Questions tagged [symmetric-polynomials]

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

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Determining Littlewood-Richardson coefficients $c_{\lambda,\mu}^{(2)}$ and $c_{\lambda,\mu}^{(1,1)}$.

We denote the Littlewood-Richardson coefficient by $c_{\lambda, \mu}^{\nu}$ where $\mu$ is a partition of $m$ and $\nu$ is a partition of $n$ and $\lambda$ is a partition of $m+n$. We know $c_{\lambda,...
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0 votes
1 answer
28 views

Antisymmetric polynomials in two variables

In Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials it is argued that a symmetric polynomial of two variables $x,y$ can be written as a sum over ...
0 votes
0 answers
15 views

About partitions and power sums

I am reading "Symmetric Functions" by Macdonald and I have troubles understanding a step in a proof (page 24). Let $\lambda$ be a partition. We define $z_\lambda=\prod_{i\geq 1}i^{m_i}m_i!$ ...
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2 votes
1 answer
81 views

How to evaluate $|a_1|^2+|a_2|^2+|a_3|^2+\dots$ using elementary symmetric polynomials.

Given the values of the elementary symmetric polynomials $$e_1(a_1,a_2,\dots,a_n),\, e_2(a_1,a_2,\dots,a_n),\, \dots,\, e_n(a_1,a_2,\dots,a_n)$$ of $n$ complex numbers $a_i\in\mathbb C,$ I want to ...
3 votes
3 answers
79 views

Prove a polynomial identity of $n$ variables

For integer $n\ge3$, let $A=x_1+x_2+\cdots +x_n$, define $S_n(k)$: $S_n(0)=A^n$ for $0<k\le n$, $\displaystyle S_n(k)=\sum_{\{i_1,i_2,\ldots,i_k\}\subseteq\{1,2,\ldots,n\}}\bigl(A-2(x_{i_1}+x_ {...
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1 vote
0 answers
84 views

Formula for coefficient of a certain polynomial to the nth power

I have a polynomial that looks like $$p_{3,10}(x_0, x_1, ... x_9) = (x_0 x_1 x_2 x_3 x_4 + x_0 x_1 x_2 x_3 x_5 + ... + x_5 x_6 x_7 x_8 x_9)^3$$ How do I determine the formula for the coefficient of ...
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3 votes
1 answer
68 views

Does $\sum_{j=1,j\neq k}^n\frac{z_k}{z_k-z_j}=\sum_{j=1,j\neq m}^n\frac{z_m}{z_m-z_j}$ implies the $(z_j)_{j=1,...,n}$ are the nth roots of $z_1^n$?

Let $(z_1,\ldots z_n)\subseteq \mathbb{C}.$ Does $$ \sum_{j=1\,,\,j\neq k}^n \frac{z_k}{z_k-z_j}=\sum_{j=1\,,\,j\neq k'}^n \frac{z_{k'}}{z_{k'}-z_j} $$ for all $k,k'\in\{1,...,n\}\,$, implies that ...
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0 votes
1 answer
67 views

$k[x_1,.\dots,x_n]$ is a free module over the ring of symmetric polyomials

I know that this question has been already discussed (e.g here: shorturl.at/coFQX) but I dont understand the proofs given there, and I found a different one. I want to show that $R[x_1,\dots,x_n]$ is ...
0 votes
1 answer
23 views

Finitely many $\alpha$ s.t. all conjugates $\leq N$ ($N \geq 1$)

Let $K$ be a number field, and let $N \geq 1$. Then there are only finitely many $\alpha \in O_K$ such that all conjugates of $\alpha$ have complex absolute value $\leq N$. The solution goes as ...
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4 votes
0 answers
49 views

What is the largest value of $e_{k}(x_1,\cdots,x_n)$ not obtainable over $(\mathbb{N}^+)^n$?

Let $k,n\in\mathbb{N}^+$, $\vec{x}=\langle x_1,\cdots,x_n\rangle$ be an $n$-tuple, and let $e_k(\vec{x})$ be the elementary symmetric polynomial of degree $k$ over $n$ variables (clearly with $k\le n$)...
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3 votes
2 answers
99 views

If $\alpha,\beta,\gamma,\delta$ are the roots of $x^4+px^3+qx^2+rx + s=0$, find in terms of $p,q,r,s$ the value of $\Sigma\frac{\alpha\beta}{\gamma }$

If $\alpha,\beta,\gamma,\delta$ are the roots of $x^4+px^3+qx^2+rx + s=0$, find in terms of $p,q,r,s$ the value of $\Sigma\frac{\alpha\beta}{\gamma }$ My general strategy was transforming the ...
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0 votes
0 answers
59 views

