Think about commutativity and noncommutativity, the exterior algebra.
Think about (A-B)^n. This relates to the Euler characteristic.
Elements of a noncyclic group can't be mapped to corresponding roots of unity if there are elements whose orders are relatively prime. For then you could combine the two roots of unity to get a generator that is a product of the two orders.
Open problem: Calculating plethysm.
What do we know about matrices whose power to some N equals 1? Then the power symmetric function equals ... ? And the other symmetric functions of the eigenvalues...?
What can we say about the representation theory of Coxeter groups?