I have implemented the Schur polynomials in a computer, following two different methods.
I was intrigued by this claim from an article:
I find this paragraph a bit unclear. I'm not sure whether it means that the equality $$ \sum_{\lambda, |\lambda|=n} s_\lambda(x) = (\sum x_i)^n \tag{$\star$} $$ holds for the Schur polynomials $s_\lambda$ (I know it holds for the zonal polynomials).
So I checked. I observed that, one indeed has
$$ \sum_{\lambda, |\lambda|=2} s_\lambda(x) = s_{(2)}(x) + s_{(1,1)}(x) = (\sum x_i)^2 $$ but one has $$ s_{(3)}(x) + {\Large\color{red}{2}}s_{(2,1)}(x) + s_{(1,1,1)}(x) = (\sum x_i)^3. $$
Are these relations correct (so $(\star)$ is wrong)? And if so, is there a general relation of the form $$ \sum_{\lambda, |\lambda|=n} c_\lambda s_\lambda(x) = (\sum x_i)^n $$ for some coefficients $c_\lambda$ which have a simple expression?