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Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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Symmetric step function

Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $J=3$ $\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$ $\mu_j\in \mathbb{R}$ $...
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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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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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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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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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Simplifying large, expanded sums

Consider the following expression: $$ D = \left[\sum_{m=1}^M \left(\frac{1}{\sqrt{R}} \sum_{r=1}^R x^{p_1}_{R(m-1)+r}\right)^2\right] \left[\sum_{m=1}^M \left(\frac{1}{\sqrt{R}} \sum_{r=1}^R x^{p_2}_{...
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Symmetric function $p_k({x_1},{x_2},…,{x_n})={x_1}^k+{x_2}^k+…+{x_n}^k$ on $GF(q)=\{0,a_1,a_2,…,a_{q-1}\}$

This is the first time for me on symmetric functions. Let us consider the symmetric function $p_k({x_1},{x_2},...,{x_n})={x_1}^k+{x_2}^k+...+{x_n}^k$ on $GF(q)=\{0,a_1,a_2,...,a_{q-1}\}$. I want to ...
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On symmetric functions

Let $\wedge^m(X)$ be the set of homogeneous symmetric function on X such that $X$ is finite i.e. $X=\{x_1,x_2,x_3, \cdots, x_n\}$. I am able to show that the dimension of $\wedge^m(X)$ is equal to ...
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Does orthogonal-invariance of a differential imply invariance of the function?

Let $U:\text{Hom}(\mathbb{R}^d,\mathbb{R}^d) \to \mathbb{R}$ be a smooth function . If $U$ is orthogonally-invariant, i.e. $U(QA)=U(A)$ for every $Q \in \text{SO}(n),A \in \text{Hom}(\mathbb{R}^d,\...
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LR-rule and Standard Young Tableau counting

given that $s_\lambda s_\mu=\sum_{\nu} C_{\lambda \mu}^\nu s_\nu$ with $\vert \lambda\vert +\vert\mu \vert=\vert \nu\vert$, why does apparently also hold that $$h(\lambda) h(\mu) {\vert \nu\vert \...
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Graded Ring Category vs Ring Category

I know that in Ring Category we have: -Objects: Rings. -Arrows: Ring homomorphisms. I do not know which are the objects and arrows in Graded Ring Category. In general, which is the definition of ...
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Determine symmetry of given function

I am browsing the internet, looking for functions to determine its symmetry (because of effect of symmetry on Fourier coefficients) and I have found this function, from khan academy video: https://...
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Macdonald's “Symmetric Functions and Hall Polynomials” Section 1.5 Example 9

I'm trying to follow Example 9 in Section 1.5 of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points. Before stating my question, I will first ...
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$n$-th Symmetric tensors as subspace of $n$-th tensor

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Let $T^n(V)=V\otimes \cdots \otimes V$ ($n$-times). Let $S_n'(V)$ be the subspace of $T^n(V)$ spanned by $$(*)\,\,\,\,\,\,\,\,\,\,\,...
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Is the following relation considered symmetric?

i'm having a little misunderstanding about how to determinate if a relation is symmetric ( in the case of a function) So i have : $\{ \langle x,y\rangle \in\Bbb N^2 \mid x = y + 10\}$ , so the ...
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Can a function be uniquely determined from its symmetric components about two different points?

Can a function $f: \mathbb{R} \to \mathbb{R}$ be uniquely determined from its symmetric components about two different points? If so how? Not sure if I'm formulating this correctly, but I've got a ...
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Reflexive, symmetric, transitive, and antisymmetric

Can there be a relation which is reflexive, symmetric, transitive, and antisymmetric at the same time? I tried to find so. If $A = \{ a,b,c \}$. Let $R$ be a relation which is reflexive, symmetric, ...
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About the symmetric multilinear maps

Consider $E$ and $F$ two vector spaces over $\mathbb{R}$, and $f : E^n \longmapsto F$ a $n$-linear map. Assume that $f$ is symmetric, i.e. for all $(v_1,...,v_n) \in E^n$, for all permutation $\...
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Symmetric Expressions in Quadratic Equation

What is Symmetric Expression in Quadratic Equation? As per the definition of my textbook, - The Symmetric Expressions of the roots $\alpha$, $\beta$ of an equation are those expressions in $\...
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Find Range of $a$ if $ \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor $ is an Even function

