Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
676 questions
0
votes
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7 views
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*}
...
0
votes
0answers
6 views
+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.
...
0
votes
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 ...
0
votes
0answers
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+...
0
votes
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 ...
2
votes
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
$$\sum_{...
1
vote
1answer
45 views
prove that $\frac{d (\ln F(t))}{dt}=\sum_{k>0}{(-1)^{k+1}(x_1^k+…+x_n^k)}t^{k-1}$ [on hold]
$F(t)=(1+x_1t)(1+x_2t)...(1+x_nt)$, prove that $\frac{d (\ln F(t))}{dt}=\sum_{k>0}{(-1)^{k+1}(x_1^k+...+x_n^k)}t^{k-1}$.
Let's $n=1$, $F(t)=1+x_1t$, than $\frac{d (\ln F(t))}{dt}= \frac {x_1}{1+...
0
votes
0answers
24 views
Symmetric Rational Functions vs Symmetric Polynomials
Let $\{z,x_1,\ldots,x_n\}$ be variables and let $e_1,\ldots,e_n\in\mathbb{Q}[x_1,\ldots,x_n]$ be the elementary symmetric polynomials defined by $$z^n-e_1z^{n-1}+\cdots +(-1)^ne_n =(z-x_1)(z-x_2)\...
3
votes
4answers
88 views
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+...
2
votes
0answers
27 views
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)...
3
votes
0answers
32 views
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=(...
1
vote
2answers
83 views
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 ...
0
votes
0answers
32 views
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 ...
1
vote
2answers
67 views
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 ...
0
votes
1answer
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 ...
0
votes
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}\,)$$
I find $...
1
vote
1answer
53 views
Does it follow that a+b+c=x+y+z=m+n+p [closed]
If
$$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}{...
2
votes
2answers
66 views
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 ...
1
vote
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 ...
1
vote
4answers
46 views
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 $...
2
votes
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)(...
1
vote
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 $$\large(a^2 + b^2 + c^2)\left(\dfrac{1}{a^2} + \dfrac{...
1
vote
1answer
46 views
Sum expansion of the elementary symmetric polynomials
Recently I stumbled upon the following equation:
$$e_k(V\cup W) = \sum_{i=0}^{|W|}e_{k-i}(V)e_i(W)$$
$V$ and $W$ are subsets of $\{x_0,x_1,...,x_n\}$, and $V\cap W = \ \varnothing$. Where both $...
0
votes
2answers
35 views
How to solve underdetermined systems of polynomial equations?
I am trying to solve under determined simultaneous non - linear equations, where the variables are multiplied, but the power of the variables is always 1, is there a formal way doing it?
For eg: $x+y+...
2
votes
2answers
53 views
Decide for which $x,y,z$ the following equation system is met: ${1+x+y=xy}$ $2+y+z=yz$ and $5+z+x=zx$
I need to decide for which $x,y,z$ the following equation system is met:
$$ 1+x+y=xy $$ $$ 2+y+z=yz $$
$$5+z+x=zx$$
I can see that $x=0, y=0, z=0$ is not a solution.
I tried to divide ...
0
votes
0answers
37 views
Characters of orthogonal groups as symmetric functions
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)={\rm Tr}\left(R_\lambda(U)\right)$. They are symmetric functions of the eigenvalues of $U$ and can be written in terms ...
2
votes
3answers
60 views
Inequalities, but working around an absolute value
So what I want to prove is
$$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$
for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$.
Now I'm able to ...
2
votes
2answers
72 views
Inequality with a+b+c=1 and $18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3)$
Let $a,b,c$ be reals with $a+b+c=1$. Show that :
$$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+...
2
votes
0answers
24 views
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 $p(x)$: $...
3
votes
2answers
55 views
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
$$...
2
votes
1answer
81 views
What is the Computational Complexity of the Elementary Symmetric Polynomials
The elementary symmetric polynomials in $n$ variables, $e_k(X_1,\dots,X_n)$, are defined implicitly by
$$(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 \...
2
votes
1answer
44 views
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 ...
1
vote
1answer
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
$$s_{\mu}^*(x_1,...,x_n)=\...
1
vote
1answer
103 views
filtered algebra vs graded algebra
BACKGROUND
When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions.
They say that $\Lambda_n^*$ is a ...
1
vote
2answers
19 views
skew Schur functions $C^{\lambda}_{\mu, \nu}$
When working with skew Schur functions, they can be defined as follows.
Let $C^{\lambda}_{\mu, \nu}$ be the integers such that
$$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$
...
4
votes
3answers
84 views
$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)
This question is related to this other question.
When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
1
vote
1answer
41 views
Some interesting systems of equations [closed]
1, Solve the system of equations:$\left\{\begin{matrix}
x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z
\\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2}
\end{matrix}\right.$
2,Solve the system of ...
0
votes
1answer
46 views
Algebra precalculus problem
I need to solve this problem and I don’t know how.
If $y^2 + z^2 + yz = a^2$
$z^2 + x^2 + zx = b^2$
$x^2 + y^2 + xy = c^2$
$yz + zx + xy = 0,$ then
$a \pm b \pm c = 0$
I can see that $a^2 + b^2 +...
0
votes
1answer
84 views
symmetric functions vs symmetric polynomials
I am doing my thesis related with symmetric functions and representations.
It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials.
When reading chapter 1....
0
votes
1answer
74 views
Solving an interesting polynomial with degree 4? [duplicate]
So the equation is as follows:
$$ 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}{...
4
votes
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 ...
0
votes
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 ...
0
votes
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 ...
32
votes
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 &...
3
votes
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 ...
4
votes
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 ...
0
votes
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.
3
votes
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
votes
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 ...
0
votes
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}{(...
