# Questions tagged [symmetric-polynomials]

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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 ...
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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 ...
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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 ...
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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 ...
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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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