Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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A proof of Pólya-Szegő inequality

Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define $$ A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}}, $$ where $\...
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Prove that $n$ hypersurfaces with cyclic permutations of the last $n-1$ variables only intersect if all the variables have the same value.

Suppose that in $\mathbb{R}^n$, I have an equation $F(x,y,z,\dots,n)=0$ with the property that it is symmetric in the last $(n-1)$ variables $\{y,z,\dots,n\}$, but not in the first variable $x$. For ...
matilda's user avatar
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Equality of two symmetric maps

Let $V$ and $W$ be two real vector spaces. I would like to show that two symmetric and multi-linear functions $$\alpha_1,\alpha_2:V^n\to W$$ are equal if and only if $$\forall v\in V:\alpha_1(v)=\...
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Why are the eigenfunctions of the Laplacian on a region with spherically symmetric boundary condition are not spherically symmetric?

For example the eigenfunctions of a spherically symmetric membrane can be found in https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane. Then again I sometimes see people in Physics saying ...
TheFibonacciEffect's user avatar
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Algebraic integers: Explicit proof that $\prod_{i,j} (x^2+\alpha_i x+\beta_j)$ (with $\alpha_i$ and $\beta_j$ conjugates) has integer coefficients.

An algebraic integer is a complex number that is the root of a monic polynomial with integer coefficients. In Exercise 3 of Chapter 6 of Ireland and Rosen's book on number theory, we are asked to show ...
Samuel Johnston's user avatar
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How to symmetrize this vector expression?

In the context of a physics calculation, I have encountered an integral of the form: $$\int d^3k_1d^3k_2F(\vec{k_1},\vec{k_2})$$ The notes that I'm reading tell me that I need to symmetrize the ...
Wild Feather's user avatar
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3 answers
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Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots

I'm trying to prove that the cubic equation $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots. The coefficients are $a_3 = - 1 - \sigma - \tau - \chi$ $a_2 = -2 (\sigma +...
Rich T's user avatar
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Pieri formula for power sum symmetric polynomial

I know Pieri's formula for elementary symmetric polynomials and for complete homogeneous symmetric polynomials, but is there an analogue for power sum symmetric polynomial? It seems that it should be ...
AndrewGap's user avatar
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Prove Cauchy-like determinant formula

How can you prove the following determinant formula, where the determinant is the same as that of the Cauchy matrix $$\det [ \frac{1}{1 - x_i y_j} ]_{i, j = 1}^n = \frac{\prod_{1 \leq i < j \leq n} ...
Jonathan McDonald's user avatar
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Symmetric PDFs for random sets

I’m studying the probability theory for finite sets (in my field we call them Random Finite Sets, but they are known as (simple) Point Processes). In this context, a random finite set is a random ...
matteogost's user avatar
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Symmetric sigmoidal function with asymptotes?

Consider the cubic function: f(x) = x + ax^3 Simple cubic function This function has several of the features that I need, which are: crosses the origin (0,0) If x > 0 then y <= x If x < 0 ...
Thiago Rangel's user avatar
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inequality about elementary symmetric polynomials with real variables

Fix $k\in \mathbb{N}$. Suppose $y_1,\ldots, y_{2k}$ are real numbers. Then $$\left(\sum_{1\le i_1<i_2<\cdots<i_k\le 2k} y_{i_1}y_{i_2}\cdots y_{i_k}\right)^2 \ge \binom{2k}{k}^2y_1y_2\cdots ...
Sayan's user avatar
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How to perform Symmetrization?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be an asymmetric function with respect to its variables. How can I transform it to become symmetric? For example, when $n=2$, take $f(x,y)=x+y^2$. Pairwise adding ...
sam wolfe's user avatar
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Fulton Harris Lemma A.28

I am trying to understand the proof of the lemma A.28 in the representation theory book by Fulton and Harris. Let $x_1,\ldots, x_k$ and $y_1,\ldots, y_k$ two distinct indipendent variables. Denote $...
Radagast's user avatar
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Skew diagram horizontal $m$-strip defintion and board strip question

In Symmetric Functions and Hall Polynomials by Manin, Manin claims that for a skew diagram $\theta = \lambda - \mu$ to be a horizontal $m$-strip, "the sequences $\lambda$ and $\mu$ are interlaced,...
Henry Yan's user avatar
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Does the given symmetry help to prove that the region in which conditions $\textbf{cond.1}$ and $\textbf{cond.2}$ hold is the same?

