Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $\rho(G\setminus H):=\sum_{g\in G\setminus H}\rho(g)$ is non-scalar. Choose such a $(\rho,V)$ and consider $(\rho^*,V^*)$, the dual of the representation $(\rho,V)$ defined by $$\rho^*:G\to GL(V^*)$$ $$\rho^*(g)=\rho(g^{-1})^T$$ where $V^*=\operatorname{Hom}(V,\mathbb{C})$ is the dual space of $V$.
My surmise is that the matrix $$T:=\rho(G\setminus H)\otimes I_{\deg(\rho)}+I_{\deg(\rho)}\otimes \rho(G\setminus H)+(\rho\otimes\rho^*)(G\setminus H)$$ has an irrational eigenvalue. Is this true?
I know that the matrices $\rho(G\setminus H)$ and $(\rho\otimes\rho^*)(G\setminus H)$ have all integer eigenvalues. To be more precise, the eigenvalues of $\rho(G\setminus H)$ are $$[\underbrace{-|H|,-|H|,\ldots,-|H|}_{\dim(V^H)~\text{times}},\underbrace{0,0,\ldots,0}_{\dim V-\dim(V^H)~\text{times}}].$$ And the eigenvalues of $(\rho\otimes\rho^*)(G\setminus H)$ are $$[|G|-|H|,\underbrace{-|H|,-|H|,\ldots,-|H|}_{\dim(V\otimes V^*)^H-1~\text{times}},\underbrace{0,0,\ldots,0}_{\dim (V\otimes V^*)-\dim(V\otimes V^*)^H~\text{times}}],$$ where $V^H:=\lbrace{v\in V~|~\rho(g)v=v, \forall v\in V}\rbrace$ is the $H$-fixed subspace of $V$ and $\dim(V^H)=\frac{1}{|H|}\sum_{h\in H}\chi_{\rho}(h)$. $(V\otimes V^*)^H$ is the $H$-fixed subspace of $V\otimes V^*$ and $\dim(V\otimes V^*)^H=\frac{1}{|H|}\sum_{h\in H}\chi_{\rho}(h)\chi_{\rho^*}(h)$.
But I don't know how to use this fact to ensure that there exists an irrational eigenvalue of $T$. Any help would be appreciated.
Cross posted in Overflow: https://mathoverflow.net/questions/503465/existence-of-irrational-eigenvalues-of-a-sum-of-representation-matrices