Articles of symmetric polynomials

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,…x_n) = \frac{p(x_1,…x_n)}{q(x_1,…x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if $$f(x_1,…,x_n) = f(x_{\sigma(1)},…,x_{\sigma(n)})$$ for any permutation $\sigma$ of $\{1,\ldots,n\}$. Let $F$ denote the field of rational functions and $S$ denote the subfield of symmetric rational functions. Suppose […]

Schur skew functions

Let $\lambda,\mu,\nu$ be some partitions. Let’s denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ then the Schur skew function is $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$ how can I prove that $s_{\lambda/\mu}=\sum_T x^T$ where $T$ is a tableaux of shape $\lambda/\mu$? (so we are supposing that $\mu\subset\lambda$) (I know that $c^\lambda_{\mu\nu}=0$ if $|\lambda|\neq […]

Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it

I got this question for homework and I’ve never seen anything similar to it. Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > x_2$, without solving the equation. $25x^2-5\sqrt{76}x+15=0$ I tried factoring it and I got $(-5x+\sqrt{19})^2-4=0$ What can I do afterwards that does […]

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete’s formulas. I know that $x_1^2+x_2^2+x_3^2 = -2$, as I already calculated that, but I can’t seem to get the cube of the roots. I’ve tried $$(x_1^2+x_2^2+x_3^2)(x_1+x_2+x_3)$$ but […]

What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?

The product of monomial symmetric polynomials can be expressed as $m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$ for some constants $c_{\lambda\mu}^{\nu}$. In the case of Schur polynomials, these constants are called the Littlewood-Richardson coefficients. What are they called for monomial symmetric polynomials, and how do I calculate them?

A non-Vandermonde matrix with Vandermonde-like determinant?

This question is related to the previous one. Consider $n$ variables $x_1,x_2,\ldots,x_n$ and the following $n\times n$ matrix: $$ A=\begin{bmatrix} 1 & \cdots & 1 \\ x_2 + x_3 + \dots + x_n & \dots & x_1 + x_2 + \dots + x_{n-1} \\ x_2{x_3} + x_2{x_4}+ \dots + x_{n-1}x_n & \dots & x_1{x_2} + […]

Quadratic equation, find $1/x_1^3+1/x_2^3$

In an exam there is given the general equation for quadratic: $ax^2+bx+c=0$. It is asking: what does $\dfrac{1}{{x_1}^3}+\dfrac{1}{{x_2}^3}$ equal?

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes (Av_n).$$ And the symmetric group $S_n$ acts on $V^{\otimes n}$ from the right as $$(v_1\otimes \cdots \otimes v_n)\sigma=v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)}.$$ The actions of ${\rm GL}(V)$ and $S_n$ commute with each other. More strongly, […]

A system of equations with 5 variables: $a+b+c+d+e=0$, $a^3+b^3+c^3+d^3+e^3=0$, $a^5+b^5+c^5+d^5+e^5=10$

Find the real numbers $a, b, c, d, e$ in $[-2, 2]$ that simultaneously satisfy the following relations: $$a+b+c+d+e=0$$ $$a^3+b^3+c^3+d^3+e^3=0$$ $$a^5+b^5+c^5+d^5+e^5=10$$ I suppose that the key is related to a trigonometric substitution, but not sure what kind of substitution, or it’s about a different thing.

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove whether there is a solution for this particular […]