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Questions tagged [symmetric-polynomials]

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

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Computing the shifted symmetric polynomial $s^*_{(1^2)}(x_1,x_2)$

By definition, Let $\lambda=(\lambda_1,...,\lambda_n)$ be a partition with $l(\lambda)\leq n$. We define the shifted Schur polynomial in $n$ variables corresponding to $\lambda$ as \begin{equation*} ...
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+100

inner product in the algebra of shifted symmetric function, $\Lambda^*$

In the ring of symmetric functions we have an inner product defined by $$\big \langle h_{\lambda}, m_{\mu} \big \rangle = \delta_{\lambda,\mu}.$$ Where $\delta_{\lambda,\mu}$ is the Kronecker delta. ...
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1answer
78 views

Why $\prod_i (1+x_i)$ is not considered a symmetric function, $\prod_i (1+x_i) \notin \Lambda$

The tittle of this question is the main question. Why it is necessary to not consider these kind of functions in $\Lambda$. My teacher told me that if one consider these functions to be in the ring ...
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37 views

Is there an error in given example?

I'm trying to understand the material from J. T. Kajiya's paper. In place of the term marked by the red rectangle, shouldn't there appear, according to Vieta's formulas: $(-1)^1a_0(\alpha_1+\alpha_2+...
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1answer
16 views

Does exponential of graph Laplacian preserve symmetry of polynomial?

Consider the graph: $1$--$2$--$3$. The graph Laplacian is $$L(\mathcal{G})=\begin{bmatrix}1& -1& 0\\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix}$$ Construct the polynomial in ...
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1answer
86 views

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$ [duplicate]

let $x,y,z,w\in R$,and such $x+y+z+w=2$.show that $$\sum_{cyc}\dfrac{x}{x^2-x+1}\le\dfrac{8}{3}$$ I have only solve when $x,y,z,w>0$, because $$\dfrac{x}{x^2-x+1}\le\dfrac{4}{3}x$$ so $$\sum_{...
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1answer
45 views

prove that $\frac{d (\ln F(t))}{dt}=\sum_{k>0}{(-1)^{k+1}(x_1^k+…+x_n^k)}t^{k-1}$ [on hold]

$F(t)=(1+x_1t)(1+x_2t)...(1+x_nt)$, prove that $\frac{d (\ln F(t))}{dt}=\sum_{k>0}{(-1)^{k+1}(x_1^k+...+x_n^k)}t^{k-1}$. Let's $n=1$, $F(t)=1+x_1t$, than $\frac{d (\ln F(t))}{dt}= \frac {x_1}{1+...
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24 views

Symmetric Rational Functions vs Symmetric Polynomials

Let $\{z,x_1,\ldots,x_n\}$ be variables and let $e_1,\ldots,e_n\in\mathbb{Q}[x_1,\ldots,x_n]$ be the elementary symmetric polynomials defined by $$z^n-e_1z^{n-1}+\cdots +(-1)^ne_n =(z-x_1)(z-x_2)\...
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Solving a linear system of reciprocals.

Solve for $\begin{cases}\frac{1}{x} +\frac{1}{y}+\frac{1}{z}=0\\\frac{4}{x} +\frac{3}{y}+\frac{2}{z}=5\\\frac{3}{x} +\frac{2}{y}+\frac{4}{z}=-4\end{cases}$ I turn the equations into $\begin{cases}yz+...
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Generalized Frobenius Formula

I would like to ask a very concrete question. First I recap what I know One can label a representation $\mathcal{R}$ of $\mathrm{U}(N)$ with a sequence of ordered integers (positive, negative or null)...
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Why does the numerator in Weyl-dimension formula asymptotically look like a Schur polynomial?

Let $G=SO(2r)$ and let $\{e_i\pm e_j\}_{i<j\leq r}$ be the positive roots and so the coroots are given by $\check{\alpha}=\frac{2\langle x,\alpha \rangle}{\langle \alpha, \alpha \rangle}$ where $x=(...
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maximum value of $\sum (a-b)^2$

If $a^2+b^2+c^2=5$ and $a,b,c \in \mathbb{R},$ find the maximum value of $(a-b)^2+(b-c)^2+(c-a)^2$. My Try: $(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2(ab+bc+ac)$ $$=10-2(ab+bc+ac)$$ Now this ...
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Elementary symmetric polynomials proof

In the book Polynomial Invariants of Finite Groups by Larry Smith, he proved the algebraic independence of elementary symmetric polynomials as follows: Suppose $g(e_1,…,e_n) =0$ where $ g $ is not ...
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A question for the preparation of internationals [closed]

Given the real numbers $a$ and $b$, for which it is true that $$a^3+b^3+3ab=1$$, evaluate $a+b$. I tried working this question out, by factorizing, but I didn't manage to reach a conclusion, which ...
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42 views

Solving in terms of z , three variable two equation system

Solve in terms of $z$ $$ \begin{cases} 4z&= x + 2y \\ 3z^2&=\frac{1}{2}x^2 + y^2 \\ \end{cases} $$ Solution: $x = 2z/3$ and $y = 5z/3$. I don't understand how they got to the solution with ...
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1answer
95 views

