Elementary polynomials

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Definition:The elementary polynomials $e_\la$ are defined by the formulas

$$e_\la = e_{\la_1}e_{\la_2}\dots,$$

$$e_r = m_\la \qquad {\rm where}\;\; \la=(1,\dots,1,0,\dots,0) \;(r \;\;{\rm ones}).$$

Lemma 2   We have

$$e_{\la'}=m_\la+\sum_{\mu<\la}a_{\la\mu}m_\mu.$$

Therefore $\{e_{\la},\;\la=(n\geq\la_1\geq\dots\geq\la_m\geq 0),\; m\in{\Bbb Z}_{\geq 0}\}$ form a basis of symmetric polynomials in $n$ variables.

Exercise:Proof the lemma. $\;\Box$

Note that one can express any symmetric polynomial as a sum of products of $e_i$, $i=0,1,\dots,n$, where $e_0=1$. In the mathematical language $e_1,\dots, e_n$ are a set of generators of our ring.