Properties of the 'forgotten' symmetric polynomials

Question

In I.G. Macdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions $f$ are introduced very briefly as the result of applying an involution $\omega$ to the monomial symmetric functions $m$. This involution $\omega$ is defined earlier as the transformation of elementary $e$ into complete homogeneous symmetric $h$ polynomials.
The relation is $\sum_{r=0}^n (-1)^r e_r h_{n-r} = 0$ for $n > 0$. It obviously works on $e$ and $h$ with integer arguments.

My specific question is whether the $f$ are (only?) defined as taking a partition as argument (like the monomials $m$ and the Schur $s$), or if they obey $f_{a,b,c}=f_a f_b f_c$ like $e$, $h$ and $p$ do. If the $f$ are only defined for partition argument, than it would suffice to substitute $f$ for $m$ and $h$ for $e$ in the following example: $m_{2,1}=e_{2,1}-3 e_3$ and thus obtain $f_{2,1}=h_{2,1}-3 h_3$.

Since the definition of $f$ is only based on the (formal) relation to $m$ by involution $\omega$, it is unclear to me what kind of reasoning leads to the correct answer.

Answer 1

I think your assumption is right. In the Introduction to Symmetric Functions by Mike Zabrocki there is a nice table on page 18, that relates all kinds of symmmetric functions. There $m_\mu$ is related to $e_\lambda$ by the same coefficients $G_{\lambda\mu}$ as $f_\mu$ and $h_\lambda$ are, but he leave[s] it as an exercise to determine some sort of formula for these ($G_{\lambda \mu}$) coefficients.

If anyone ever comes to know these coefficients let me know...

Answer 2

If I understand correctly, the question is whether the forgotten symmetric functions are multiplicative, i.e, is $$f_{\lambda}=\prod_{i=1}^{l(\lambda)}f_{(\lambda_i)}$$ for every partition $\lambda$?

This is false. For example, $$f_{(1)}^2 = 2f_{(1,1)}+f_{(2)}$$ since the corresponding equation with $m$ in place of $f$ is true.

Basically, $f$ is not multiplicative since $\omega(f)=m$ is not multiplicative.