Tagged Questions

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

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Symmetrizing a sequence of vectors

Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
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Existence of a Root of Elementary Monomials

Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$. For example: $$m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3$$ ...
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Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.

Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
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Name these symmetric polynomials over non-commutative variables…

Given a set of $N$ non-commutative variables $x_k$. Is there a special name for symmetric polynomials of homogenous degree $d$ of the form that all $x_k$s appear with exponent at most $1$ at a time? ...
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Simple proof of Newton identities

The functions $s_1=x_1+x_2+\cdots +x_n$, $s_2=\sum_{i<j} x_ix_j$, $\cdots$, $s_n=x_1x_2\cdots x_n$ are elementary symmetric functions in $x_1,x_2,\cdots,x_n$ (or more precisely, elementary ...
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Symmetrized monomials under Weyl group?

Consider a given partition $\lambda=(\lambda_1,\lambda_2,...,\lambda_N)$ and start with the monomial $$z_1^{\lambda_1}z_2^{\lambda_2}...z_N^{\lambda_N}$$ in $N$ variables $z_1,z_2,...,z_N$. Now we ...
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terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $e_1,\cdots,e_n$ and $$\mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n].$$ We define a ...
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Looking for the name of particular collection of polynomials

I came across the following algebraic structure when working on a seemingly unrelated problem and am unable to find a name for it. Let $R$ be a commutative ring with identity. Given ...
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Find the coefficients in $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$

Use evaluation homomorphisms $F[x_1,x_2, \dots, x_n] \to F$ to obtain the coefficients in: $$(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$$ where the $s_i$ ...
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How would you approach this problem? I need all the possible solutions $$(x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4)$$ which satisfy $$\frac{(x_1-x_2)^2-(y_1-y_2)^2}{(x_1-x_2)^3(y_1-y_2)^3}+\frac{(x_1-x_3)^... 0answers 97 views An identity with determinant and trace of a matrix How to prove the following identity:$$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A) where $\mathrm{Tr}_{\sigma}(A)$ is defined as following if $\sigma$ is ...
Let $E$ be a free module, we define the $r$-th divided power as the dual of the symmetric power $D_r(E):=(S_r(E^*))^*$. For every $u \in E$ we can define its $r$-th divided power $u^{(r)} \in D_r$ by ...