# Questions tagged [symmetric-polynomials]

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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-...
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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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### 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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### 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 ...
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### 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 ...
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### 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
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### 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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### 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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### 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 ...
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### 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 ...
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 ...