Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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33 views

Cycle indicator symmetric function and Polya's theorem

Let $G$ be a subgroup of the symmetric group $S_n$. Given a partition $\lambda$ of $n$, denote by $n_G(\lambda)$ the number of elements in $G$ of cycle-type $\lambda$. The cycle indicator function of $...
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1answer
86 views

If X and Y have both the same mean and a symmetrical distribution, do we have that $P(X<Y)=P(X>Y)$? [closed]

Let X and Y be two continuous independent real-valued random variables. Assume that X and Y have the same expectation $\mu\in\mathbb R$ and that X and Y have symmetrical density functions with respect ...
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14 views

Functions orthogonal to linear span when considering a normal distribution

Let $x \sim N(0, 1)$ be a random variable distributed as standard normal. I am looking for functions that satisfy $$E[x f(x)] = 0 $$. In particular, functions whose best approximation is a constant ...
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30 views

Minors of Jacobians of elementary symmetric polynomials

Let $e_1,\ldots,e_n\in \mathbb{C}[x_1,\ldots,x_n]$ denote the elementary symmetric polynomials and $J_{e_1,\ldots,e_n}$ the Jacobian matrix. Then, the determinant of $J_{e_1,\ldots,e_n}$ is equal to $\...
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0answers
15 views

Generating function of column-strict plane partitions with fixed support

A plane partition is a set $P$ of triples $(x,y,z)$ such that if in the lexicographic order we have $(x',y',z')\leq (x,y,z)$, then $(x',y',z')\in P$. We can define a partition $\lambda(P)$ that plays ...
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18 views

Given a nondegenerate, symmetric bilinear form, and linearly independent vectors, find a set of vectors s.t. $g\left(v_{i},w_{j}\right)=\delta_{ij}$

I'm trying to prove the following theorem: Let $V$ be a $n$-dimensional vector space over $\mathbb{F}$, $g:V\times V\to \mathbb{F}$ a symmetric and non-degenerate bilinear form. If $v_1,\ldots ,v_k \...
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1answer
34 views

Generalization of the formula representing power sums as sums of Schur polynomials

Let $m_1\ge m_2\ge\ldots\ge m_k$ are nonnegative integers. Then, we can consider the following product of power sums: $$ p_m(x_1,\ldots,x_n)=\prod_{i=1}^{k}\sum_{j=1}^{n}x_j^{m_i}. $$ Since Schur ...
3
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1answer
33 views

Understanding an isomorphism between direct limit of the character group and character group of the inverse limit

I am struggling with the following general setup from Chapter IV(page 269) in Macdonald's book on symmetric functions and Hall polynomials. Let $$K=\varprojlim M_n$$ be their inverse limit, which is ...
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1answer
21 views

Show that $\prod_{i=1}^n\frac{z-tx_i}{z-x_i}=1+\sum_{i=1}^n\frac{(1-t)x_i}{z-x_i}\prod_{\substack{j\ne i,\\j\in [n]}}\frac{x_i-tx_j}{x_i-x_j}.$

Let $x_1,...,x_n,t,z$ are independent indeterminants over the ring of integer $\mathbb Z$, how to prove that: $$\prod_{i=1}^n\frac{z-tx_i}{z-x_i}=1+\sum_{i=1}^n\frac{(1-t)x_i}{z-x_i}\prod_{\substack{...
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1answer
39 views

Show that Hessian matrix reduces to Laplacian in spherically symmetric integral

TLDR: read the final paragraph. For my course I have to evaluate $$\int_{S^2}d\vec R\ R_iR_j\frac{\partial^2G(\vec R)}{\partial R_i\partial R_j}$$ Here the Einstein summation convention is used, $G$ ...
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18 views

Why do the power sum symmetric functions form a set of orthogonal idempotents in the ring of symmetric functions under the internal product?

I am reading Macdonald's book on symmetric functions and Hall polynomials. On page 116, it is claimed in (7.12) that $$ p_\lambda*p_\mu=\delta_{\lambda\mu}z_\lambda p_\lambda. $$ But I did not see why....
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1answer
64 views

If $f(x+h)=-f(x-h)$ for all $(x,h)\in(x_0-\delta,x_0+\delta)\times[0,\infty)$, can we conclude $f=0$?

