Let π ∈ S n {\displaystyle {}\pi \in S_{n}} be a permutation, and let M π {\displaystyle {}M_{\pi }} denote the corresponding permutation matrix over a field K {\displaystyle {}K} . For J ⊆ { 1 , … , n } {\displaystyle {}J\subseteq \{1,\ldots ,n\}} , let
a) Show that V J {\displaystyle {}V_{J}} is M π {\displaystyle {}M_{\pi }} -invariant if and only if π ( J ) ⊆ J {\displaystyle {}\pi (J)\subseteq J} .
b) Show that there might exist M π {\displaystyle {}M_{\pi }} -invariant subspaces that are not of the form V J {\displaystyle {}V_{J}} .
be a
permutation,
and let 
denote the corresponding
permutation matrix
over a
field
. For
,
let
is
-invariant
if and only if
.
-invariant subspaces that are not of the form .
