Homework SourseLet G Ds a b a4 b2 1bab1 a1 Consider the representatio
Let G = Ds = {a, b : a^4 = b^2 = 1,bab^-1 = a^-1}. Consider the representation rho : G rightarrow GL(2,C) given by p(a) = [5 -6 4 -5] and rho (b) = [-5 6 -4 5]. Determine whether the representation is irreducible or not. ![Let G = Ds = {a, b : a^4 = b^2 = 1,bab^-1 = a^-1}. Consider the representation rho : G rightarrow GL(2,C) given by p(a) = [5 -6 4 -5] and rho (b) = [-5 6 -4 5]](/WebImages/24/let-g-ds-a-b-a4-b2-1bab1-a1-consider-the-representatio-1058344-1761552485-0.webp "Let G = Ds = {a, b : a^4 = b^2 = 1,bab^-1 = a^-1}. Consider the representation rho : G rightarrow GL(2,C) given by p(a) = [5 -6 4 -5] and rho (b) = [-5 6 -4 5]")
Solution
The unique 2 -dimensional irreducible representation of D has character values
2,-2,0,0,0 on Id, a2 ,a,b,ab respectively.
As the given representation also has the identical values on the elements specified (computing the trace), it follows that the given representation is irreducible.
