Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $
Since the monomials form a basis of the vector space of multinomials, there is also a scalar product $ c_k=\frac{1}{N}\langle m_{\lambda_k} \mid P\rangle, $ where $N=\langle m_{\lambda_k}\mid m_{\lambda_k}\rangle$ would be a normalization constant.
My question is: Does the Scalar Product, such that the $m_λ$ are mutually orthogonal or better orthonormal, have an elementary expression?
An application could allow calculation of Kostka number, since $ s_{\lambda} = \sum K_{\lambda\mu} m_\mu, $ where $s_\lambda$ is a Schur polynomial. If this is an efficient way or not, is a different question. First I thought that I had to deal with something like square integrable functions, but then I found what I posted below $\dots$