Let (V,rho_V), (W,rho_W) be irreducible representations of a group G. The tensor product representation
may or may not be irreducible

Let (V, rho_V) be a representation of G. Fix an element g from  G and let M = (2 3 // 1 1) denote the matrix of rho_V(g) with respect to the standard basis of V.
Write the matrix of rho_(V ? V*)(g).
(-2 2 -3 3// 6 -4 9 -6// -1 1 -1 1// 3 -2 3 -2)

Let G be a 2-element cyclic group, that is G = {1,g}. Consider the following functions f_i: G --> C, where
f_1(1)=1, f_1(g)=1/2, 
f_2(1)=4, f_2(g)=3, 
f_3(1)=6, f_3(g)=4. 
For each of the functions decide whether it can be a character of some complex representation of G.
never, never, can (diagonalni matice 1 1 1 1 1 -1)

Consider the natural 2-dimensional representation of the dihedral group D_16. Calculate the sum of chi(g) over all reflections g.
0

Compute the character chi of a reflection representation of S_5. It is enough to write your answer for representatives of conjugacy classes.
chi((1)(2)(3)(4)(5))= 4
chi((12)(3)(4)(5))= 2
chi((123)(4)(5))= 1
chi((12)(34)(5))= 0
chi((123)(45))= -1
chi((1234)(5))= 0
chi((12345))= -1


Let (V, rho) be a complex finite dimensional representation of G and let chi be its character. Consider the statement "for any g in G rho(g)=id_V if and only if chi(g)=dim V".
It is an incorrect statement, as only => holds.
Counterexample: G=Z, rho(1)=(1 1 // 1 0)
For finite groups, both directons are true. (This is somewhat tricky. We counted both answers.)