example of reducible and irreducible -modules
Clearly both and are stable under the action of and thus in fact form submodules of .
If the characteristic of divides , then obviously . Otherwise, is a simple (irreducible) -module. For suppose has a nontrivial submodule , and choose a nonzero . Then some pair of coordinates of are unequal, for if not, then and then because of the restriction on the characteristic of forces . So apply a suitable element of to get another element of , where (note here that we use the fact that is a submodule and thus is stable under the action of ).
But now is also in , so . It is obvious that by multiplying by elements of and by permuting, we can obtain any element of and thus . Thus is simple.
It is also obvious that .
|Title||example of reducible and irreducible -modules|
|Date of creation||2013-03-22 16:37:50|
|Last modified on||2013-03-22 16:37:50|
|Last modified by||rm50 (10146)|