Questions tagged [symmetric-polynomials]
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80
questions
2
votes
0answers
126 views
The Galois resolvent in Lagrange
In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...
4
votes
2answers
165 views
Schur positivity of a polynomial
Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...
4
votes
1answer
195 views
proof of result from Ian Macdonald's paper “A New Class of Symmetric Functions”
I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
1
vote
0answers
51 views
sum of squares of the coefficients of a monic polynomial [closed]
Consider the monic polynomial $$f: = X^n + a_1 X^{n-1} + \cdots + a_n$$ over $\mathbb Z$. Let $p_i$ be the $i$_th power sum ($1 \le i \le n$) of the roots of $f$ (in some extension of $\mathbb Z$). ...
10
votes
1answer
321 views
Generalization of symmetric functions
A $n$-variable function $f$ is a symmetric function if
$$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$
for every permutation $\sigma \in S_n$.
In particular, if $f$...
20
votes
2answers
873 views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
5
votes
1answer
141 views
Frobenius algebras from symmetric polynomials
Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
0
votes
0answers
89 views
Generalization of elementary symmetric polynomials
The elementary symmetric polynomials (ESPs) are defined as -
\begin{align*}
E_{1}^{1}
&= X_1,
\\
E_{1}^{2}
&= X_1 + X_2,
\\
E_{2}^{2}
&= X_1 X_2,
\\
E_{2}^{3}
&= X_1 X_2 + X_1 X_3 + ...
5
votes
0answers
162 views
Expressing the elementary symmetric polynomials in the $(x_i-x_j)^2$ variables in term of the elementary polynomials in the $x_k$ variables
Let $n>1$ be an integer and let $P$ be a multivariate symmetric polynomial in $n(n-1)/2$ variables.
Let us define the multivariate polynomial $Q_P$ in n variables as:
$Q_P(x_1,...,x_n)=P\left(\{(...
0
votes
0answers
40 views
How to construct such a series of “partial” symmetric polynomials?
As we know, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric ...
0
votes
0answers
158 views
Symmetry of weights in Newton-Cotes quadrature formula
Why do the weights of a generic Newton-Cotes quadrature formula show symmetry w.r.t. the midpoint of the integration interval? I understand that, given $f:[a,b]\mapsto\mathbb{R}$ and being $n>0$ ...
3
votes
1answer
338 views
Derivatives of Riemann $\xi$ and traces of zeros
Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
1
vote
0answers
103 views
Symmetric functions for multidimensional variables
I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to make permutation invariant.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. ...
10
votes
3answers
464 views
Polynomial inequality of sixth degree
There is the following problem.
Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$
Prove that:
$$\sum_{cyc}\...
5
votes
3answers
394 views
Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$
Let $n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $ and $\sum\limits_{1\leq i<j<k\leq n}...
2
votes
0answers
109 views
Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
4
votes
1answer
116 views
Applying a simple involution to Hall-Littlewood polynomials
Consider the Hall-Littlewood polynomial
$$
P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-...
5
votes
1answer
347 views
Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
1
vote
0answers
241 views
An explicit formula for characteristic polynomial of matrix tensor product [closed]
Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...
4
votes
1answer
295 views
Possible values of symmetric functions evaluated on quaternions
$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be the units of quaternions, in particular $i^2=j^2=k^2=-1$, $ijk=-1$.
We will use non commutative variables $x$, $y$, $z$. Define $\sym_{a,b,c}$ to be ...
0
votes
1answer
290 views
Symmetric polynomials in two sets of variables
Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
7
votes
2answers
503 views
Maximize $L^p$ norm over sphere
For $p \in \mathbb{R}$, consider the function
$$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda_1^2 +...
2
votes
0answers
55 views
Classes of curves with “determinant-like operation”
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
1
vote
1answer
37 views
Does stability of equilibrium point preserved by permutation matrix (symmetry)?
Given the following differential equations:
\begin{equation}
\begin{aligned}
\dot{x}_1 &= f_1(x_1,\ldots,x_n) \\
\vdots \\
\dot{x}_n &= f_n(x_1,\ldots,x_n)
\end{aligned}
\end{equation}
In ...
3
votes
0answers
54 views
Bounds related to Monomial Symmetric Functions
Two of the well studied bases for symmetric functions over $\mathbb{Q}$ are the monomial symmetric functions $\{ m_{\lambda} \}_{\lambda \text{ a partition}}$ and the power sum symmetric functions $\{ ...
3
votes
0answers
234 views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
5
votes
1answer
209 views
When is a linear combination of the elementary symmetric polynomials reducible?
Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
3
votes
1answer
385 views
Symmetry in the triangular distribution
A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$
The mean of ...
4
votes
1answer
494 views
Symmetric functions of eigenvalues
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables).
Let $M_{n\times n}$...
7
votes
0answers
209 views
Characterizing $n$-exceptions of the ring of symmetric polynomials
(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials)
We say that an homogeneous symmetric polynomial ...
2
votes
0answers
44 views
Different term's contribution in zonal polynomial
I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal ...
4
votes
1answer
422 views
Homogeneous polynomials and symmetric binary forms
Let $f\in k[x_0,...,x_n]_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables.
Is there a way to associate to $f$ a form $g(y_1,...,y_m)$ which is symmetric in the sets of binary variables $...
10
votes
1answer
400 views
Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\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
130 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
328 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
634 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
141 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
175 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
135 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
239 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
294 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
208 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
70 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$, ...
5
votes
1answer
170 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
245 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
284 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
252 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
289 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
247 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
173 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 ...
