Questions tagged [symmetric-polynomials]
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2
votes
0answers
50 views
Classes of curves with “determinant-like operation”
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
1
vote
1answer
28 views
Does stability of equilibrium point preserved by permutation matrix (symmetry)?
Given the following differential equations:
\begin{equation}
\begin{aligned}
\dot{x}_1 &= f_1(x_1,\ldots,x_n) \\
\vdots \\
\dot{x}_n &= f_n(x_1,\ldots,x_n)
\end{aligned}
\end{equation}
In ...
3
votes
0answers
46 views
Bounds related to Monomial Symmetric Functions
Two of the well studied bases for symmetric functions over $\mathbb{Q}$ are the monomial symmetric functions $\{ m_{\lambda} \}_{\lambda \text{ a partition}}$ and the power sum symmetric functions $\{ ...
0
votes
0answers
59 views
Dual space of polynomial one-form
Recently I read a paper "Quasi-particles models for the representations of Lie algebras and geometry of flag manifold". In section 2, author gives a fact without proof. Now I rephrase this fact as ...
2
votes
0answers
195 views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
5
votes
1answer
141 views
When is a linear combination of the elementary symmetric polynomials reducible?
Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
3
votes
1answer
170 views
Symmetry in the triangular distribution
A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$
The mean of ...
4
votes
1answer
210 views
Symmetric functions of eigenvalues
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables).
Let $M_{n\times n}$...
7
votes
0answers
178 views
Characterizing $n$-exceptions of the ring of symmetric polynomials
(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials)
We say that an homogeneous symmetric polynomial ...
2
votes
0answers
33 views
Different term's contribution in zonal polynomial
I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal ...
4
votes
1answer
226 views
Homogeneous polynomials and symmetric binary forms
Let $f\in k[x_0,...,x_n]_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables.
Is there a way to associate to $f$ a form $g(y_1,...,y_m)$ which is symmetric in the sets of binary variables $...
10
votes
1answer
339 views
Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y),
$$
where $s_\lambda(x)$ is the Schur function.
Is there a known decomposition of the ...
2
votes
0answers
95 views
Macdonald polynomials: existence and specializations
I am reading Macdonald's Symmetric Functions and Orthogonal Polynomials lecture notes, and have several related questions:
In both the type A case (chapter 1) and the general irreducible root system ...
3
votes
1answer
226 views
Singular locus of zero set of elementary symmetric polynomial
Let $\sigma_{m, r}$ be the degree-$r$ elementary symmetric polynomial in $m$ variables. Let $X_{m, r}$ be the zero set of $\sigma_{m, r}$ and $S_{m, r}$ its singular locus. I.e.,
$S_{m,r}$ is the set ...
1
vote
5answers
496 views
Inequality with symmetric polynomials [closed]
How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
3
votes
0answers
119 views
How to Prove that the Eigenvalues of this Matrix are Zero, $\pm b$ for $b$ Real or Imaginary?
The matrix, $M$, is defined (below) in terms of elementary symmetric polynomials of a set. For some special cases the matrix is nilpotent; for other (sufficiently small) cases the characteristic ...
5
votes
0answers
172 views
Extracting the “positive” part of a polynomial
Let $p$ be an even polynomial of degree $2n$ such that all its roots are real; hence, it can be written as $p(x)=q(x)q(-x)$, where $q$ is a polynomial of degree $n$ will all roots non-negative. I am ...
4
votes
1answer
101 views
Normalization of Jack polynomial integral-scalar product?
In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are ...
9
votes
1answer
226 views
Hyper-symmetric polynomials (reference request)
Let $M_n$ be the linear space of $n\times n$ matrices. The product of symmetric groups $S_n\times S_n$ acts naturally on $M_n$, and thus induces an action on the coordinate algebra $k[M_n]$. Is there ...
6
votes
0answers
290 views
Coefficients in expansion of a classical symmetric polynomial
If we expand \begin{equation} P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k), \end{equation} then \begin{equation} P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\...
5
votes
0answers
191 views
A question on symmetric functions
Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
1
vote
0answers
65 views
Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space
I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".
We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
5
votes
1answer
155 views
Symmetric polynomial separating points
I've been looking for references/answers to this problem for several days and I couldn't find anything.
If we consider the closed unit ball $B$ in $\mathbb C^2$ then for any point $(z_1,z_2)\notin B$ ...
5
votes
2answers
183 views
sum of squares of Schur polynomials indexed over partition valued functions on a set
Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...
7
votes
1answer
260 views
PDE characterisation of elementary symmetric functions?
