The following question illustrates the confusion I have (from The Symmetric Group by Bruce Sagan):
Let be a reducible matrix representation with block form given by where are square matrices of the same size.
Let be a module for with submodule corresponding to . Consider the quotient vector space .
Show that is a -module with corresponding matrices . Furthermore, show that we have .
My confusion is in the definition of being a submodule (i.e. and the requirement that it is a module in its own right corresponding to matrices i.e. .
How does one do this question? I have trouble getting started and confusion with all the definitions. Any help would be greatly appreciated!