Articles of symmetric functions

Inequalities of expressions completely symmetric in their variables

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 […]

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 […]

Minima of symmetric functions given a constraint

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 – […]

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?

How much does $f \circ f$ determine $f$?

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 […]

How do I find out the symmetry of a function?

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 […]

Representation of symmetric functions

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 […]

Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?

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 + … = […]