I have two conditions $\textbf{cond.1}$ and $\textbf{cond.2}$ for two variables $x,y\in(0,2\pi)$ $$ \text{cond.1}:=\qquad f_2<f_1<f_3 $$ $$ \text{cond.2}:=\qquad g_2<g_1<g_3 $$ where ...
Phys96's user avatar
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Basis of Symmetric Polynomials on Variables with Conditions

It is well know that there are many bases for the ring of symmetric polynomials on $k$ variable. For example, if we define $p_n = \sum_{i=1}^k x_i^n$, then every symmetric polynomial can be written as ...
Mastrel's user avatar
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Variation of the generating function of elementary symmetric functions

Let $e_k(x_1, \dots, x_n)$ be the $k$th elementary symmetric polynomial. It is known that $$ \sum_{k=0}^n e_k(x_1, \dots, x_n) t^k = \prod_{i=1}^n (1 + x_i t). $$ I've encountered the following ...
Vandenman's user avatar
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Symmetric sub-planes in multidimensional case?

Consider the following function: $$f(c_1,c_2) = \min((y-c_1)^2, (y-c_2)^2),$$ where $y$ is a fixed value, while $c_1, c_2$ are scalar arguments. The function is symmetric along $c_1 = c_2$ plane. The ...
entropy's user avatar
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Sum of absolute values of real values

Let $n\in\mathbb{N}$ and $X=\{x_1,\dotsb,x_n\}\in\mathbb{R}^n$ be given values. We define symmetrical sums and power sums of $X$ as in Newton identities. So let: $$s_i(X)=\displaystyle\sum_{\substack{ ...
Aryan's user avatar
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Cauchy identity for Schur functions

PRELIMINARY. The Cauchy identity for Schur polynomials reads $$ \sum_{\lambda}s_\lambda(x_1,...,x_n)s_\lambda(y_1,...,y_n) =\prod_{i,j=1}^n\frac 1{1-x_iy_j}, $$ where $s_\lambda$ are the Schur ...
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Find a symmetric function with certain gradient

Given $m \in (0,1)$ and $n \in \mathbb{N}$, find a symmetric differentiable function $f : (0, \infty)^n \rightarrow \mathbb{R}$ that satisfies $$ \frac{\partial f(x)}{\partial x_i} = 0 \...
Abheek Ghosh's user avatar
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Plethysm Substitution Rule issue

I am looking at the plethysm "negation rule," Theorem 6 If $g\in\Lambda^n$ is homogeneous of degree $n$ and $A$ is any plethystic alphabet, then $$g[-A]=(-1)^n\big(\omega(g)\big)[A].$$ In ...
J-anon's user avatar
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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,...
Riju's user avatar
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Why is $ \sum_\mu [s_\lambda](s_\lambda s_\mu)_{/\mu} = \sum_\rho [s_\rho](s_{\rho /\mu}) s_\mu$ for any partition $\lambda$?

Say $\lambda \vdash n$ and $\mu \vdash m$ and $|\rho|=|\lambda|+|\mu|$ then for all $\lambda$ and $m$ it appears to hold that $$ \sum_\mu [s_\lambda](s_\lambda s_\mu)_{/\mu} = \sum_\rho [s_\rho](s_{\...
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Saddle point in a non-convex objective function

I am investigating the following minimization problem as follows. It is the summation of two fractional functions that mirror each other around $(y_c,y_n)=(\frac{1}{2},\frac{1}{2})$: $Min\ z(y_c,y_n)=\...
Reza Farahani's user avatar
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About the definition of pullback (in diff. geom.)

Consider $(d<\infty)$-dimensional ${ \Bbb R}$-vector spaces $V$ and $W$, and their dual spaces $V^*:=Hom(V,{\Bbb R})$ and $W^*:=Hom(W,{\Bbb R})$. Naturally, $V,W,V^*,W^*$ are all isomorphic. Now, ...
PSB's user avatar
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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 ...
user1072285's user avatar
1 vote
1 answer
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Equivalent conditions for differentiability of radially symmetric functions

Let $f : B(0, 1) \subset \mathbf R^n \to \mathbf R$ be a $C^k$-function, namely all partial derivatives up to order $k$ are continuous. Suppose that $f$ is also radially symmetric with respect to the ...
QA Ngô's user avatar
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1 answer
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Symmetric optimization problem

Consider a linear or non-linear optimization problem of the form: $$\min x_1$$ $$x_1 \ge x_2 \ge x_3$$ $$f_1(x_1,x_2,x_3,x_{12},x_{13},x_{23},x_{123}) \ge 0$$ $$\ldots$$ $$f_n(x_1,x_2,x_3,x_{12},x_{13}...
Fabius Wiesner's user avatar
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Frobenius series for the $S_n$-module $k[X]$

I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants (which can be found here). In what follows, all vector spaces and algebras are over the field of rational ...
Albert's user avatar
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How to write proof using the symetrical structure of the equation?

