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Revision as of 18:14, 1 May 2009
, 18:14, 1 May 2009→Commentary Note 12.1
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It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> at this point, not the least because it dispels the mystery of the name ''involution''.
It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> at this point, not the least because it effectively dispels the mystery of the name ''involution''. First, let us make the following observation. To say that <math>\mathrm{J}\!</math> is a lover of every woman is to say that <math>\mathrm{J}\!</math> loves <math>\mathrm{K}\!</math> if <math>\mathrm{K}\!</math> is a woman. This can be rendered in symbols as follows:
To say that <math>\mathrm{J}\!</math> is a lover of every woman is to say that <math>\mathrm{J}\!</math> loves <math>\mathrm{K}\!</math> if <math>\mathrm{K}\!</math> is a woman. This can be rendered in symbols as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
\end{bmatrix}
\end{bmatrix}
</math>
</math>
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{| align="center" cellspacing="6" width="90%"
| height="60" | <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}) ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \mathfrak{L}^\mathfrak{W}</math>
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| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
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