Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,217
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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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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 ...
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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 $z_\lambda=\prod_{i\geq 1}i^{m_i}m_i!$ ...
- 117
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1
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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
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3
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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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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 ...
- 163
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1
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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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$k[x_1,.\dots,x_n]$ 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 ...
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1
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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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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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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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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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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 ...
- 596
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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 ...
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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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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{...
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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
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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
$$\...
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When does $\det\left(\begin{smallmatrix}1&1&1\\x_1&x_2&x_3\\x_2&x_3&x_1\end{smallmatrix}\right)=0$ imply $x_1=x_2=x_3$?
Original question. Does the condition $$\det\begin{pmatrix}1&1&1\\x_1&x_2&x_3\\x_2&x_3&x_1\end{pmatrix}=0$$ imply $x_1=x_2=x_3$?
a) Is this true if we assume $x_{1,2,3}\in\...
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$3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.
Problem: Let $a,b,c$ be positive numbers satisfied $a+b+c=3$. Prove that $$\dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \leq \dfrac{3}{4}$$
I've tried U.C.T method but it doesn'...
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Number of terms of the polynomial $\prod_{1\leq i<j\leq n}( x_{i}+x_{j})$
The number of terms of the polynomial:
$$\prod_{1\leq i<j\leq n}\left( x_{i}+x_{j}\right)$$
coincides for $n=2,3,4,5,6$ with the sequence OEIS A001858. One of the things it counts is the number of ...
- 2,188
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Number of classes of terms in the $n$-th power of the elementary symmetric polynomial
Consider the $n$-th power of the elementary symmetric polynomial $e_2(x_1,\ldots,x_n)$:
$$e_2^n(x_1,\ldots,x_n)=\left( \sum_{1 \le i \lt j \le n}{x_ix_j} \right)^n$$
and the number of classes of terms ...
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Condition for two multivariate polynomials to be equal up to a permutation of variables
Consider two multivariate polynomials with integer coefficients $P(x_1,\ldots,x_n)$ and $Q(x_1,\ldots,x_n)$.
Let's evaluate the two polynomials over all the permutations of $(x_1,\ldots,x_n)$ and ...
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Proving that $\sum_{i=0}^{n/2} (-1)^i \frac{n}{n-i} {n-i \choose i} = 2\cos(\frac{\pi n}{3})$
I was investigating the Girard-Waring identity, specifically for two variables:
$$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 $$
This lead ...
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Being given a cyclic summation on 3 letters equal to $1$, deduce the value of another cyclic summation
If
$$
\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} = \;?
$$
I tried to manipulated the equation above using some properties as
...
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Approximation of symmetric non-negative function, which is component-wise increasing by a symmetric polynomial with non-negative co-efficients
Let $f\colon [0,1]^n\to [0,\infty)$ be a symmetric (continuous) function, i.e., $f(x_1,...,x_n)=f\left(x_{\sigma(1)},...,x_{\sigma(n)}\right)$ for every permutation $\sigma$ of $\{1,...,n\}$ such that ...
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Prove an identity for elementary and complete symmetric homogeneous polynomials
I stumbled upon this formula while I was playing around with some equations, and I was wondering if anyone had any deeper insight into this problem. Let $E_i(x_1,\ldots,x_k)$ be the elementary ...
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A smarter (not bashy) way to solve this roots of unity problem?
(Mandelbrot) Let $\xi = \cos \frac{2\pi}{5} + i \sin \frac{2pi}{5}$ be a complex fifth root of unity. Set $a = 20\xi^2 + 13 \xi, b = 20\xi^4 + 13\xi^2, c = 20\xi^3 + 13\xi^4, \text{and } d = 20\xi + ...
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$x_1^n + \cdots + x_{\ell} ^n$ an integer for all $n \in \mathbb{N}$
Let $x_1$, $\ldots$, $x_{\ell} $ complex numbers such that
$$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} $$
...
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Finding Galois group of polynomial
$x^3+x^2+1$
$x^3-12x+8$
$x^4+4x^2+9$
$x^4+3x^3-3x+3$
I started taking class in Galois theory this fall and here are some problems I was told to solve. The problem is I feel huge lack of practice and ...
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Schur functors are pairwise non isomorphic
In Fulton-Harris Part I Weyl's construction there a characterization of some of the irreducible representation of $GL(V)$, with $V$ a finite complex vector space. In particular, Theorem $6.3$ point $(...
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Algebraic Independence of symmetric power sum polynomials
Let $P_k(X_1,...,X_n) = X_1^k+...+X_n^k$. My Question is how to proof that the Polynomials $(P_1,...,P_n)$ are algebraically independent. My first try was to imitate the proof of the algebraic ...
