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I am stuck on an exercise.

(a) Let $\mathfrak{X}$ be an irreducible $F$-representation of $G$ over an arbitrary field. Show that $\sum_{g \in G} \mathfrak{X}(g) = 0$ unless $\mathfrak{X}$ is the principal representation.

(b) Let $H \subset G$ and $g \in G$ be such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb{C}$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show that $\chi(g) = 0$.

For the second part, the hint given is to compute the trace of $\sum_{h \in H} \mathfrak{X}(hg)$, where $\mathfrak{X}$ affords $\chi$. I see that this is equal $|H| \chi(g)$, so it suffices to show that it vanishes, but I have not been able to do this.

Any hints or help would be appreciated, thanks.

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1 Answer 1

I guess $G$ is a finite group. Let $V$ denote the vector space of the representation $\mathfrak{X}$ (for both questions).

a) Let $\varphi = \sum_{g \in G} \mathfrak{X}(g)$. Then $\mathfrak{X}(g) \varphi = \varphi$ for all $g \in G$, so $\mathrm{Im} \varphi$ is contained in the subspace of $V$ consisting of the vectors on which $G$ acts trivially. Since the representation is irreducible and nontrivial, this subspace is zero.

b) Following the hint, let $\psi = \sum_{h \in H} \mathfrak{X}(hg)$, then $\mathfrak{X}(h) \psi = \psi$ for all $h \in H$, so $\mathrm{Im} \psi$ is contained in the subspace of $V$ consisting of the vectors on which $H$ acts trivially. By hypothesis this subspace is zero.

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