# Questions tagged [symmetric-polynomials]

The symmetric-polynomials tag has no usage guidance.

116
questions

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### Kostka-Jack numbers with the zero Jack parameter

Define the Kostka-Jack number $K_{\lambda,\mu}(\alpha)$ as the coefficient of the monomial symmetric polynomial $m_\mu$ in the expression of the Jack $P$-polynomial $P_\lambda(\alpha)$ as a linear ...

2
votes

0
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42
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### Skew Jack polynomial when the Jack parameter is zero

According to Macdonald's book, when the Jack parameter $\alpha$ is $0$,then the Jack $P$-polynomial $P_\lambda(\alpha)$ is the elementary symmetric polynomial $e_{\lambda'}$ where $\lambda'$ is the ...

7
votes

1
answer

249
views

### Existence of a linear map resulting in the determinant being an elementary symmetric polynomial

Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...

0
votes

0
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### "Degenerate" Schur polynomials

Let's say that a Schur polynomial is degenerate if its number of variables is less than the weight of the partition it is associated to. For example, according to Sage, the Schur polynomial of the ...

3
votes

1
answer

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### Eigenvalues of the Jack polynomials for the Calogero-Sutherland operator

The Calogero-Sutherland operator on the space of homogeneous symmetric polynomials in $n$ variables is defined by
$$
\frac{\alpha}{2}\sum_{i=1}^n x_i^2\frac{\partial^2}{\partial x_i^2} + \frac{1}{2}\...

3
votes

0
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### Explicit basis of symmetric harmonic polynomials

An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...

5
votes

1
answer

271
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### The coefficients of the Jack polynomials are polynomials in the Jack parameter

I implemented the Jack polynomials with a symbolic Jack parameter $\alpha$ in their coefficients ($\alpha=1$ for Schur polynomials, $\alpha=2$ for zonal polynomials). From my implementation (and also ...

11
votes

0
answers

563
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### Monotonicity of ratio of symmetric polynomials

The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...

1
vote

1
answer

78
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### Multidimensional power series with coefficients equal to an order of stabilizer of a set of powers

I have encountered a necessity to work with a series of the following form.
There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They ...

5
votes

3
answers

482
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### Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials.
Is it known how to write the result of the application of $L_0$, ...

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0
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72
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### Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...

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0
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### Matrix transform of the bivariate Narayana polynomials into the arithmetic and geometric means of the two indeterminates

The matrix identity presented below is a specialization of the more general result displayed in the MSE-Q "Lah and associahedra partition polynomials and symmetric functions (reference request)&...

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87
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### Hall-Littlewood polynomials with sage

I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...

2
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92
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### Lower & upper bound on the maximal component given the system of power sums

Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like)
$$
\begin{cases}
x_1+x_2+\...

-1
votes

1
answer

293
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### Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?

The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...

4
votes

1
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364
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### Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?

I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials ...

3
votes

1
answer

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### Alternative bases of symmetric polynomials in cohomology ring of flag varieties and coinvariant algebras

$\DeclareMathOperator\Fl{Fl}$It is known that $H^*(\Fl(m)) \cong R^{\mathbb Z}(m)$, where $\Fl(m)$ denotes the variety of complete flags in $\mathbb C^m$, and $R^{\mathbb Z}(m)$ is the coinvariant ...

7
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1
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333
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### Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?

For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...

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### Applications of Jack polynomials

I developed four libraries (Julia, R, Python, Haskell) for the computation of Jack polynomials. I developed them for fun because I found this was programmatically interesting. But now I'd like them to ...

7
votes

0
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### Generalization of Lucas sequences to order 3 (and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...

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votes

2
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### Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers

Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$.
Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...

3
votes

1
answer

320
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### Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?

Consider a set $N$ with elements $n_1, n_2, \dots, n_k$ which are distinct integers. Introduce the notation $N_{i=1,2,\dots,s}$ for the $s$ blocks of a set partition of $N$. Consider a supplementary ...

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1
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525
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### Construction of a symmetric polynomial in the roots that acts like the discriminant

The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...

8
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1
answer

579
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### When the Littlewood-Richardson rule gives only irreducibles?

Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:
$$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$
is there a classification of the cases where the LR ...

1
vote

0
answers

114
views

### Schur polynomial with integer values

There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...

2
votes

0
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### Symmetric polynomial constructed from symmetric group

Let $n$ be a positive integer, $S_n$ be the symmetric group. For a permutation $p=[p_1,\dots,p_n]\in S_n$, define $x^p := x_1^{p_1}\cdots x_n^{p_n}$. It can be seen that the following polynomial is ...

3
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148
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### Bounds on symmetric polynomials in power-sum form with bounded coefficients

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$.
For a partition $\lambda= (j_1,\...

3
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0
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### Different generating sets for conjugation invariants of several matrices

There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...

2
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0
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### Positive values of Schur polynomials

Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials
$$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...

4
votes

1
answer

311
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### Nonnegativity locus of Schur polynomials

Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...

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1
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416
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### Symmetric polynomials that detect positivity

Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you ...

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### Symmetric polynomial inequality arising from the fixed-point measure of a random permutation

A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows.
Given $n$ non-negative reals $a_1, ...

5
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0
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### A generalization of the difference of squares identity

Let us find explicit integer functions for the coefficients of the monomial expansion of
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...

3
votes

1
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### Polynomial function defined recursively by a resultant - is it well defined?

Preliminaries
Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| ...

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1
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### A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$

If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\...

3
votes

1
answer

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### Conjecture on some combinatorial constant

In the process of computing Shapley values, I observed an interesting combinatorial constant. I am not exactly sure where such behavior comes. And here is the conjecture.
Notations
For any finite non-...

3
votes

0
answers

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### Multiplicities of irreducible $U(n)$-modules in the tensor product $V_{\lambda}\otimes V_{\mu}$

It is known that every irreducible representation of the unitary group $U(n)$ can be uniquely described by the non-increasing sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of integers (denote the ...

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votes

1
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### Representing a symmetric polynomial as a conical sum of squares

This question in inspired by the recent solution to another question.
The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$:
$$m_{(4, 3, 2, 1)} + m_{(4, ...

2
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0
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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
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### 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
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1
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### 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
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0
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### 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 $\mathbb Z$). ...

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### 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 $f$...

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2
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### 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 ...

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votes

1
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### 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
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0
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### 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

0
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### 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(\{(...

4
votes

1
answer

524
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### 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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0
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### 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. ...

11
votes

3
answers

634
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### 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}\...