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Revision as of 04:38, 1 May 2009
, 04:38, 1 May 2009→Commentary Note 12.1
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math>
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math>
|}
The matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> is very instructive to examine at this point, and even shows the sense behind the name ''involution''.
{| 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>
|-
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
|}
|}
