Scalar Product for Vector Space of Monomial Symmetric Functions

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 $\dots$

Solution 1:

It seems like the real question is what one can say about (the computation of) the coordinate functions for the basis of monomial symmetric functions; this does not involve a scalar product for which the monomial symmetric functions are orthonormal. (Of course once one knows the coordinate functions, the scalar product making the monomial symmetric functions an orthonormal basis can be computed as usual by multiplying corresponding coordinates and then summing them all). Whether computing these coordinate functions is difficult depends on how the symmetric polynomials are given (for instance they could be given as expressions in the elementary symmetric polynomials, or as combinations of Schur functions). For the most obvious way to give a symmetric polynomial, namely as an expression expanded in the monomials in the $X$s, the problem is almost trivial: the coefficient of $m_\lambda$ is equal to the coefficient of any monomial in the $X$s that occurs in $m_\lambda$, for instance the monomial $X^\lambda=X_1^{\lambda_1}\ldots X_n^{\lambda_n}$ (all these coefficients are equal by definition of a symmetric function). For other ways to specify symmetric functions, the problem therefore reduces to that of expanding them into monomials, which may be tedious, but is entirely straightforward.