Let's consider the Littlewood-Richardson coefficients $c^{\lambda}_{\mu \nu}$ so that \begin{equation} V_\mu \otimes V_\nu = \bigoplus_\lambda V_\lambda^{\oplus c^{\lambda}_{\mu \nu}} \end{equation} where $V_\mu$ are representations of $GL_n$.

Usually the basis elements of the (infinite dimensional) vector space of irreducible representations of $GL_n$ are labelled by partitions. I have two questions:

Is there a basis (with basis elements labelled by $i, j, k, \cdots$) that "diagonalizes" the Littlewood-Richardson coefficients? In other words, $c^{i}_{jk} = 0$ unless $i=j=k$?

If so, is there an elegant way of relating such a basis to the basis labelled by partitions?