So, this should be a simple task, but I'm not able to find an argument: Assume $f\in C^2(\mathbb R)$ and $(x_0,\delta)\in\mathbb R\times(0,\infty)$ with $$f(x+h)=-f(x-h)\tag1$$ for all $(x,h)\in(x_0-\...
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1answer
19 views

Negative Symmetric Functions

For vectors $x$ and $y$, is there any differentiable function $f(\cdot)$, such that $f(x, y) = -f(y, x)$? For 3d vectors, the cross-product is one such function. But what about arbitrary dimensional ...
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8 views

Find the symmetries of graph:

I have a point $P(a,b)$ and I know that the graph of a function could be symmetric with respect to the point $P$ if the following condition is respected: $f:A\rightarrow R, A\subset R, f(a-x)+f(a+x)=...
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40 views

How to evaluate $s_\lambda(q,q^2,\cdots,q^m)$? (principal specialisation of the schur function)

It is required to show that $$ s_\lambda(q,q^2,\cdots,q^m) = q^{m(\lambda)}\prod_{i,j \in \lambda}\frac{1-q^{c_{i,j}+m}}{1-q^{h_{i,j}}} $$ where $c_{i,j}=j-i$ is the content of cell $(i,j)$, and $h_{i,...
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1answer
40 views

Maximum value of a symmetric function on a square

In a paper of Frits Beukers "A note on the irrationality of $\zeta(2)$ and $\zeta(3)$", he says It is a matter of straightforward computation to show that $$\frac{y(1-y)x(1-x)}{1-xy} \le \...
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1answer
26 views

Show that the $f(x)=\frac{1}{2} +\frac{1}{2^x+1}$

I am struggling with this question. Seems simple enough right but NO. I might be just overthinking it, but someone please help. To know if it is an odd function if $$f(-x)=-f(x).$$
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24 views

Schur function in linerly shifted power symmetric basis

Previously I have aksed the following question about schur function in power symmetric basis Schur function principal specialisation Let $s_{\lambda}(p_1,p_2,)$ denote schur function in power-sum ...
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2answers
139 views

Can $(x+y)^2$ be written in the form $f(x) g(y) + g(x)f(y)$?

Are there functions $f$ and $g$ such that $f(x)g(y) + g(x)f(y) = (x+y)^2$. I'm asking this because in quantum mechanics, the symmetric wavefunction of two identical particles is written as $\psi(x_1, ...
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1answer
64 views

Hall-Littlewood polynomials and elementary symmetric functions— Chapter III (2.8) in Macdonald's “Symmetric Functions and Hall Polynomials”

I'm confused about the proof of Chapter III (2.8), page 209 in Macdonald's book, see proof of (2.8). Here is the background. Let $\Lambda_n$ be the ring of symmetric polynomials in $r$ variables, i.e. ...
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1answer
30 views

Is the function $f(x,y)$ describing the given graph is symmetric for any $x\neq y$? Or, only for $x=y$?

We have an infinite periodic graph with the two lengths $x$ and $y$ and we have a function $f(x,y)$ that describes a specific property of this periodic structure. Then, according to the symmetry of ...
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0answers
18 views

Identity for the $L^1$ norm between nonnegative symmetric functions

Consider two symmetric nonnegative functions $f_1:\mathbb{R}^k \mapsto [0,\infty)$ and $f_2:\mathbb{R}^k \mapsto [0,\infty)$, i.e., for all $x \in \mathbb{R}^k$ and all permutations $\sigma(x)$ of $x$ ...
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1answer
112 views

How to express symmetric polynomials in terms of elementary symmetric polynomials?

Firstly, I know there are very similar questions (How to express a symmetric polynomial in terms of elementary symmetric polynomials and Expressing a symmetric polynomial in terms of elementary ...
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70 views

How does the Lipshitz norm of bounded and permutation invariant functions depend on the number of arguments?

Take a Lipshitz function on a compact subset of $R^N$ where we want to scale up $N$. If we know that (1) the functions are invariant to permutations (2) that there is a bound that is independent of ...
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1answer
26 views

symmetric difference function

Let $X$ be a given set and let $A$ be its subset. Define $D$ as a map from the power set of $X$ to itself such that $D(B)=(A \setminus B) \cup (B\setminus A)$. I have already proved that $D$ is ...
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1answer
297 views

Find out functions of the form $g(x,y) = \int f(x,t) f(y,t) \lambda(dt)$

I am interested in the following question. Given a symmetric function $g: \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R$ or $\mathbb R_{+}^{n}\times \mathbb R_{+}^{n} \rightarrow 0$. I am ...
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0answers
34 views

Schur function form a basis and skew-symmetric functions

I'm trying to prove that Schur functions (in their algebraic definition) form a basis of symmetric functions, in the following way: Show that $a_{\lambda+\delta}$ form a basis of skew-symmetric ...
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2answers
72 views

Solving for $x$ and $y$ when $(3x + y)(x + 3y)\sqrt{xy} = 14$

Solve for $x$ and $y$ when $$(3x + y)(x + 3y)\sqrt{xy} = 14$$ $$(x+y)(x^2 + 14xy + y^2) = 36.$$ I was thinking of squaring the first equation and moving on from there, but I think it'll be a bit too ...
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2answers
61 views

Solving system of equations with fraction

I am having difficulties solving the following system : $u \neq t$ and $(t, u) \in \mathbf{R} - \{-1, 1\}$ $\frac{t}{t^2-1}-\frac{u}{u^2-1}=0$ $\:\frac{t^2}{t-1}-\frac{u^2}{u-1}=0$ I tried expanding ...
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2answers
59 views

Ring structure on $R = \oplus_N R^n$ where $R^n:=R(S_n)$ is the vector space of class functions defined on $S_n$.

