Representations And Characters Of Groups Second Edition

Question

1. Let $G=\left\langle x: x^{3}=1\right\rangle \cong C_{3}$, and let $V$ be the 2-dimensional $\mathbb{C G}$ module with basis $v_{1}, v_{2}$, where $$ v_{1} x=v_{2}, v_{2} x=-v_{1}-v_{2} $$ (This is a $\mathbb{C} G$-module, by Exercise 3.2.) Express $V$ as a direct sum of irreducible $\mathbb{C} G$-submodules.

Let G kx: x3 1l C3, and let V be the 2-dimensional CGmodule with basis v1, v2, where v1x v2, v2x ÿv1 ÿ v2: (This is a CG-module, by Exercise 3.2.) Express V as a direct sum of irreducible CG-submodules.

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Answer 1

Solution: representation in Exersise 8.1

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