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

(

v in V?).

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!

Many thanks,

Atticus