Define $R = \oplus_N R^n$ where $R^n:=R(S_n)$ is the vector space of class functions defined on $S_n$. According to Sagan, we can define a ring structure on $R$ that makes $R$ into a graded algebra, ...
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1answer
29 views

Solve parametric equations satisfying that the equations have positive roots…

Solve parametric equations satisfying that the equations have positive roots: $\left\{\begin{matrix} x_{1}+x_{2}+x_{3}+...+x_{m} &=9 \\ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}...
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0answers
24 views

Creating a modified symlog scale function that is symmetric around 1 (10^0) instead of 0

I want to create a symlog function that is symmetric around 1 (10^0) instead of 0. The range of output values for the function will be positive: [10^-6 - 10^6] and the input values will be from a ...
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0answers
37 views

Composition of binary symmetric functions

I'm interested in symmetric functions of two variables $f(x,y)$ with the property that $f(f(x,y),z)$ is symmetric in $x,y,$ and $z$ (or equivalently, symmetric functions such that $f(f(x,y),z) = f(f(x,...
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4answers
113 views

Is this alternative proof of the inequality $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{3}{2}$ correct?

Prove that for all positive real numbers: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\geq\dfrac{3}{2}$$ This is same as this question but a different approach is used there whereas I want to verify ...
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1answer
26 views

Question envolving bilinear forms (symmetric and skew symmetric).

Let $f: V \times V \to F $ be a bilinear form and $f$ has the follow property: Whenever $f(u,v)=0$ we have $f(v,u)=0$. The goal is to prove that $f$ is symmetric or $f$ is skew - symmetric, for that ...
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5answers
440 views

Can every symmetric function be written as some function of a sum?

I am looking for a simple counter-example to a "theorem" about symmetric functions claimed in a published paper. The claim asserts, among many other things, that there are functions $\sigma$ ...
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1answer
138 views

prove that $\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$

The question is, prove that $$\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$$ where \begin{eqnarray} \label{eqpdf} h(x) &=&(x^2)\times \left\{ \begin{array}{cc} \int_{0.5}^{1}...
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5answers
139 views

Simplifying $\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}$

Simplify $$\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}\,.$$ I tried very hard but I am not being able to solve it easily I opened up everything and ...
3
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0answers
99 views

Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?

I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
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0answers
15 views

Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
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3answers
45 views

Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
1
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1answer
71 views

What are the asymptotics of the $q$-binomial?

I have a rather basic question regarding the $q$-binomial $\begin{bmatrix}N \\ r \end{bmatrix}=\frac{(1-q^N)(1-q^{N-1} ) \dots (1-q^{N-r+1})}{ (1-q)(1-q^2)\dots(1-q^r) }$ as $N$ goes to infinity. ...
2
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1answer
94 views

Solving Functional Equation Involving Substitution

Suppose we seek a function of two variables $f(x,y)$ such that $f(x,y) = f(y,x)$ $f(x,y) = f\left(\frac{1+y}{x}, ~ y \right)$ Are there known techniques for approaching such questions? I already ...
0
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2answers
74 views

Relation between the roots and coefficient.

Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$. I got it done by evaluate the sum and difference of $a^2b+b^2c+c^2a$ and $ab^...
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2answers
36 views

Help to give/improve a proof on simple problem of Riemann integration

Let $K$ be a nonnegative, symmetric around zero, real function satisfying $$\int_{-1}^1K(u)du=1.$$ I want to show that $0<\int_0^1 K(u)u^2du$. This is intuitive but I'm struggling to show it. I ...
6
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1answer
176 views

Is the minimum of this optimization problem essentially unique?

Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Let $s&...
2
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2answers
82 views

Expansion of $(x+y)^n+(x+z)^n+(y+z)^n-x^n-y^n-z^n$ in terms of elementary symmetric polynomials

Consider the symmetric polynomial in $3$ variables $$ f_n(x,y,z)=(x+y)^n + (x+z)^n+(y+z)^n - x^n-y^n-z^n $$ where $n\geq 0$ is an integer. I'm inquiring if there is a closed formula for the ...
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1answer
32 views

Can Weierstrass's theorem be specialized to symmetric functions and symmetric polynomials?

Weierstrass's theorem says that continuous functions can be uniformly approximated by polynomials. Can one have a similar theorem saying that symmetric functions can be uniformly approximated by ...
16
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2answers
337 views

Symmetric functions written in terms of the elementary symmetric polynomials.

[A recent post reminded me of this.] How can we fill in the blanks here: For any _____ function $f(x,y,z)$ of three variables that is symmetric in the three variables, there is a _____ function $...
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1answer
41 views

If $f(x,y)$ is symmetric, do the left and right derivatives need to be equal at $f(a,a)$ (in absolute value)

For $a$ in the domain, and the function being defined at $a$. And by $f_1$ I mean the derivative of the function w.r.t to its first argument (so $f_1(x,y)$ means the derivative w.r.t the first ...

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