An inequality with the $constant= \frac{1}{2}+ \frac{5}{18}\,\sqrt{3}$

Given $a,\,b,\,c> 0$ such that$:$ $a+ b+ c= 3$$.$ Prove$:$ $$\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}\geqq \left ( \frac{1}{2}+ \frac{5}{18}\,\sqrt{3} \right )(\,a^{\,2}+ b^{\,2}+ c^{\,2}\,)$$ I find $...
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Does it follow that a+b+c=x+y+z=m+n+p [closed]

If $$abc=xyz=mnp$$ $a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2=x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2=m^4+n^4+p^4-2m^2n^2-2n^2p^2-2p^2m^2$ $$\frac{x}{m}=\frac{n}{b}=\frac{c}{...
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Is value of $\alpha$ defined?

Consider three quadratic functions: $$P_1(x)=ax^2-bx-c$$ $$P_2(x)=bx^2-cx-a$$ $$P_3(x)=cx^2-ax-b$$ Where $a,b,c \in \mathbb{R}\backslash \left\{0\right\}$ If there exists real number $\alpha$ such ...
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1answer
24 views

Roots of polynomial in $\mathbb{F}_{101}$ using symmetric polynomials.

I want to calculate the linear factorisation of the polynomial $f=x^5+5x^4+10x^3+10x^2+5x+70$ in the ring $\mathbb{F}_{101}[x]$. Using Eisenstein $p=5$ and Gauss' lemma, the polynomial is irreducible ...
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4answers
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Express the sum of three roots as combination of quotients

I'm trying to do an exercise relative to symmetrical polynomials. We're given the following polynomial: $ X^3 + pX +q = 0$ With $x_1, x_2, x_3 $ its roots. We're asked to give an expression with $...
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3answers
203 views

Solve the following system of equations - (3)

Solve the following system of equations: $$\large \left\{ \begin{align*} 3x^2 + xy - 4x + 2y - 2 = 0\\ x(x + 1) + y(y + 1) = 4 \end{align*} \right. $$ I tried writing the first equation as $(x - 2)(...
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1answer
118 views

Prove an inequality with positives $a$, $b$ and $c$.

If $a$, $b$ and $c$ are positives such that $(a + b + c)\left(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}\right) = x$ ($x \ge 9$) then prove that $$\large(a^2 + b^2 + c^2)\left(\dfrac{1}{a^2} + \dfrac{...
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1answer
46 views

Sum expansion of the elementary symmetric polynomials

Recently I stumbled upon the following equation: $$e_k(V\cup W) = \sum_{i=0}^{|W|}e_{k-i}(V)e_i(W)$$ $V$ and $W$ are subsets of $\{x_0,x_1,...,x_n\}$, and $V\cap W = \ \varnothing$. Where both $...
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How to solve underdetermined systems of polynomial equations?

I am trying to solve under determined simultaneous non - linear equations, where the variables are multiplied, but the power of the variables is always 1, is there a formal way doing it? For eg: $x+y+...
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2answers
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Decide for which $x,y,z$ the following equation system is met: ${1+x+y=xy}$ $2+y+z=yz$ and $5+z+x=zx$

I need to decide for which $x,y,z$ the following equation system is met: $$ 1+x+y=xy $$ $$ 2+y+z=yz $$ $$5+z+x=zx$$ I can see that $x=0, y=0, z=0$ is not a solution. I tried to divide ...
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Characters of orthogonal groups as symmetric functions

The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)={\rm Tr}\left(R_\lambda(U)\right)$. They are symmetric functions of the eigenvalues of $U$ and can be written in terms ...
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3answers
60 views

Inequalities, but working around an absolute value

So what I want to prove is $$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$ for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$. Now I'm able to ...
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Inequality with a+b+c=1 and $18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3)$

Let $a,b,c$ be reals with $a+b+c=1$. Show that : $$18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3).$$ I have tried to something like this: $$18a^4-24a^3+6a^2-12a+12\geq 0$$ $$18b^4-24b^3+6b^2-12b+...
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Vieta's Formula for Chebyshev basis

Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
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Find $K=a^2b+b^2c+c^2a$ for roots $a>b>c$ of a cubic.

If $a>b>c$ are the roots of the polynomial $P(x)=x^3-2x^2-x+1$ find the value of $K=a^2b+b^2c+c^2a$. Using Vièta's formulas: $$a+b+c=2$$ $$ab+bc+ca=-1$$ $$abc=-1$$ Using those I found that $$...
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1answer
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What is the Computational Complexity of the Elementary Symmetric Polynomials

The elementary symmetric polynomials in $n$ variables, $e_k(X_1,\dots,X_n)$, are defined implicitly by $$(X-X_1)(X-X_2) \cdots (X-X_n)=\sum_{k=0}^n (-1)^k e_k(X_1,\dots,X_n) X^{n-k}, \quad 1 \leq k \...
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1answer
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Write formula in elementary symmetric polynomials

