The Macdonald symmetric functions (or Macdonald polynomials) $P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar product $$ \langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}z_\lambda\prod_{i=1}^{\ell(\lambda)} \frac{1-q^{\lambda_i}}{1-t^{\lambda_i}}. $$ I am using standard symmetric function notation as found in Macdonald, Symmetric Functions and Hall Polynomials, second ed. The $P_\lambda(x)$'s are uniquely determined by this orthogonality and by the triangularity condition $$ P_\lambda(x) = m_\lambda(x) +\sum_{\mu<_L\lambda}u_{\lambda\mu}m_\mu(x),\ (*) $$ where $<_L$ denotes lexicographic order on partitions of $n$ (with $\langle 1^n\rangle$ first and $(n)$ last). It is a nontrivial fact that (*) continues to hold if $<_L$ is replaced by dominance order $<$. (See the remark on page 323 of Macdonald's book.) What is the most general scalar product with this property? Does it have to coincide with the Macdonald scalar product or some limits thereof (such as the Jack scalar product, where we replace $\frac{1-q^{\lambda_i}}{1-t^{\lambda_i}}$ with $\alpha^{\ell(\lambda)}$)?
Another viewpoint is obtained by defining $$ \langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu} y_\lambda, $$ where $y_\lambda$ is an indeterminate. If we restrict to partitions of $n$, then the condition that triangularity with respect to $<_L$ implies triangularity with respect to $<$ defines an algebraic variety. What is the structure of this variety?