Characterization of finite fields

Let $\mathbb{K}$ be a finite field of order $q$ and with $p = char(\mathbb{K})$. Let $\mathbb{K} = \{\lambda_1 , \dots , \lambda_q\}$. I know that $x^q - x = \prod_{i=1}^{q}{(x-\lambda_i)}$ So if I ...
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1 vote
0 answers
80 views

Schur function of the conjugate representation

Consider the Schur function for irrep $(1)$ of $\mathrm{U}(3)$: $s_{(1)}(z_1,z_2,z_3)=z_1+z_2+z_3$. What is the Schur function for the irrep conjugate to this, i.e. the irrep $(1)^*$ where all the ...
2 votes
0 answers
52 views

Ring of invariants of symmetric group acting on ring of formal power series

Let $k$ be a field and $R=k[[x_1, \dots, x_n]]$ be the ring of formal power series in $n$ variables. Let $S_n$ be the symmetric group of order $n!$. Then $S_n$ acts on $R$ by $k$-automorphisms by ...
0 votes
0 answers
31 views

Is there a name for the polynomials of the form $\sum_{i = 0}^a x^{a - i}y^i - 1$? [duplicate]

I am trying to find out what is known about polynomials of the form $$ \sum_{i = 0}^a x^{a - i} y^i - 1 = x^a + x^{a-1}y + \dots + y^a - 1. $$ I tried searching for anything regarding the sums of ...
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2 votes
0 answers
15 views

Derivation of the Macdonald operator $D_{n}(X;q,t)$

Since I first encountered Equation (3.2) on p.315 of Macdonald's Symmetric functions and Hall polynomials, I have wanted to know where it comes from. So how does one derive the operator \begin{...
0 votes
0 answers
11 views

Schur polynomials: Showing that $m_ie_j=s_{(i,1^j)}+s_{(i+j,1^{j-1})}$

How to show that $m_ie_j=s_{(i,1^j)}+s_{(i+j,1^{j-1})}$, for all $i,j \in \lbrace{1,\ldots...n\rbrace}$ where $s$ are the schur polynomials, $m_i$ the monomial symmetric polynomials and $e_j$ the ...
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2 votes
0 answers
71 views

Polynomial ring of invariants and graphs

Let $G$ be a finite simple graph with nodes enumerated as $1, 2, ..., n$. Assign a variable $x_i$ to node number $i$. Consider the action of $\mathrm{Aut}\, G \subset S_n$ on the polynomial ring $$\...
6 votes
1 answer
130 views

When does $\det\left(\begin{smallmatrix}1&1&1\\x_1&x_2&x_3\\x_2&x_3&x_1\end{smallmatrix}\right)=0$ imply $x_1=x_2=x_3$?

Original question. Does the condition $$\det\begin{pmatrix}1&1&1\\x_1&x_2&x_3\\x_2&x_3&x_1\end{pmatrix}=0$$ imply $x_1=x_2=x_3$? a) Is this true if we assume $x_{1,2,3}\in\...
2 votes
2 answers
132 views

$3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.

Problem: Let $a,b,c$ be positive numbers satisfied $a+b+c=3$. Prove that $$\dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \leq \dfrac{3}{4}$$ I've tried U.C.T method but it doesn'...
2 votes
0 answers
57 views

Number of terms of the polynomial $\prod_{1\leq i<j\leq n}( x_{i}+x_{j})$

The number of terms of the polynomial: $$\prod_{1\leq i<j\leq n}\left( x_{i}+x_{j}\right)$$ coincides for $n=2,3,4,5,6$ with the sequence OEIS A001858. One of the things it counts is the number of ...
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1 vote
0 answers
28 views

Number of classes of terms in the $n$-th power of the elementary symmetric polynomial

Consider the $n$-th power of the elementary symmetric polynomial $e_2(x_1,\ldots,x_n)$: $$e_2^n(x_1,\ldots,x_n)=\left( \sum_{1 \le i \lt j \le n}{x_ix_j} \right)^n$$ and the number of classes of terms ...
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0 votes
1 answer
48 views

Condition for two multivariate polynomials to be equal up to a permutation of variables

Consider two multivariate polynomials with integer coefficients $P(x_1,\ldots,x_n)$ and $Q(x_1,\ldots,x_n)$. Let's evaluate the two polynomials over all the permutations of $(x_1,\ldots,x_n)$ and ...
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3 votes
2 answers
75 views

Proving that $\sum_{i=0}^{n/2} (-1)^i \frac{n}{n-i} {n-i \choose i} = 2\cos(\frac{\pi n}{3})$

