Schur polynomials

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Definition:A Schur function $s_\la$ is the sum of function

$$x_1^{\la_1}\dots x_n^{\la_n}\prod_{i<j}\frac{x_i}{x_i-x_j}$$

with all functions obtained from it by a permutation of variables.

Equivalently, $s_\la$ is the antisymmetrization of monomial $x_1^{\la_1}x_2^{\la_2+1}\dots x_n^{\la_n+n-1}$ divided by the Vandermond function $\prod_{i<j}(x_i-x_j)$,

$$s_\la=\left(\sum_{w}(-1)^{sgn(w)}x_{w(1)}^{\la_1}x_2^{\la_2+1}\dots x_{w(n)}^{\la_n+n-1}\right)/\prod_{i<j}(x_i-x_j),$$

where the sum is over all permutations of $n$ elements.

Exercise:Show that $s_\la$ is a symmetric polynomial. $\;\Box$

Lemma 5  

$$s_\la=m_\la+\sum_{\mu<\la}K_{\la\mu}m_\mu.$$

Therefore $\{s_{\la},\; \la=(\la_1\geq\dots\geq\la_n\geq 0)\}$ form a basis of symmetric polynomials.

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

In fact $K_{\la\mu}$ are very important nonnegative integers called Kostka numbers.