Monomial polynomials

Let $\la=(\la_1,\dots,\la_n)$.

Definition:The monomial symmetric polynomial $m_\la$ is the sum of monomial $x_1^{\la_1}\dots x_n^{\la_n}$ and all distinct monomials obtained from it by a permutation of variables.

For example, if $\la=(2,1,1)$ then $m_\la=x_1^2x_2x_3+x_1x_2^2x_3+x_1x_2x_3^2$. The total degree of $m_\la$ is $\sum_i\la_i$, the degree of $m_\la$ in each variable $x_i$ is $\la_1$. In order to avoid repetitions among $m_\la$ we will always assume that $\la_1\geq\dots\geq\la_n$. A basis is the smallest set of polynomials through which you can express all the others.

Definition:A set of symmetric polynomials $S$ is called a basis, if 1) any symmetric polynomial can be expressed as a sum of polynomials from $S$ with some coefficients. 2) No polynomial from $S$ can be expressed as a sum of other polynomials from $S$.

Exercise:The monomial polynomials $\{m_\la, \la=(\la_1\geq,\dots\geq\la_n\geq 0)\}$ form a basis. $\;\Box$