# Questions tagged [symmetric-polynomials]

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

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### Proof of Newton's Formulas.

This is question 22 in section 14.6 in Dummit and Foote, I am trying to understand its solution: (Newton's Formulas)Let $f(x)$ be a monic polynomial of degree $n$ with roots $\alpha_1, \dots, \alpha_n$...
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### Relation between Fourier series and Schur polynomials

I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the variables $x_j=e^{i\theta_j}$. ...
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### Solving system of power sum symmetric polynomials

I'm interested in solving the following system of $n$ equations in the unknowns $x_i, i=1, ..., n$ $$\sum_{i=1}^n x^k_i = \alpha_k$$ where $k=1, ..., n$. The LHS is the power sum symmetric polynomial ...
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### Isometric automorphisms of the ring of symmetric functions

I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation ...
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### Prove $\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$

Let $a,b,c>0: abc=1.$ Prove that$$\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$$ I've tried to use equivalent steps but it is quite complicated. ...