I am reading the paper https://arxiv.org/pdf/1611.03265.pdf
Let $\mathcal{J}$ be the commutative subalgebra of the Yokonuma Hecke algebra $Y_{r,n}(q)$ geberated by $t_1,\dots t_n $ which is isomorphic to the group algebra of $(\mathbb{Z}/r\mathbb{Z})^n$ over a field $\mathbb{K}$. A character $\chi$ of $\mathcal{J}$ is determined by the choice of $\chi(t_j)\in \{\xi_1, \dots, \xi_r\}$ for $1\leq j \leq n$, where $\{\xi_1, \dots, \xi_r\}$ is the set of roots of unity of order $r$. Let $Irr\mathcal(J)$ be the set of characters of $\mathcal{J}$.
Let $E_{\chi}=\prod_{1\leq i \leq n}(\frac{1}{r}\sum_{0\leq s\leq r-1}\chi(t_i)^s{t_i}^{-s})$. Then the set $\{E_{\chi}\}$ is a complete set of primitive orthogonal idempotents of $\mathcal{J}$.
My question is (a) how to check that $\sum_{\chi\in Irr{\mathcal(J)}}E_{\chi}=1?$ By Schur orthogonality relations?
(b) Let $P_{k}(X)$ be the Lagrange polynomial $P_{k}(X)=\prod_{1\leq l \leq r,l\neq r}\frac{X-{\xi}^l}{{\xi}^k-{\xi}^l}$. Then $E_{\chi}=P_{c_1}(t_1)\dots P_{c_n}(t_n)$, where $\chi(t_k)=\xi^{c_k}$. I don't know how to check it. Any help will be appreciated.