Prove that if \(G\) is finite group and \(\lambda(x)\) is a linear character of \(G\), then for any irreducible character \(\chi\) of \(G\), the function \(\chi^*\) defined by \(\chi^*(\sigma)=\lambda(\sigma)\chi(\sigma)\) for all \(\sigma\in G\) is also an irreducible character of \(G\).
Let \(A:G\to GL_n(\mathbb{C})\) and \(B:G\to GL_n(\mathbb{C})\) be two representations of a finite group \(G\). Show that if for all \(\sigma\in G\), there exists a matrix \(P(\sigma)\) such that \[ \bigl(P(\sigma)\bigr)^{-1}A(\sigma)P(\sigma)=B(\sigma), \] then there exist a nonsingular matrix \(T\) such that for all \(\sigma\), \[ T^{-1}A(\sigma)T=B(\sigma). \]
Solution
(a).
1. \(\chi^*\) is the character of a representation of \(G\).
1.1. Let \(\rho: G \to \operatorname{GL}(V)\) be an irreducible representation affording the character \(\chi\).
\(\chi\) is an irreducible character.
1.2. Since \(\lambda\) is a linear character, \(\lambda: G \to \mathbb{C}^\times\) is a 1-dimensional representation.
definition of a linear character.
1.3. The 1-dimensional representation \(\lambda\) and representation \(\rho\) give a tensor product representation \(\lambda \otimes \rho: G \to \operatorname{GL}(V)\) defined by \((\lambda \otimes \rho)(\sigma) = \lambda(\sigma) \rho(\sigma)\).
standard tensor product of representations (specifically with a 1-dimensional factor).
1.4. The character of \(\lambda \otimes \rho\) is \(\operatorname{Tr}(\lambda(\sigma)\rho(\sigma)) = \lambda(\sigma)\operatorname{Tr}(\rho(\sigma)) = \lambda(\sigma)\chi(\sigma) = \chi^*(\sigma)\).
linearity of trace.
1.5. Hence \(\chi^*\) is the character of \(\lambda \otimes \rho\).
4.
2. The character \(\chi^*\) is irreducible.
2.1. A character \(\psi\) of a finite group over \(\mathbb{C}\) is irreducible if and only if \(\langle \psi, \psi \rangle = 1\).
character theory of finite groups over \(\mathbb{C}\).
2.2. Since \(G\) is finite, for each \(\sigma \in G\), \(\sigma^{|G|} = e\), so \(\lambda(\sigma)^{|G|} = \lambda(e) = 1\).
group homomorphism from a finite group.
2.3. Thus \(\lambda(\sigma)\) is a root of unity in \(\mathbb{C}\), so \(|\lambda(\sigma)| = 1\) and \(|\lambda(\sigma)|^2 = \lambda(\sigma)\overline{\lambda(\sigma)} = 1\).
roots of unity have absolute value \(1\).
2.4. Compute the inner product:\[ \langle \chi^*, \chi^* \rangle = \frac{1}{|G|} \sum_{\sigma \in G} \chi^*(\sigma)\overline{\chi^*(\sigma)} = \frac{1}{|G|} \sum_{\sigma \in G} \lambda(\sigma)\chi(\sigma)\overline{\lambda(\sigma)\chi(\sigma)} = \frac{1}{|G|} \sum_{\sigma \in G} |\lambda(\sigma)|^2 |\chi(\sigma)|^2. \]
definition of the inner product of class functions.
2.5. Substituting \(|\lambda(\sigma)|^2 = 1\) gives:\[ \langle \chi^*, \chi^* \rangle = \frac{1}{|G|} \sum_{\sigma \in G} |\chi(\sigma)|^2 = \langle \chi, \chi \rangle. \]
2.4.
2.6. Since \(\chi\) is irreducible, \(\langle \chi, \chi \rangle = 1\), so \(\langle \chi^*, \chi^* \rangle = 1\).
5 and irreducibility of \(\chi\).
2.7. Therefore \(\chi^*\) is an irreducible character of \(G\).
1, 1, and 6.
(b).
1. Pointwise similarity implies equality of characters \(\chi_A = \chi_B\).
1.1. The characters of \(A\) and \(B\) are given by \(\chi_A(\sigma) = \operatorname{Tr}(A(\sigma))\) and \(\chi_B(\sigma) = \operatorname{Tr}(B(\sigma))\).
definition of the character of a matrix representation.
1.2. For each \(\sigma \in G\), \(B(\sigma) = P(\sigma)^{-1}A(\sigma)P(\sigma)\).
hypothesis.
1.3. Since the trace is invariant under cyclic permutations and similarity transformations:\[ \operatorname{Tr}(B(\sigma)) = \operatorname{Tr}\bigl(P(\sigma)^{-1}A(\sigma)P(\sigma)\bigr) = \operatorname{Tr}(A(\sigma)). \]
cyclic property of trace: \(\operatorname{Tr}(XY) = \operatorname{Tr}(YX)\) with \(X = P(\sigma)^{-1}A(\sigma)\), \(Y = P(\sigma)\).
1.4. Hence \(\chi_A(\sigma) = \chi_B(\sigma)\) for all \(\sigma \in G\).
1 and 3.
2. Two complex representations of a finite group are isomorphic if and only if their characters are identical.
2.1. By Maschke’s theorem, every complex representation of a finite group is completely reducible (a direct sum of irreducible representations).
Maschke’s theorem for finite groups over \(\mathbb{C}\).
2.2. The multiplicity of an irreducible representation \(V_i\) in \(V\) is uniquely determined by the character via \(m_i = \langle \chi_V, \chi_i \rangle\).
orthogonality of irreducible characters.
2.3. Since \(\chi_A = \chi_B\), \(A\) and \(B\) have the same irreducible constituents with the same multiplicities.
2.2.
2.4. Thus \(A \cong B\) as \(\mathbb{C}[G]\)-modules.
2.3.
2.5. Consequently, there exists an invertible matrix \(T \in \operatorname{GL}_n(\mathbb{C})\) such that \(T^{-1}A(\sigma)T = B(\sigma)\) for all \(\sigma \in G\).
isomorphism of matrix representations.
3. Q.E.D.
2 (a) and 2 (b).