# Questions tagged [symmetric-polynomials]

The symmetric-polynomials tag has no usage guidance.

80
questions

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votes

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### 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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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 ...

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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

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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(\{(...

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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

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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

**1**answer

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 ...

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vote

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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

**3**answers

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

**3**answers

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

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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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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 ...

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votes

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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 $\{ ...

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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

**1**answer

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

**1**answer

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

**1**answer

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 $M_{n\times n}$...

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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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**5**answers

634 views

### Inequality with symmetric polynomials [closed]

How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?

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votes

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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 ...

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votes

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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

**1**answer

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 ...

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votes

**1**answer

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 ...

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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^{\...

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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 $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...

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vote

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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 $F$, ...

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**1**answer

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

**2**answers

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 $x_1,\dots,x_d$. ...

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votes

**1**answer

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

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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 ...

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votes

**1**answer

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

**2**answers

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

**1**answer

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 ...