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Filling in the gap of a proof of the Fundamental Theorem of Symmetric Polynomials.
Suppose we can assume that the set of symmetric rational functions in $k(X_1,...,X_n)$, where $k$ is a field, is the same as the field generated by $k(a_1,...,a_n)$ where the $a_i$ are the elementary ...
- 345
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0
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Image of permutations $\lambda$ with $l(\lambda) \leq 2$ under RSK
Consider $p,q \in SSYT(\lambda)$ such that $l(p) = l(q) \leq 2$. I was hoping to find the image under the rsk algorithm of $(p,q)$? In other words the two lines-arrays (or generalized permutations) ...
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$\frac{3}{b+c+d}+ \frac{3}{c+d+a}+\frac{3}{d+a+b}+\frac{3}{a+b+c} \ge \frac{16}{a+b+c+d}$ for $a,b,c,d>0$ [duplicate]
Prove that
$$\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}$$
if $a,b,c,d>0$.
My attempt:
I've put this in the answers.
See also comments under this ...
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2
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What's the meaning of setting a power symmetric polynomial to a given value
I was reading an involutive introduction to symmetric functions of Mark Wildon. In particular, I was trying to understand how to derive the derangements of $S_n$ with symmetric functions. In the book ...
- 5,334
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0
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Fixed Field of Polynomials in 3 Variables
$K=k(x,y,z)$ be the field of fractions of the integral domain $R=k[x,y,z]$. Hence the field comes equipped with the automorphisms of $G=S_3$ by permuting $x,y$ and $z$.
I want to prove the following
$...
- 1,269
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0
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Symmetric function and roots of equation
Let $$x+y=p$$ $$xy=q$$
Easy to proof that any fraction $F(p,q)=Z(p,q)/R(p,q)$, where $Z(p,q)$ and $R(p,q)$ are polynomials of $p$ and $q$ is symmetric with respect to $x$, $y$.
But let's say I can ...
- 1
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Given $P \div z^{n-k} $ evaluated in the roots of $P$, find $P$.
Say there is some polynomial $P=(z-a_1)(z-a_2)\cdots(z-a_n) = Q(z) z^{n-k} + R(z)$ where the degree of $R$ is less than $n-k$.
Now say we are given $q_1=Q(a_1), q_2=Q(a_2), \dots, q_n=Q(a_n)$, but not ...
- 4,326
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Expression for symmetric part of alternating polynomial
Given an alternating polynomial $f$ in n variables $x_1, \dots, x_n$, i.e. with $f(\sigma. \textbf x) = \text{sgn}(\sigma)f(\textbf x)$, it is possible to write $f = gV$ with $V$ the Vandermonde ...
- 162
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0
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Applications of Jack polynomials
I developed a Julia package for the computation of Jack polynomials. The zonal polynomials are particular cases ($\alpha=2$) of Jack polynomials (up to a renormalization), and they have some ...
- 1,589
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Subset of $\mathbb{C}^{n}$ that are related to polynomial in $\mathbb{Q}[x]$
Let $n\in\mathbb{N}$. Is there a way to study the set $\mathcal{S}_{n}$ of $(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n})\in\mathbb{C}^{n}$ (resp. $\in\mathbb{R}^{n}$), such that the polynomial $$(x-\...
- 45
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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
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1
answer
31
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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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votes
0
answers
189
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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
$$
s_k=\...
- 57.1k
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2
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Given the values of $abc$, $ab + ac + bc$ and $a + b + c$, how do we find the value of $a^{3} + b^{3} + c^{3}$?
Given the system of equations
\begin{align*}
\begin{cases}
\hfil abc &=\;\; -6\\[6pt]
\hfil a + b + c &=\;\; \phantom{-}2\\[6pt]
\hfil ab + bc + ca &=\;\; -5
\end{cases}
\end{align*}
How ...
user1071833
1
vote
6
answers
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views
Proving $(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ with various solutions.
$(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$
Solutions in the answers.
$\ \\ \ \\ \ \\ \ \\$
Edit) Since this question is closed, I'll add more contexts for this question.
This identity is called "...
- 2,392
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votes
1
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Problem with Veblen's proof for the transcendence of $\pi$
I'm trying to understand the following proof, (no need to read all of it), but there's one point where I'm stuck. The proof is by contradiction, so they use the definition of an algebraic number (a is ...
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0
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What polynomial theorem was used here?
I was going through a proof and couldn't understand this part about homogeneous symmetric polynomials.
$$x^4(y − z) + y^4(z − x) + z^4(x − y) = −(x − y)(y − z)(z − x) $$
Viewing the left-hand side as ...
- 11
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1
answer
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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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