13. Symmetries of the Square

   D8
┌──┬──┬──┬──┬──┬──┬──┬──┐
│⊢ │⍒ │⍒⍒│⍋⌽│⌽ │⍋ │⍋⍒│⍒⌽│
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍒ │⍒⍒│⍋⌽│⊢ │⍒⌽│⌽ │⍋ │⍋⍒│
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍒⍒│⍋⌽│⊢ │⍒ │⍋⍒│⍒⌽│⌽ │⍋ │
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍋⌽│⊢ │⍒ │⍒⍒│⍋ │⍋⍒│⍒⌽│⌽ │
├──┼──┼──┼──┼──┼──┼──┼──┤
│⌽ │⍋ │⍋⍒│⍒⌽│⊢ │⍒ │⍒⍒│⍋⌽│
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍋ │⍋⍒│⍒⌽│⌽ │⍋⌽│⊢ │⍒ │⍒⍒│
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍋⍒│⍒⌽│⌽ │⍋ │⍒⍒│⍋⌽│⊢ │⍒ │
├──┼──┼──┼──┼──┼──┼──┼──┤
│⍒⌽│⌽ │⍋ │⍋⍒│⍒ │⍒⍒│⍋⌽│⊢ │
└──┴──┴──┴──┴──┴──┴──┴──┘

A permutation can be represented as an integer vector or as a square boolean matrix with exactly one 1 in each row and each column. Functions pm←⊢∘.=⍳∘≢ and mp←⍳∘1⍤1 transform from one to the other [3b].

   p                     x
6 3 2 1 5 4 0         96 84 59 5 19 47 36
   pm p      
0 0 0 0 0 0 1            x[p]
0 0 0 1 0 0 0         36 5 59 84 47 19 96
0 0 1 0 0 0 0
0 1 0 0 0 0 0            (pm p)+.×x
0 0 0 0 0 1 0         36 5 59 84 47 19 96
0 0 0 0 1 0 0
1 0 0 0 0 0 0
   mp pm p
6 3 2 1 5 4 0

⊢ ⍋ ⍒ ⌽ on permutations produce permutation results. They can be identified with ⊢ ⍉ ⊖⍉ ⊖ on square matrices.

   (⊢ pm p) ≡ pm ⊢p
1
   (⍉ pm p) ≡ pm ⍋p
1
   (⊖⍉pm p) ≡ pm ⍒p
1
   (⊖ pm p) ≡ pm ⌽p
1

Since ⊢ ⍉ ⊖⍉ ⊖ on matrices are transpositions of the square, then so are ⊢ ⍋ ⍒ ⌽ on permutations. The group table D8 is a compact presentation of numerous identities involving ⊢ ⍋ ⍒ ⌽ on permutations — D8[i;0] composed with D8[0;j] is D8[i;j] . For example:

That is, the 2 2 entry asserts that ⍒⍒⍒⍒p ←→ ⊢p ; the 1 5 entry asserts that ⍒⍋p ←→ ⌽p ; and so on. The veracity of these assertions can be checked by

   (⍎¨D8,¨'p') ≡ ⍎¨(∘.,⍨0⌷D8),¨'p' ⊣ p←?⍨n

Finally, per the previous chapter where r←,⍳⊢ relabels the group elements,

   (r D8) ≡ r ∘.{⍺[⍵]}⍨ ↓r D8

Earlier versions of these ideas appeared in [49a, 50].