"Diagonalizing" Littlewood-Richardson coefficients

If a diagonal basis existed, tensoring with a fixed representation would kill all but finitely many basis elements. This is not the case because e.g. tensoring with the $1$-dimensional trivial representation doesn't kill anything.

The vector space spanned by the irreps of $G=GL_n$ can be identified, by means of the character, with the vector space of $G$-invariant algebraic functions on $G$, for the adjoint action.

If you disregard the distinction between various kinds of functions (algebraic functions, smooth functions, distributions,...), then the Dirac delta functions at the various points of $G/G_{ad}$ can be thought of as a basis of this vector space.