ONT Re: Differential Logic -- Series A
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DLOG. Note A21
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We've seen a couple of groups, V_4 and S_3, represented in various ways, and
we've seen their representations presented in a variety of different manners.
Let us look at one other stylistic variant for presenting a representation
that is frequently seen, the so-called "matrix representation" of a group.
Recalling the manner of our acquaintance with the symmetric group S_3,
we began with the "bigraph" (bipartite graph) picture of its natural
representation as the set of all permutations or substitutions on
the set X = {A, B, C}.
Table 1. Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| A B C | A B C | A B C | A B C | A B C | A B C |
| | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | |
| v v v | v v v | v v v | v v v | v v v | v v v |
| | | | | | |
| A B C | C A B | B C A | A C B | C B A | B A C |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
Then we rewrote these permutations -- since they are
functions f : X -> X they can also be recognized as
2-adic relations f c X x X -- in "relative form",
in effect, in the manner to which Peirce would
have made us accustomed had he been given
a relative half-a-chance:
e = A:A + B:B + C:C
f = A:C + B:A + C:B
g = A:B + B:C + C:A
h = A:A + B:C + C:B
i = A:C + B:B + C:A
j = A:B + B:A + C:C
These days one is much more likely to encounter the natural representation
of S_3 in the form of a "linear representation", that is, as a family of
linear transformations that map the elements of a suitable vector space
into each other, all of which would in turn usually be represented by
a set of matrices like these:
Table 2. Matrix Representations of the Permutations in Sym(3)
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 |
| 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 |
| 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
The key to the mysteries of these matrices is revealed by noting that their
coefficient entries are arrayed and overlayed on a place mat marked like so:
[ A:A A:B A:C |
| B:A B:B B:C |
| C:A C:B C:C ]
Of course, the place-settings of convenience at different symposia may vary.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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