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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?

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  • $\begingroup$ I believe you're looking for Newton's identities? $\endgroup$
    – Jakobian
    Mar 3, 2021 at 0:23
  • $\begingroup$ @Jakobian I don't see where monomial symmetric functions are expressed in terms of power sum symmetric functions. $\endgroup$
    – user567863
    Mar 3, 2021 at 0:32
  • $\begingroup$ Explaining more instead of acting mysterious would help $\endgroup$
    – Jakobian
    Mar 3, 2021 at 0:33
  • $\begingroup$ @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 $\endgroup$
    – user567863
    Mar 3, 2021 at 0:43

2 Answers 2

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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.

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  • $\begingroup$ Thanks! That's what I was looking for and yes, it's not pretty. :) $\endgroup$
    – user567863
    Mar 3, 2021 at 2:33
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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.

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