Consider the expression $$ \prod_{i\in I, j\in J} (x-\alpha_i - \beta_j) $$ as polynomial in $x$. As this expression is symmetric in the $\alpha_i$ and in the $\beta_j$, you should be able to write ...
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44 views

shifted symmetric polynomials

BACKGROUND When defining shifted symmetric polynomials we do it in the following way: Let $\mu=(\mu_1,..., \mu_n)$ be a partition with length less or equal to $n$. Then $$s_{\mu}^*(x_1,...,x_n)=\...
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1answer
103 views

filtered algebra vs graded algebra

BACKGROUND When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions. They say that $\Lambda_n^*$ is a ...
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2answers
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skew Schur functions $C^{\lambda}_{\mu, \nu}$

When working with skew Schur functions, they can be defined as follows. Let $C^{\lambda}_{\mu, \nu}$ be the integers such that $$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$ ...
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$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)

This question is related to this other question. When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
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1answer
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Some interesting systems of equations [closed]

1, Solve the system of equations:$\left\{\begin{matrix} x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z \\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2} \end{matrix}\right.$ 2,Solve the system of ...
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Algebra precalculus problem

I need to solve this problem and I don’t know how. If $y^2 + z^2 + yz = a^2$ $z^2 + x^2 + zx = b^2$ $x^2 + y^2 + xy = c^2$ $yz + zx + xy = 0,$ then $a \pm b \pm c = 0$ I can see that $a^2 + b^2 +...
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1answer
84 views

symmetric functions vs symmetric polynomials

I am doing my thesis related with symmetric functions and representations. It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials. When reading chapter 1....
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1answer
74 views

Solving an interesting polynomial with degree 4? [duplicate]

So the equation is as follows: $$ 6x^2 -\ 25x \ + 12 \ +\ \frac6{x^2}\ + \frac{25}{x} = 0$$ So one thing that is immediately observable is that pairs of roots will be of the from $$x_1=-\frac{1}{...
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1answer
98 views

Relation of complete homogeneous symmetric polynomials and the elementary symmetric polynomials

I was reading about the symmetric polynomials and saw the following relation: $$\sum _{{i=0}}^{m}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{{m-i}}(X_{1},\ldots ,X_{n})=0\text{ for } m>0$$ The proof ...
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0answers
107 views

Symmetric group action on polynomial ring

Let the symmetric group $S_4$ act on $\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$ by permuting the 1st $4$ variables and again permuting the last $4$ variables. We can restrict the action to the ...
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1answer
39 views

On coefficients of $(t-x_1)(t-x_2)..(t-x_i)$

I was reading Emil Artin's Galois Theory (2nd edition). On pp.39-40, Artin defines a number of things as follows. Let $k$ be a field and $E=k(x_1,\ldots,x_n)$ be the field of all rational functions ...
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4answers
2k views

Super hard system of equations

Solve the system of equation for real numbers \begin{split} (a+b) &(c+d) &= 1 & \qquad (1)\\ (a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\ (a^3+b^3)&(c^3+d^3) &= 7 &...
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2answers
87 views

Advice on solving these simultaneous (quadratic/cubic) equations?

I have the following simultaneous equations: $$a x^2 + (b+2ay)x - c_1 = 0$$ $$ay^2 + (b+2ax)y - c_2 = 0$$ Where I'd like to solve for $x$ and $y$. Obviously $a,b,c_1,c_2$ are known constants. They ...
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1answer
94 views

chebyshev's inequality - Question

I had a question in my exam and they asked to prove that prove that: $$3(1+a^2+a^4)\geq(1+a+a^2)^2$$ for all $a\in\mathbb R$. Now , I solved it , but the problem is that in the answer they wrote ...
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2answers
46 views

How to solve the following system of equations?

$\left\{ \begin{aligned} xy + 2x + 2y &= -8\\ yz + 2y + 2z &= 24\\ xz + 2x + 2z &= -11 \end{aligned} \right.$ I need to solve it over the set of real numbers.
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3answers
50 views

Solve the system $x^2(y+z)=1$ ,$y^2(z+x)=8$ and $z^2(x+y)=13$

Solve the system of equations in real numbers \begin{cases} x^2(y+z)=1 \\ y^2(z+x)=8 \\z^2(x+y)=13 \end{cases} My try: Equations can be written as: \begin{cases}\frac{1}{x}=xyz\left(\frac{1}{y}+\...
2
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0answers
42 views

Are elementary symmetric polynomials evaluated in powers of double cosines of rational multiples of $\pi$ integer?

For $n,k,m\ge 1$ integer, define $$S(n,m,k)=\sum_{A\subset\{1,\ldots,n\},\\\#A=k}\left(\prod_{a\in A}(-1)^a2^m\cos^m\left(\frac{a\pi}n\right)\right).$$ In other words, $S(n,m,k)$ is the $k$'th ...
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1answer
111 views

For $x$, $y$, $z$ the sides of a triangle, show $\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$

in $\triangle ABC$, let $AB=z,BC=x,AC=y$,show that $$\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$$ by well kown Iran 96 inequality $$(xy+yz+xz)\left(\frac{1}{(...