# Questions tagged [symmetric-polynomials]

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

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• 584
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### Formula for coefficient of a certain polynomial to the nth power

I have a polynomial that looks like $$p_{3,10}(x_0, x_1, ... x_9) = (x_0 x_1 x_2 x_3 x_4 + x_0 x_1 x_2 x_3 x_5 + ... + x_5 x_6 x_7 x_8 x_9)^3$$ How do I determine the formula for the coefficient of ...
• 163
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### Does $\sum_{j=1,j\neq k}^n\frac{z_k}{z_k-z_j}=\sum_{j=1,j\neq m}^n\frac{z_m}{z_m-z_j}$ implies the $(z_j)_{j=1,...,n}$ are the nth roots of $z_1^n$?

Let $(z_1,\ldots z_n)\subseteq \mathbb{C}.$ Does $$\sum_{j=1\,,\,j\neq k}^n \frac{z_k}{z_k-z_j}=\sum_{j=1\,,\,j\neq k'}^n \frac{z_{k'}}{z_{k'}-z_j}$$ for all $k,k'\in\{1,...,n\}\,$, implies that ...
• 55
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### $k[x_1,.\dots,x_n]$ is a free module over the ring of symmetric polyomials

I know that this question has been already discussed (e.g here: shorturl.at/coFQX) but I dont understand the proofs given there, and I found a different one. I want to show that $R[x_1,\dots,x_n]$ is ...
23 views

### Finitely many $\alpha$ s.t. all conjugates $\leq N$ ($N \geq 1$)

Let $K$ be a number field, and let $N \geq 1$. Then there are only finitely many $\alpha \in O_K$ such that all conjugates of $\alpha$ have complex absolute value $\leq N$. The solution goes as ...
• 648
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### What is the largest value of $e_{k}(x_1,\cdots,x_n)$ not obtainable over $(\mathbb{N}^+)^n$?

Let $k,n\in\mathbb{N}^+$, $\vec{x}=\langle x_1,\cdots,x_n\rangle$ be an $n$-tuple, and let $e_k(\vec{x})$ be the elementary symmetric polynomial of degree $k$ over $n$ variables (clearly with $k\le n$)...
• 8,174
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### If $\alpha,\beta,\gamma,\delta$ are the roots of $x^4+px^3+qx^2+rx + s=0$, find in terms of $p,q,r,s$ the value of $\Sigma\frac{\alpha\beta}{\gamma }$

If $\alpha,\beta,\gamma,\delta$ are the roots of $x^4+px^3+qx^2+rx + s=0$, find in terms of $p,q,r,s$ the value of $\Sigma\frac{\alpha\beta}{\gamma }$ My general strategy was transforming the ...
• 1,865
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### Characterization of finite fields

Let $\mathbb{K}$ be a finite field of order $q$ and with $p = char(\mathbb{K})$. Let $\mathbb{K} = \{\lambda_1 , \dots , \lambda_q\}$. I know that $x^q - x = \prod_{i=1}^{q}{(x-\lambda_i)}$ So if I ...
• 985
1 vote
80 views

### Schur function of the conjugate representation

Consider the Schur function for irrep $(1)$ of $\mathrm{U}(3)$: $s_{(1)}(z_1,z_2,z_3)=z_1+z_2+z_3$. What is the Schur function for the irrep conjugate to this, i.e. the irrep $(1)^*$ where all the ...
• 596
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### Ring of invariants of symmetric group acting on ring of formal power series

Let $k$ be a field and $R=k[[x_1, \dots, x_n]]$ be the ring of formal power series in $n$ variables. Let $S_n$ be the symmetric group of order $n!$. Then $S_n$ acts on $R$ by $k$-automorphisms by ...
31 views

### Is there a name for the polynomials of the form $\sum_{i = 0}^a x^{a - i}y^i - 1$? [duplicate]

I am trying to find out what is known about polynomials of the form $$\sum_{i = 0}^a x^{a - i} y^i - 1 = x^a + x^{a-1}y + \dots + y^a - 1.$$ I tried searching for anything regarding the sums of ...
• 1
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### Derivation of the Macdonald operator $D_{n}(X;q,t)$

Since I first encountered Equation (3.2) on p.315 of Macdonald's Symmetric functions and Hall polynomials, I have wanted to know where it comes from. So how does one derive the operator \begin{...
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### Schur polynomials: Showing that $m_ie_j=s_{(i,1^j)}+s_{(i+j,1^{j-1})}$

How to show that $m_ie_j=s_{(i,1^j)}+s_{(i+j,1^{j-1})}$, for all $i,j \in \lbrace{1,\ldots...n\rbrace}$ where $s$ are the schur polynomials, $m_i$ the monomial symmetric polynomials and $e_j$ the ...
• 523
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1 vote
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### McKay correspondence for Irreps of $G \subset SU(2)$

We follow Alexander Kirilov's book "Quiver Representations and Quiver Varieties", Section 8.3: McKay correspondence: Let $G$ be a nontrivial finite subgroup in $SU(2)$. Let $Q(G)$ be the ...
1 vote
31 views

### The relationship between $f_{i}$

Let $\lambda_{i}$ be a real number for any $1 \le i \le n$, defining $f_{i}$ as following $$f_{i} = \sum^{n}_{j=1} (\lambda_{j})^{i}$$ Suppose we know $f_{1},f_{2},f_{3}$ and $f_{4}$, then can we ...
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