We often encounter inequalities of symmetric expressions, i.e. the expression doesn’t change if the variables in it are interchanged, with the prior knowledge of a certain relation between those variables. In all such cases that I have encountered thus far, we can find the extremum of the expression by letting the variables equal. Here are […]

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 […]

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 […]

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint $g(x,y,z,\ldots)=0$, $$\bf\text{When is it true that the extrema is achieved when }\ x=y=z=\ldots?$$ An example where this claim is true: $$ g(x,y,z) = x+y+z – […]

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 […]

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?

I ran into this little problem somewhere online: if $g(x) = f(f(x)) = x^2 – x + 1$, what is $f(0)$? Plugging first $x=1$ and then $x=0$ into the identity $g(f(x)) = f(g(x))$, it is not hard to see that f(0) = f(1) = 1. But that made me wonder: what else can we really […]

For example, how do I know that with: $$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$ $f$ has the property: $$f(x_1,x_2,x_3,x_4)=f(x_2,x_1,x_4,x_3)=f(x_4,x_3,x_2,x_1)=f(x_3,x_4,x_1,x_2)$$ I mean it’s not that easy to discover all of them, right? I myself used to know only the first part of that symmetry until I find the completed one today. Is there […]

Let $n \in \mathbb{N}$. Show that every symmetric function $f\colon E^n \rightarrow \mathbb{R}$ can be written in the form $f(x) = g\Bigl(\frac{1}{n}\sum_{i=1}^n \delta_{x_i} \Bigr)$, where $g$ has to be chosen appropriately (dependent on $f$). $E$ is an arbitrary Polish space. First I asked about the meaning of “$g$”. Since we have concluded that it is […]

Given the elementary symmetric polynomials $e_k(X_1,X_2,…,X_N)$ generated via $$ \prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N. $$ How can one get the monomial symmetric functions $m_\lambda(X_1,X_2,…,X_N)$ as products and sums in $e_k$? For example: $N=4$ $$ m_{(2,1,1,0)}=X_1^2X_2X_3 + \text{all permutations}= e_3\cdot e_1 – 4 e_4 , $$ $$ m_{(2,2,0,0)}=X_1^2X_2^2 + … = […]

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