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| ===Note 21=== | | ===Note 21=== |
| | | |
− | <pre>
| + | We've seen a couple of groups, ''V''<sub>4</sub> and ''S''<sub>3</sub>, 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. |
− | 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, | + | Recalling the manner of our acquaintance with the symmetric group ''S''<sub>3</sub>, 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''}. |
− | 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}. | |
| | | |
| + | <pre> |
| Table 1. Permutations or Substitutions in Sym {A, B, C} | | Table 1. Permutations or Substitutions in Sym {A, B, C} |
| o---------o---------o---------o---------o---------o---------o | | o---------o---------o---------o---------o---------o---------o |
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| | | | | | | | | | | | | | | | | |
| o---------o---------o---------o---------o---------o---------o | | o---------o---------o---------o---------o---------o---------o |
| + | </pre> |
| | | |
− | Then we rewrote these permutations -- since they are | + | Then we rewrote these permutations — since they are functions ''f'' : ''X'' → ''X'' they can also be recognized as 2-adic relations ''f'' ⊆ ''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: |
− | 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: | |
| | | |
| + | <pre> |
| e = A:A + B:B + C:C | | e = A:A + B:B + C:C |
| | | |
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| | | |
| j = A:B + B:A + C:C | | j = A:B + B:A + C:C |
| + | </pre> |
| | | |
− | These days one is much more likely to encounter the natural representation | + | These days one is much more likely to encounter the natural representation of ''S''<sub>3</sub> 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: |
− | 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: | |
| | | |
| + | <pre> |
| Table 2. Matrix Representations of the Permutations in Sym(3) | | Table 2. Matrix Representations of the Permutations in Sym(3) |
| o---------o---------o---------o---------o---------o---------o | | o---------o---------o---------o---------o---------o---------o |
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| | | | | | | | | | | | | | | | | |
| o---------o---------o---------o---------o---------o---------o | | o---------o---------o---------o---------o---------o---------o |
| + | </pre> |
| | | |
− | The key to the mysteries of these matrices is revealed by noting that their | + | 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: |
− | coefficient entries are arrayed and overlayed on a place mat marked like so: | |
| | | |
| + | <pre> |
| [ A:A A:B A:C | | | [ A:A A:B A:C | |
| | B:A B:B B:C | | | | B:A B:B B:C | |
| | C:A C:B C:C ] | | | C:A C:B C:C ] |
| + | </pre> |
| | | |
| Of course, the place-settings of convenience at different symposia may vary. | | Of course, the place-settings of convenience at different symposia may vary. |
− | </pre>
| |
| | | |
| ==Differential Logic : Series B== | | ==Differential Logic : Series B== |