3#     1#     2#

2+     |         2+     0+     1+     2#     3#     1#

1#     |         1#     2#     3#     0+     1+     2+              

2#     |         2#     3#     1#     2+     0+     1+

3#     |         3#     1#     2#     1+     2+     0+

Pivot has these laws:

Recall:                          a # a  =  a

Commutativity: a # b  =  b # a

Cancellation:               a # (a # b)  =   b    

Level associativity:    (a#b) # (c#d)  =  (a#c) # (b#d)    

Transposition:             a#b = c   if and only if     a = b#c

Self-distribution:         a # (b#c)   =   (a#b) # (a#c)

If R is a permutation in S₃, then R distributes over #:

S₃ symmetry:               R(a#b) =  Ra # Rb     

This is because pivot self-distributes, and every permutation in S₃ derives from pivot.

          Pivot operates on triple ratios thus:

          a#(x₁, x₁, x₃)        =       (x_a#1, x_a#2, x_a#3)

          For any triple ratios x and y, and any index a:

          a#(x*y)       =       (a#x)*(a#y)

          a#(x/y)       =       (a#x)/(a#y)

          a#1             =       1

          a#0_n            =       0_a#n

          a#∞_n           =       ∞_a#n

          a#-1_n          =       -1_a#n

          a#(x+_ny)     =       (a#x)+_a#n(a#y)

          (1#x)* (2#x)* (3#x)     =       1

          x * (1+(x))* (2+(x))     =       1

          Since (a;b;c) corresponds, in the 3 arithmetic, to the dual number ((a/c);(b/c)), then 3#(a;b;c) corresponds to its conjugate ((b/c);(a/c)) . Likewise, 2# corresponds, in the 2 arithmetic, to dual conjugation, and 1# is dual conjugation in the 1 arithmetic.

          So the permutations are conjugations within each of the three arithmetics. But they also turn the arithmetics into each other.