# Questions tagged [symmetric-polynomials]

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

656 questions
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### Decide for which $x,y,z$ the following equation system is met: ${1+x+y=xy}$ $2+y+z=yz$ and $5+z+x=zx$

I need to decide for which $x,y,z$ the following equation system is met: $$1+x+y=xy$$ $$2+y+z=yz$$ $$5+z+x=zx$$ I can see that $x=0, y=0, z=0$ is not a solution. I tried to divide ...
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### Characters of orthogonal groups as symmetric functions

The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)={\rm Tr}\left(R_\lambda(U)\right)$. They are symmetric functions of the eigenvalues of $U$ and can be written in terms ...
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### Inequalities, but working around an absolute value

So what I want to prove is $$|xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3|$$ for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$. Now I'm able to ...
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### filtered algebra vs graded algebra

BACKGROUND When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions. They say that $\Lambda_n^*$ is a ...
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### skew Schur functions $C^{\lambda}_{\mu, \nu}$

When working with skew Schur functions, they can be defined as follows. Let $C^{\lambda}_{\mu, \nu}$ be the integers such that $$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$ ...
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### $\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)

This question is related to this other question. When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
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### Proofs of Newton's identities

I am reading the article "Newton's Identities" by D. G. Mead. Newton's Identities capture the the relation between the elementary symmetric functions of $x_1, x_2,\ldots,x_n$ and the sums of the ...
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### Some interesting systems of equations [closed]

1, Solve the system of equations:$\left\{\begin{matrix} x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z \\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2} \end{matrix}\right.$ 2,Solve the system of ...
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### Prove $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$

If $a.b,c \in \mathbb{R^+}$ and $ab+bc+ca=1$ Then Prove $$S=\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$$ My try we have S=\sum \frac{a}{\sqrt{a^2+ab+bc+...
Let $A$ be a $k$-algebra and let $V$ be a subalgebra of $A$ that is a polynomial algebra. Is $A$ a free module over $V$? One example is when $A=\mathbb{C}[x_1, x_2, \ldots x_n]$ and $V$ is the ring ...
### If $a^2+b^2 \gt a+b$ and $a,b \gt 0$ Prove that $a^3+b^3 \gt a^2+b^2$
I'm not too sure about this, I have been working on for some time and I reached a solution (not really too sure about) Question: If $a^2+b^2 \gt a+b$ and $a,b \gt 0$ Prove that $a^3+b^3 \gt a^2+b^2$ ...