Find Range of $a$ if $$ f(x)= \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor $$ is an Even function My try: we have $$f(-x)=f(x)$$ $\implies$ $$ \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor=\lfloor \frac{1}{...
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splitting fields of shifted generic polynomials

I have come across the following expression in some research papers when they want to show that splitting fields are disjoint. Let $K$ be an algebraically closed field of characteristic $p>0$, let ...
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Taylor expansion of symmetric function knowing its form at coincident points

Let a 2-variable symmetric function $F(x,y)$ be analytic around the point $x=a=y$. It admits a Taylor expansion $$ F(x,y)=\sum_{m,n\geq0}\frac{1}{m!n!} F_{m,n} (x-a)^m(y-a)^n\,. $$ Now, I know the ...
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reference request: Type A crystal proof of Schur-positivity

In this question: https://mathoverflow.net/questions/272306/what-techniques-are-there-to-prove-schur-positivity, one of the techniques listed to prove Schur-positivity is called a Type-A crystal proof,...
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positivity in bases of symmetric functions

I've read conjectures concerning the positivity of certain symmetric polynomials, which ask if a symmetric polynomial has all nonnegative coefficients when written in some basis. I'm curious what ...
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Symmetric function of the roots is a rational function of the coefficients.

I'm reading through Stewart's Galois Theory. In chapter one, the author asserts that for a cubic polynomial with roots $\alpha_0,\alpha_1,\alpha_2$, and $\omega$ a principal cube root of unity, ...
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Trivial induced representation and the Frobenius Character

I am given $\lambda$ a partition of n and I am asked to show that $\mathcal{F}(1_{triv,\lambda}\uparrow_{S_\lambda}^{S_n}) = h_\lambda$ where $\mathcal{F}$ is the Frobenius character. I don't really ...
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The Uniqueness of Symmetrization of functions

When I discuss a question with my colleague, he said he has a feeling that the symmetrization of functions should be unique up to a coefficient. However, I keep being critical at the point. ...
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Find distinct real numbers satisfying $\frac{xy}{x-y} = \frac{1}{30}$ and $\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$

Figure out with distinct real numbers the system of equations. $$\frac{xy}{x-y} = \frac{1}{30}$$ $$\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$$ I multiplied x-y both side on the first equation ...
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Solve $x+\frac{1}{y-x}=1$, $y+\frac{1}{x-y}=2$

I've got this problem: Solve for pairs of reals, $$ \left \{ \begin{array}{rcl} x+\dfrac{1}{y-x} & = & 1 \\ y+\dfrac{1}{x-y} & = & 2 \end{array} \right. $$ I've tried ...
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Proving an identity for complete homogenous symmetric polynomials

Probably everybody knows the expression: $$ \sum_{k_1,k_2\ge0}^{k_1+k_2=k}a_1^{k_1}a_2^{k_2}=\frac{a_1^{k+1}-a_2^{k+1}}{a_1-a_2}, $$ where $a_1\ne a_2$ is assumed. It seems that it can be further ...
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Name for the relationship $f\left(\frac{1}{x}\right) = \sqrt{x}f(x)$

In part of a larger problem, I have shown that I have a function $f$ which satisfies \begin{align*} f\left(\frac{1}{x}\right) = \sqrt{x}f(x), \;\; x>0. \end{align*} I am just wondering if such a ...
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Is this symmetric rational function known?

In the context of mathematical physics, a collaborator and I stumbled upon a physical quantity expressed in terms of the following rational function: $$ f(x_1, \dots, x_m;y_1,\dots, y_n) = \frac{\...
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Functions from $\mathbb{R} \rightarrow \mathbb{R}$ with 2 centers of symmetry.

"The graph of a function $f: \mathbb{R}\to\mathbb{R}$ has two has at least two centres of symmetry. Prove that $f$ can be represented as sum of a linear and periodic function." Source: sum of a linear ...
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Trouble interpreting a generating function equation

Given this generating function equation: $\prod^n_{i=1}\frac{x_iz}{e^{zx_i}-1} = \sum^\infty_{j=0}B_j^{(n)}(x_1, x_2, ...x_n)\frac{z^j}{j!}$ I am having trouble breaking down the pieces of this. I ...
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Moments of Elementary symmetric functions

For the elementary symmetric function $$e_k(\mathbf{x}) = \sum_{\substack{\mathbf{y} : y_j \in {0,1},\\\sum_j y_j = k}} \prod_{j=1}^n x_j^{y_j}$$ for a fixed $\mathbf{x}$ where all elements are ...
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Sigmoid function symmetry property

In Wikipedia, I found the symmetry equation for a sigmoid function as $g(x) + g(-x) = 1$, where $g(x) = 1/1 + \exp(-x)$. As per the property stated above, $g(x)$ becomes a symmetric function. But, ...
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Optimizing a symmetric function.

