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1answer
182 views

Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...
6
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0answers
161 views

Find a symmetric polynomial with a projection divisible by a known polynomial

Consider the polynomial $Q$, a homogeneous quartic in seven variables: $$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\ ...
8
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1answer
310 views

Combinatorics and symmetric functions

No answers from stackexchange, so I'll try this here: (The actual questions in this posting are at the bottom.) Occasionally someone asks on stackexchange how to show that every nonempty finite set ...
1
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0answers
64 views

Irreducibility of a certain matrix variety

Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ ...
4
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1answer
228 views

Examples of specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
3
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1answer
129 views

Values of symmetric polynomials determining inputs

I've got a question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots \in ...
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0answers
70 views

Different bases of symmetric polynomials

Let the symmetric group $S_n$ act on $R^n$ by permuting coordinates and consider the ring of symmetric polynoials. These symmetric polynomials can be written fro example as polynomials in the ...
3
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0answers
141 views

Significance behind splitting of (x+y)^7-(x^7+y^7)? [closed]

A quick check via third roots of unity establishes$$(x+y)^7-(x^7+y^7) = 7xy(x+y)(x^2+xy+y^2)^2$$ Some questions: What is the significance of this factorization? Does it have any connection to the ...
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2answers
243 views

When powers of matrices are represented as a sum of integral matrices

There is a ring $R$ and its subring $K$ with unit. We have a matrix $A$ of order $n$ over $R$. Someone said, that if $A^m$ for $m=1,...,n$ can be represented as a sum of matrices over $R$ which a ...
4
votes
1answer
322 views

Normal forms for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$

Is there a standard normal form for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$? Or, put another way, is there a nice way to describe the orbit space of the natural (diagonal) action ...
4
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2answers
159 views

Reference request on symmetric polynomials

A version of this question on stackexchange got a few comments from one person and no answers. Let $e_k$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,\ldots,x_n$ (so in ...
4
votes
1answer
237 views

A nice generating set for the symmetric power of an algebra

I'm looking for a reference for the following fact. Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ ...
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0answers
91 views

Multivariate polynomial with positive coefficients

This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...
5
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0answers
143 views

algebra of endomorphisms over the diagonal invariants

Let $k$ be a field of characteristic 0 (say $\mathbb{C}$). Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$. It is known that the ...
1
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1answer
223 views

Majorization of power sum symmetric functions

I'm wondering if there is a characterization for whether $$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \mathbb{R}^+$}.$$ (Or ...
6
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2answers
534 views

Generalizing the Fundamental Theorem of Symmetric Polynomials

The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
5
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2answers
733 views

Diagonal invariants of the symmetric group on $k[X_1,X_2,…,X_n,Y_1,Y_2,…,Y_n]$

This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
0
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0answers
253 views

Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...