# Essays/Symmetries of the Square

The following are tables of the group of transpositions (reflections and rotations) of the square, using two sets of labels:

 ] |.@|: |."1@|. |."1@|: |. |: |."1 |.@|:@|. |.@|: |."1@|. |."1@|: ] |.@|:@|. |. |: |."1 |."1@|. |."1@|: ] |.@|: |."1 |.@|:@|. |. |: |."1@|: ] |.@|: |."1@|. |: |."1 |.@|:@|. |. |. |: |."1 |.@|:@|. ] |.@|: |."1@|. |."1@|: |: |."1 |.@|:@|. |. |."1@|: ] |.@|: |."1@|. |."1 |.@|:@|. |. |: |."1@|. |."1@|: ] |.@|: |.@|:@|. |. |: |."1 |.@|: |."1@|. |."1@|: ]
 ] \: \:@\: /:@|. |. /: /:@\: \:@|. \: \:@\: /:@|. ] \:@|. |. /: /:@\: \:@\: /:@|. ] \: /:@\: \:@|. |. /: /:@|. ] \: \:@\: /: /:@\: \:@|. |. |. /: /:@\: \:@|. ] \: \:@\: /:@|. /: /:@\: \:@|. |. /:@|. ] \: \:@\: /:@\: \:@|. |. /: \:@\: /:@|. ] \: \:@|. |. /: /:@\: \: \:@\: /:@|. ]
 ] ] identity |.@|: \: rotate counterclockwise 90° |."1@|. \:@\: rotate counterclockwise 180° |."1@|: /:@|. rotate counterclockwise 270° |. |. reflect along x-axis |: /: reflect along main diagonal |."1 /:@\: reflect along y-axis |.@|:@|. \:@|. reflect along back diagonal

The second set of labels are due to considerations discovered independently by Thomson [1979], Hui [1981], and Benkard and Seebe [1983]:

{= and i."1&1 are an inverse pair, mapping integer permutation vectors to boolean permutation matrices and vice versa. Let F be a composition of ] \: /: |. , a function on permutations, and T be a composition of ] |: |. |."1 , a transposition of the square. Identify F and T , if

```(F -: T&.({=)    ) p
(T -: F&.(i."1&1)) m
```

In other words, the group may be viewed as a group of transpositions of the square, or isomorphically as a group of functions on permutations.

For example:

```   NB. reflect along y-axis
F=: /:@\:
T=: |."1
] p=: ?.~ 5
1 4 0 3 2

({=) p
0 1 0 0 0
0 0 0 0 1
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
T ({=) p
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
i."1&1 T ({=) p
3 0 4 1 2
F p
3 0 4 1 2
(F -: T&.({=)) p
1

] m=: ({=) p
0 1 0 0 0
0 0 0 0 1
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
i."1&1 m
1 4 0 3 2
F i."1&1 m
3 0 4 1 2
({=) F i."1&1 m
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
T m
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
(T -: F&.(i."1&1)) m
1
```

The tables are a compact presentation of numerous identities involving functions ] |: |. |."1 on square matrices, or ] \: /: |. on permutations: The entry (<i,j){G is equivalent to the result of composing (<i,0){G and (<0,j){G . For example:

 i,j row column composition simplification 1 6 \: /:@\: \:@/:@\: /: |.@|: |."1 |.@|:@:(|."1) |: 5 5 /: /: /:@/: ] |: |: |:@|: ] 2 1 \:@\: \: \:@\:@\: /:@|. |."1@|. |.@|: |."1@|.@|.@|: |."1@|: 2 2 \:@\: \:@\: \:@\:@\:@\: ] |."1@|. |."1@|. |."1@|.@:(|."1)@|. ]

## References

• Iverson, Kenneth E., Formalism in Programming Languages, Communications of the ACM, Volume 7, Number 2, 1964-02, Table 9.
• Thomson, Norman D., The Geometric Primitives of APL, APL79 Conference Proceedings, 1978-05-30.
• Hui, Roger, The N Queens Problem, APL Quote Quad, Volume 11, Number 3, 1981-03.
• Benkard, J. Philip, and John N. Seebe, Reflections on Grades, APL83 Conference Proceedings, 1983-04-10.