I encountered a problem recently, Let $f(x_1,x_2,x_3)=e^{-(x_1+x_2+x_3)},0<x_1,x_2,x_3<\infty$ be the joint pdf of random variables $X_1,X_2,X_3$. Find $P(x_1<x_2<x_3)$. I want to state ...
BlackSheep1048576's user avatar
2 votes
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Sum of Schur functions incorporating the length statistic

Macdonald's Symmetric Functions and Hall Polynomials includes the following identities (listed as exercises on pages 76 and 78 respectively): \begin{equation} \begin{array}{ll} \text{Ex.} \,(4) \quad \...
Jeanne Scott's user avatar
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Linear involution for Specht modules

Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$. Here the Specht modules are ...
Albert's user avatar
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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{...
BatsOnASwing's user avatar
2 votes
1 answer
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Is the Ring of Symmetric Functions complete?

So, this question came up in a discussion today and I thought I'd post it here. Given that the ring of symmetric functions $\Lambda$ can be equipped with the Hall scalar product. Is it also a complete ...
BatsOnASwing's user avatar
1 vote
1 answer
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Computing elementary symmetric polynomials of degree $3$

My question is related to page 4 of http://math.uchicago.edu/~may/REU2020/REUPapers/Graham.pdf Let $e_j=\sum \limits_{1\leq i_1<...<i_j \leq n}x_{i_1}\dots x_{i_j}$ be the elementary symmetric ...
Gurterz's user avatar
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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 ...
Sumanta's user avatar
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1 vote
1 answer
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Prove that this differential equation has reflective symmetric about a line

Context I am studying analytic mechanics using [1]. In the context of orbital mechanics the authors arrive at the following initial-value problem. \begin{align} \frac{d^2 u{(\theta)}}{d\theta^2 } + ...
Michael Levy's user avatar
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Equivalent symmetric sums proof [closed]

Let $a$ and $b$ be integers greater than one which have no common divisors. Prove that $$\sum_{i=1}^{b-1}\left\lfloor\frac{ai}{b}\right\rfloor=\sum_{j=1}^{a-1}\left\lfloor\frac{bj}{a}\right\rfloor.$$ ...
Justintime's user avatar
1 vote
1 answer
104 views

Is there a symmetric real valued function on ${\mathbb R}^n$ that lies between sum and product?

I am seeking a sequence of functions $f_n:{\mathbb R}^n \rightarrow {\mathbb R}$ where $n=1, 2, 3, \ldots$ which have the following properties: Within the unit cube, for each point $(x_1, x_2, \ldots,...
Bilal Khan's user avatar
1 vote
1 answer
81 views

Does this 3D function need to be symmetric for $\frac {\partial f(x,y)}{\partial x}= |\frac {\partial f(x,y)}{\partial y}|$ for all $x < y$?

Suppose I have a 3D function $f(x,y):\mathbb {R^2_{\ge 0}} \to \mathbb {R_{\le0}}$ such that $$ \frac {\partial f(x,y)}{\partial x}= \begin{cases} >0&\text{if}\, x<y \\\ 0&\text{if}\, ...
Eli J's user avatar
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1 vote
2 answers
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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 ...
Paul Epstein's user avatar
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0 answers
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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) ...
jacopoburelli's user avatar
4 votes
2 answers
93 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 ...
jacopoburelli's user avatar
3 votes
2 answers
60 views

algebraic binary operations that are associative and nice in a way.

It is well known that $$ a*b = ab + a + b$$ and $$ x*y = \dfrac{x+y}{1-xy} $$ are commutative associative binary operations. What rational function $f$ is a commutative associative binary operations ...
kazuki's user avatar
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Strange property of the functional equation $f(x+y)=ax+by$.

Suppose $$\tag{*} f(x+y)=ax+by $$ holds for some function $f:\mathbb{R}\to\mathbb{R}$ and some $a,b\in\mathbb{R}$. Given the function $g:\mathbb{R}^2\to\mathbb{R}$ defined as $$ g(x,y)=f(x+y) $$ we ...
sam wolfe's user avatar
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2 votes
2 answers
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Finding Solutions to a Symmetric Non-Linear Equation (for Some Cases Beside $x = y = z$)

Find all values $x,y,z$ (whether real or complex) such that : $2x-2y+z^{-1} = 2022^{-1}\\ 2z-2x+y^{-1}=2022^{-1}\\ 2y-2z + x^{-1} = 2022^{-1}$ I know that for case when $x=y=z$ i can easily find : $...
JangoHypno's user avatar
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How to formulate the symmetry in Ramsey's theorem for $c$ colours?

Ramsey's theorem for $c$ colours states that (this statement is not copied from anywhere): Let $c\ge 2$ be an integer, $[c]=\{1,2,\ldots,c\}$ represent $c$ colours, and let $(r_1,r_2,\ldots,r_c)$ be ...
Favst's user avatar
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Symmetric pairing functions?

Most common pairing functions $\pi: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, like the Cantor pairing function, are not symmetric (i.e $\pi(k_1, k_2) \neq \pi(k_2, k_1)$), it seems to me. Xie 2021 ...
GrueEmerald's user avatar

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