Questions tagged [symmetric-polynomials]
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Irrational elements can always be moved
Let $x_1,x_2,x_3,\ldots,x_n$ be the roots of a polynomial $P_n(x)$. Let $F$ be the field $\mathbb{Q}[x_1,x_2,x_3,\ldots,x_n]$, i. e. all the possible combinations of rational numbers with $x$'s.
It's ...
- 111
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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 ...
- 2,540
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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 ...
- 2,540
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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 ...
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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 ...
- 2,540
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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}\...
- 2,540
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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 ...
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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 ...
- 2,540
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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} \...
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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 ...
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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$, ...
- 1,519
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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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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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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 ...
- 6,259
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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+\...
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