I was investigating the Girard-Waring identity, specifically for two variables: $$x_1^n + x_2^n = \sum_{i=0}^{\frac{n}{2}} (-1)^i \frac{n}{n-i} {n-i \choose i}(x_1+x_2)^{n-2i}(x_1x_2)^i $$ This lead ...
1 vote
1 answer
45 views

Being given a cyclic summation on 3 letters equal to $1$, deduce the value of another cyclic summation

If $$ \dfrac{a}{b+c} + \dfrac{b}{a+c} + \dfrac{c}{a+b} = 1 $$ then $$ \dfrac{a^2}{b+c} + \dfrac{b^2}{a+c} + \dfrac{c^2}{a+b} = \;? $$ I tried to manipulated the equation above using some properties as ...
1 vote
0 answers
30 views

Approximation of symmetric non-negative function, which is component-wise increasing by a symmetric polynomial with non-negative co-efficients

Let $f\colon [0,1]^n\to [0,\infty)$ be a symmetric (continuous) function, i.e., $f(x_1,...,x_n)=f\left(x_{\sigma(1)},...,x_{\sigma(n)}\right)$ for every permutation $\sigma$ of $\{1,...,n\}$ such that ...
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6 votes
1 answer
92 views

Prove an identity for elementary and complete symmetric homogeneous polynomials

I stumbled upon this formula while I was playing around with some equations, and I was wondering if anyone had any deeper insight into this problem. Let $E_i(x_1,\ldots,x_k)$ be the elementary ...
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6 votes
4 answers
298 views

A smarter (not bashy) way to solve this roots of unity problem?

(Mandelbrot) Let $\xi = \cos \frac{2\pi}{5} + i \sin \frac{2pi}{5}$ be a complex fifth root of unity. Set $a = 20\xi^2 + 13 \xi, b = 20\xi^4 + 13\xi^2, c = 20\xi^3 + 13\xi^4, \text{and } d = 20\xi + ...
8 votes
1 answer
205 views

$x_1^n + \cdots + x_{\ell} ^n$ an integer for all $n \in \mathbb{N}$

Let $x_1$, $\ldots$, $x_{\ell} $ complex numbers such that $$h_n \colon =\sum_{k=1}^{\ell} x_k^n \in \mathbb{Z}$$ for all $n \in \mathbb{N}$. To show that $$e_k \colon = \sum x_{i_1} \cdots x_{i_k} $$ ...
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1 vote
1 answer
85 views

Finding Galois group of polynomial

$x^3+x^2+1$ $x^3-12x+8$ $x^4+4x^2+9$ $x^4+3x^3-3x+3$ I started taking class in Galois theory this fall and here are some problems I was told to solve. The problem is I feel huge lack of practice and ...
1 vote
1 answer
47 views

Schur functors are pairwise non isomorphic

In Fulton-Harris Part I Weyl's construction there a characterization of some of the irreducible representation of $GL(V)$, with $V$ a finite complex vector space. In particular, Theorem $6.3$ point $(...
3 votes
3 answers
176 views

Algebraic Independence of symmetric power sum polynomials

Let $P_k(X_1,...,X_n) = X_1^k+...+X_n^k$. My Question is how to proof that the Polynomials $(P_1,...,P_n)$ are algebraically independent. My first try was to imitate the proof of the algebraic ...
1 vote
2 answers
72 views

Filling in the gap of a proof of the Fundamental Theorem of Symmetric Polynomials.

Suppose we can assume that the set of symmetric rational functions in $k(X_1,...,X_n)$, where $k$ is a field, is the same as the field generated by $k(a_1,...,a_n)$ where the $a_i$ are the elementary ...
1 vote
0 answers
22 views

Image of permutations $\lambda$ with $l(\lambda) \leq 2$ under RSK

Consider $p,q \in SSYT(\lambda)$ such that $l(p) = l(q) \leq 2$. I was hoping to find the image under the rsk algorithm of $(p,q)$? In other words the two lines-arrays (or generalized permutations) ...
1 vote
1 answer
58 views

$\frac{3}{b+c+d}+ \frac{3}{c+d+a}+\frac{3}{d+a+b}+\frac{3}{a+b+c} \ge \frac{16}{a+b+c+d}$ for $a,b,c,d>0$ [duplicate]

Prove that $$\frac{3}{b+c+d}+ \frac{3}{c+d+a}+\frac{3}{d+a+b}+\frac{3}{a+b+c} \ge \frac{16}{a+b+c+d}$$ if $a,b,c,d>0$. My attempt: I've put this in the answers. See also comments under this ...
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4 votes
2 answers
85 views

What's the meaning of setting a power symmetric polynomial to a given value

I was reading an involutive introduction to symmetric functions of Mark Wildon. In particular, I was trying to understand how to derive the derangements of $S_n$ with symmetric functions. In the book ...
0 votes
0 answers
35 views