For $k\leq{}n$ the elementary symmetric polynomials are defined by:
$$e_k(x_1,\ldots,x_n)=\sum_{1\leq{}i_1<...<i_k\leq{}n}x_{i_1}\cdots x_{i_k}$$
I believe I can prove (by a complex brute force ...
2
votes
0answers
185 views
what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.
I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...
3
votes
1answer
250 views
Connection between the Chebyshev polynomials and the Faber polynomials
From a comment on this question:
@draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), which 'invert" the cyclic partition polynomials ...
5
votes
2answers
220 views
Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
I am looking at a polynomial of the entries of a matrix, and this polynomial is invariant under permutation of the rows or columns of the matrix. Is there a similar characterization as in the case of ...
5
votes
1answer
148 views
Alternating elements in free graded-commutative algebras
It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...
7
votes
2answers
300 views
Schur polynomial, change of variable
Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...
6
votes
2answers
377 views
Free $k[x_1, \dots, x_n]^{S_n}$-module?
Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{...
1
vote
0answers
210 views
Irreducible representations of $S_n$ inside the ring of symmetric polynomials
I will describe two ways to associate irreducible representations of $S_n$ with polynomials inside the ring of symmetric polynomials and I want to know if there is any connection between the two.
$\...
37
votes
4answers
3k views
The sum of squared logarithms conjecture
I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
13
votes
4answers
860 views
$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial
Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and $y=(...
2
votes
0answers
157 views
counting how many boxes from a given Young tableau contribute to hook length made out of two YTs
Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...
3
votes
1answer
142 views
Polynomial that is symmetric in some variables
I would like to construct (or determine the existence/inexistence) of a polynomial $p(x_1,...,x_k, y_1,...,y_n)$ satisfying the following properties:
$p$ is symmetric with relation to the variables $...
4
votes
2answers
606 views
On a positivity property of Hall-Littlewood polynomials
Here's the new, more thought through version.
Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
2
votes
0answers
139 views
Behavior of elementary symmetric polynomials near zero sets
It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with $\sigma_{k}(...
4
votes
2answers
656 views
Isotypic components of the action of the symmetric group on polynomials
The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...
4
votes
0answers
177 views
Strategy to prove formula for top chern class from knowlege of chern character
I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...
10
votes
2answers
501 views
A particular specialization of symmetric polynomials: is it bijective?
Let $\Lambda^d_n$ the space of symmetric polynomials
in $n$ variables, with maximum 'partial degree' of each variable $d$.
A basis for this space is the set of symmetrized monomials $m_\lambda$,
where ...
1
vote
1answer
523 views
Decomposition of symmetric homogeneous polynomials
Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...
6
votes
0answers
199 views
Find a symmetric polynomial with a projection divisible by a known polynomial
Consider the polynomial $Q$, a homogeneous quartic in seven variables:
$$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\
(d_1^2-(R+s_1+s_2-s_3-s_4)(R+s_1-s_2+s_3+s_4))\\(d_1^2-(R-s_1-s_2-s_3-s_4)(R-s_1+s_2+...
9
votes
1answer
370 views
Combinatorics and symmetric functions
No answers from stackexchange, so I'll try this here:
(The actual questions in this posting are at the bottom.)
Occasionally someone asks on stackexchange how to show that every nonempty finite set ...
1
vote
0answers
75 views
Irreducibility of a certain matrix variety
Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ ...
10
votes
3answers
520 views
Examples of specializations of elementary symmetric polynomials
Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$
indeterminates. The $h^{th}$elementary symmetric polynomial is the
sum of all monomials with $h$ factors
\begin{eqnarray*}
e_{h}(\...
3
votes
1answer
225 views
Values of symmetric polynomials determining inputs
I've got a question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots \in \mathbb{Z}[X]$,...
1
vote
0answers
105 views
Different bases of symmetric polynomials
Let the symmetric group $S_n$ act on $R^n$ by permuting coordinates and consider the ring of symmetric polynoials. These symmetric polynomials can be written fro example as polynomials in the ...
3
votes
0answers
162 views
Significance behind splitting of (x+y)^7-(x^7+y^7)? [closed]
A quick check via third roots of unity establishes$$(x+y)^7-(x^7+y^7) = 7xy(x+y)(x^2+xy+y^2)^2$$
Some questions:
What is the significance of this factorization?
Does it have any connection to the ...
1
vote
2answers
272 views
When powers of matrices are represented as a sum of integral matrices
There is a ring $R$ and its subring $K$ with unit. We have a matrix $A$ of order $n$ over $R$. Someone said, that if $A^m$ for $m=1,...,n$ can be represented as a sum of matrices over $R$ which a ...
