Changes

|}

|}

It is instructive at this point to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</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%"

{| align="center" cellspacing="6" width="90%"

| <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}) ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \mathfrak{L}^\mathfrak{W}</math>

| 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>

|}

|}

0  1  2  3  4  5  6  7  8  9

0  1  2  3  4  5  6  7  8  9

</pre>

</pre>

===Commentary Note 12.2===

===Commentary Note 12.2===