Line 3,621: |
Line 3,621: |
| ==Note 21== | | ==Note 21== |
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| + | To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table: |
| + | |
| + | {| align="center" cellpadding="6" width="90%" |
| + | | align="center" | |
| <pre> | | <pre> |
− | To construct the regular representations of S_3,
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− | we pick up from the data of its operation table:
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− |
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| Table 1. Symmetric Group S_3 | | Table 1. Symmetric Group S_3 |
| + | o-------------------------------------------------o |
| + | | | |
| + | | ^ | |
| + | | e / \ e | |
| + | | / \ | |
| + | | / e \ | |
| + | | f / \ / \ f | |
| + | | / \ / \ | |
| + | | / f \ f \ | |
| + | | g / \ / \ / \ g | |
| + | | / \ / \ / \ | |
| + | | / g \ g \ g \ | |
| + | | h / \ / \ / \ / \ h | |
| + | | / \ / \ / \ / \ | |
| + | | / h \ e \ e \ h \ | |
| + | | i / \ / \ / \ / \ / \ i | |
| + | | / \ / \ / \ / \ / \ | |
| + | | / i \ i \ f \ j \ i \ | |
| + | | j / \ / \ / \ / \ / \ / \ j | |
| + | | / \ / \ / \ / \ / \ / \ | |
| + | | ( j \ j \ j \ i \ h \ j ) | |
| + | | \ / \ / \ / \ / \ / \ / | |
| + | | \ / \ / \ / \ / \ / \ / | |
| + | | \ h \ h \ e \ j \ i / | |
| + | | \ / \ / \ / \ / \ / | |
| + | | \ / \ / \ / \ / \ / | |
| + | | \ i \ g \ f \ h / | |
| + | | \ / \ / \ / \ / | |
| + | | \ / \ / \ / \ / | |
| + | | \ f \ e \ g / | |
| + | | \ / \ / \ / | |
| + | | \ / \ / \ / | |
| + | | \ g \ f / | |
| + | | \ / \ / | |
| + | | \ / \ / | |
| + | | \ e / | |
| + | | \ / | |
| + | | \ / | |
| + | | v | |
| + | | | |
| + | o-------------------------------------------------o |
| + | </pre> |
| + | |} |
| | | |
− | | ^
| + | Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before: |
− | | e / \ e
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− | | / \
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− | | / e \
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− | | f / \ / \ f
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− | | / \ / \
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− | | / f \ f \
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− | | g / \ / \ / \ g
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− | | / \ / \ / \
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− | | / g \ g \ g \
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− | | h / \ / \ / \ / \ h
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− | | / \ / \ / \ / \
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− | | / h \ e \ e \ h \
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− | | i / \ / \ / \ / \ / \ i
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− | | / \ / \ / \ / \ / \
| |
− | | / i \ i \ f \ j \ i \
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− | | j / \ / \ / \ / \ / \ / \ j
| |
− | | / \ / \ / \ / \ / \ / \
| |
− | | ( j \ j \ j \ i \ h \ j )
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− | | \ / \ / \ / \ / \ / \ /
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− | | \ / \ / \ / \ / \ / \ /
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− | | \ h \ h \ e \ j \ i /
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− | | \ / \ / \ / \ / \ /
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− | | \ / \ / \ / \ / \ /
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− | | \ i \ g \ f \ h /
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− | | \ / \ / \ / \ /
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− | | \ / \ / \ / \ /
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− | | \ f \ e \ g /
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− | | \ / \ / \ /
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− | | \ / \ / \ /
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− | | \ g \ f /
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− | | \ / \ /
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− | | \ / \ /
| |
− | | \ e /
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− | | \ /
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− | | \ /
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− | | v
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| | | |
− | Just by way of staying clear about what we are doing,
| + | It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. |
− | let's return to the recipe that we worked out before:
| |
| | | |
− | It is part of the definition of a group that the 3-adic
| + | Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators: |
− | relation L c G^3 is actually a function L : G x G -> G.
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− | It is from this functional perspective that we can see
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− | an easy way to derive the two regular representations.
| |
| | | |
− | Since we have a function of the type L : G x G -> G,
| + | # <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.</math> |
− | we can define a couple of substitution operators:
| + | # <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math> |
− | | |
− | 1. Sub(x, <_, y>) puts any specified x into
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− | the empty slot of the rheme <_, y>, with
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− | the effect of producing the saturated
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− | rheme <x, y> that evaluates to x·y.
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− | | |
− | 2. Sub(x, <y, _>) puts any specified x into
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− | the empty slot of the rheme <y, >, with
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− | the effect of producing the saturated
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− | rheme <y, x> that evaluates to y·x.
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| | | |
| + | <pre> |
| In (1), we consider the effects of each x in its | | In (1), we consider the effects of each x in its |
| practical bearing on contexts of the form <_, y>, | | practical bearing on contexts of the form <_, y>, |