# Monomial into power sum

This might be a routine problem but I'm unable to express monomial symmetric functions $$m_\lambda$$ into power sum symmetric functions $$p_\mu$$'s. Could you please help me with it?

• I believe you're looking for Newton's identities? Mar 3, 2021 at 0:23
• @Jakobian I don't see where monomial symmetric functions are expressed in terms of power sum symmetric functions. Mar 3, 2021 at 0:32
• Explaining more instead of acting mysterious would help Mar 3, 2021 at 0:33
• @Jakobian I'm sorry I don't understand what's mysterious here. I can't find an expression of monomial symmetric functions in terms of power sum symmetric functions Mar 3, 2021 at 0:43

## 2 Answers

There is a formula, but it is not pretty. See Mike Zabrocki's symmetric function notes, page 38. The definition of $$z_\lambda$$ is elsewhere, I believe it is $$z_\lambda=1^{m_i}m_i!\cdot 2^{m_2}m_2!\cdots$$, where $$m_i$$ is the multiplicity of $$i$$. They reference a chapter not in that document for an idea of the proof, but it is included here, page 43=44.

• Thanks! That's what I was looking for and yes, it's not pretty. :) Mar 3, 2021 at 2:33

In MacDonald's $$\textit{Symmetric Functions and Hall Polynomials}$$, Chapter I, Section 6 discusses all cases between the fundamental bases for symmetric functions. Included is a complete Table based on Kostka numbers and related matrices.