I am wondering if there is any literature on relations between fractional power symmetric polynomials. For a particular example, with the variables $\textbf{x} = (x_1,x_2,\dots x_n),$, can we reasonably express $$e^{(n)}_{\frac 12}(\textbf{x}) = \sum_{i=1}^nx^{\frac 12}_i$$ in terms of $e_1,e_2,\dots e_n,$ where: $$e^{(n)}_j(\textbf{x}) = \sum_{1\le j_1 < j_2 < \cdots < j_k \le n} x_{j_1} \dotsm x_{j_k} $$ ? Square roots or radicals are to be expected if possible at all since for $n = 2$, we can simply use $$e^{(2)}_{\frac 12} = \sqrt{x_1} + \sqrt{x_2} = \sqrt{x_1+x_2+2\sqrt{x_1x_2}} = \sqrt{e^{(2)}_1 + 2\sqrt{e^{(2)}_2}}$$ and technically this idea can be repeated enough to obtain some formula but it gets really tedious after $n = 3.$
I came upon this while trying to answer this question where it is asked whether there is a nice formula for: $$\sum_{k=1}^n\tan\dfrac{k\pi}{2n+1}.$$ Interesting thing is the above is an easy problem if the terms are squared: $$\sum_{k=1}^n\tan^2\dfrac{k\pi}{2n+1} = n(2n+1),$$ which follows from the fact that $x_i = \tan^2\dfrac{k\pi}{2n+1}$ are roots of the polynomial: $$p(x) = \sum_{j=0}^n\binom{2n+1}{2j}(-z)^{n-j}.$$ This means that one way to find the sum of tangents would be to express $$x_1^{1/2}+x_2^{1/2}+\dots +x_n^{1/2}$$ by a combination of the $n$ elementary symmetric polynomials $e^{(n)}_j,$ but this also where I am stuck right now.