I have a set of $n$ variables $(x_1,\ldots, x_n)$ satisfying $x_i < n$ and $\sum x_i < n^{3/2}$. I want to maximize: $$\sum\limits_{i,j, i \neq j} \sqrt{x_i^{1/3} x_j^{1/3} (x_i^{1/3} + x_j^{1/3}...
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Schur symmetric functions

Let $s_{\lambda}$ denotes the Schur function associated to a partition $\lambda$ of $n$. Let $\lambda = (3,2)$, then by Giambelli's formula $s_{(3,2)} =det\begin{pmatrix} s_{3} & s_{4} \\ s_1 &...
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Solve a nonlinear system of equations in 3 variables

I need to solve this system of equations $$\frac 1x+\frac{1}{y+z}=-\frac2{15}$$ $$\frac 1y+\frac{1}{x+z}=-\frac2{3}$$ $$\frac 1z+\frac{1}{x+y}=-\frac1{4}$$ I've tried to express $x$ in terms of $y$, ...
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Minimum value of $S=\frac{3a}{b+c}+\frac{4b}{a+c}+\frac{5c}{a+b}$ [duplicate]

Find Minimum value of Minimum value of $$S=\frac{3a}{b+c}+\frac{4b}{a+c}+\frac{5c}{a+b}$$ My Try: we have $$S=3 \times \left (\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)+\frac{b}{a+c}+\frac{2c}{...
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$n a_n \frac{\partial D}{\partial a_{n-1}} + (n-1) a_{n-1} \frac{\partial D}{\partial a_{n-2}} + \cdots + a_1 \frac{\partial D}{\partial a_0} = 0$

Would you please help me solve Problem 6 of Appendix A.2, "Symmetric Functions," in An Introduction to the Theory of Numbers, Niven, Zuckerman, Montgomery, 5th ed., Wiley (New York), 1991: Suppose ...
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Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
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Is this alternative odd function equation valid?

An odd function is a function where: $$f(-x) = -f(x)$$ Is this alternative form valid ? $$f(x) = -f(-x)$$ I'm asking because an odd function graphically is symmetrical with respects to the ...
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Not the same concept as colexicographic order, although agreeing in a case of interest. Is there a name for this?

Earlier I posted this question: Is there some standard name for this particular order of terms in an elementary symmetric polynomial? This present question will further refine the definition of what ...
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Lagrange Method of Diagonalization and Another Method

I'm studying symmetric forms, and I'm stuck on understanding the Lagrange Method of Diagonalization. To be specific, when I diagonalize a symmetric matrix I thought that I should find that the ...
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Is there some standard name for this particular order of terms in an elementary symmetric polynomial?

The elementary symmetric polynomial $e_{n,k}$ in variables $x_1,\ldots,x_n$ is the sum of all products of $k$ of those $n$ variables, thus: $$ e_{n,k} = \sum_{\begin{smallmatrix} A \, \subseteq\, \{1,\...
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Is there some general way to characterize real symmetrical functions?

I am looking for a way to rewrite a function $f: \mathbb{R}^2 \mapsto \mathbb{R}$ that is symmetric in its two arguments. That is: $$ f(x,y) = f(y,x) $$ At first I was thinking that since $x$ and $y$ ...
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Product of Schur functions

Given two sets of variables $Y=\{y_1,\cdots,y_n\}$, $Z=\{z_1,\cdots,z_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s_{\lambda}(Y) s_{\mu}(Z)$...
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Why must a symmetric polynomial of the conjugates lie in $\mathbb{Q}$?

Let $K/\mathbb{Q}$ be an algebraic number field of degree $n$, and suppose $\alpha \in K.$ Let $\sigma_i \colon K \to \mathbb{C}$ for $i = 1,\dots,n$ be the $n$ embeddings. Why is it that symmetric ...