Fixed Field of Polynomials in 3 Variables

$K=k(x,y,z)$ be the field of fractions of the integral domain $R=k[x,y,z]$. Hence the field comes equipped with the automorphisms of $G=S_3$ by permuting $x,y$ and $z$. I want to prove the following $...
0 votes
0 answers
49 views

Symmetric function and roots of equation

Let $$x+y=p$$ $$xy=q$$ Easy to proof that any fraction $F(p,q)=Z(p,q)/R(p,q)$, where $Z(p,q)$ and $R(p,q)$ are polynomials of $p$ and $q$ is symmetric with respect to $x$, $y$. But let's say I can ...
3 votes
1 answer
80 views

Given $P \div z^{n-k} $ evaluated in the roots of $P$, find $P$.

Say there is some polynomial $P=(z-a_1)(z-a_2)\cdots(z-a_n) = Q(z) z^{n-k} + R(z)$ where the degree of $R$ is less than $n-k$. Now say we are given $q_1=Q(a_1), q_2=Q(a_2), \dots, q_n=Q(a_n)$, but not ...
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1 vote
0 answers
39 views

Expression for symmetric part of alternating polynomial

Given an alternating polynomial $f$ in n variables $x_1, \dots, x_n$, i.e. with $f(\sigma. \textbf x) = \text{sgn}(\sigma)f(\textbf x)$, it is possible to write $f = gV$ with $V$ the Vandermonde ...
1 vote
0 answers
28 views

Applications of Jack polynomials

I developed a Julia package for the computation of Jack polynomials. The zonal polynomials are particular cases ($\alpha=2$) of Jack polynomials (up to a renormalization), and they have some ...
0 votes
1 answer
55 views

Subset of $\mathbb{C}^{n}$ that are related to polynomial in $\mathbb{Q}[x]$

Let $n\in\mathbb{N}$. Is there a way to study the set $\mathcal{S}_{n}$ of $(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n})\in\mathbb{C}^{n}$ (resp. $\in\mathbb{R}^{n}$), such that the polynomial $$(x-\...
1 vote
0 answers
42 views

McKay correspondence for Irreps of $G \subset SU(2)$

We follow Alexander Kirilov's book "Quiver Representations and Quiver Varieties", Section 8.3: McKay correspondence: Let $G$ be a nontrivial finite subgroup in $SU(2)$. Let $Q(G)$ be the ...
1 vote
1 answer
31 views

The relationship between $f_{i}$

Let $\lambda_{i}$ be a real number for any $1 \le i \le n$, defining $f_{i}$ as following $$f_{i} = \sum^{n}_{j=1} (\lambda_{j})^{i}$$ Suppose we know $f_{1},f_{2},f_{3}$ and $f_{4}$, then can we ...
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14 votes
0 answers
189 views

Fermat's little Theorem and root power sums.

Let $f\in\mathbb{Z}[x]$ be monic of degree $n\ge 1$, and suppose $f$ factors as $$ f=(x-r_1)\cdots (x-r_n) $$ where $r_1,...,r_n$ are algebraic integers. For each positive integer $k$, let $$ s_k=\...
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0 votes
2 answers
165 views

Given the values of $abc$, $ab + ac + bc$ and $a + b + c$, how do we find the value of $a^{3} + b^{3} + c^{3}$?

Given the system of equations \begin{align*} \begin{cases} \hfil abc &=\;\; -6\\[6pt] \hfil a + b + c &=\;\; \phantom{-}2\\[6pt] \hfil ab + bc + ca &=\;\; -5 \end{cases} \end{align*} How ...
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1 vote
6 answers
187 views

Proving $(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ with various solutions.

$(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ Solutions in the answers. $\ \\ \ \\ \ \\ \ \\$ Edit) Since this question is closed, I'll add more contexts for this question. This identity is called "...
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6 votes
1 answer
178 views

Problem with Veblen's proof for the transcendence of $\pi$

I'm trying to understand the following proof, (no need to read all of it), but there's one point where I'm stuck. The proof is by contradiction, so they use the definition of an algebraic number (a is ...
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1 vote
0 answers
57 views

What polynomial theorem was used here?

I was going through a proof and couldn't understand this part about homogeneous symmetric polynomials. $$x^4(y − z) + y^4(z − x) + z^4(x − y) = −(x − y)(y − z)(z − x) $$ Viewing the left-hand side as ...
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4 votes
1 answer
37 views

Can the product of two quasisymmetric functions (that are not symmetric) be symmetric?

Given two quasisymmetric functions (see definition below) that are not symmetric, must their product also not be symmetric? From Stanley's Enumerative Combinatorics 2, recall that a function $f\in \...
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