Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,349
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Polynomial factorization using determinant
Algebraic identities
$$
P_2(a,b)=a^3+b^3=(a+b)(a^2+b^2-ab)
$$
$$
P_3(a,b,c)=a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)
$$
$$
P_4(a,b,c,d)=a^3+b^3+c^3+d^3-3abc-3abd-3acd-3bcd=(a+b+c+d)(a^2+b^2+c^2+...
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Basis for skew-symmetric polynomials in $n$ variables
I am reading from Introduction to Group Characters by Walter Ledermann [2nd edition]
On page $113$, the proposition $4.4$ states
The set $V=\{V_l\}$ where $l$ ranges over all strictly decreasing ...
- 326
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A system of equations of power sums for 3 variables
I am currently interested in the following problem:
Find $x, y, z$ such that
$$
\begin{cases}
x + y + z = 2 \\
x^2 + y^2 + z^2 = 6 \\
x^3 + y^3 + z^3 = 8
\end{cases}
$$
I noticed how the solutions ...
- 411
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General form of symmetric function of 3 dimensional vectors
In the paper Deep Sets by Zaheer et al, they prove a theorem (eq 18 in appendix) that states that any general scalar symmetric function of $M$ variables can be written as
$$f(x_1, x_2, \dots, x_M) = \...
- 11
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Linear independence of elementary symmetric functions over the ring of polynomials
Let $R:=\mathbb{C}[x_1,\ldots,x_n]$ and $R_{\text{Sym}}$ denote the ring of symmetric polynomials in $x_{1},\ldots,x_{n}$ over $\mathbb{C}$. Consider the elementary symmetric polynomials $e_1,\ldots,...
- 31
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Regarding the number of variables in Symmetric Functions
I'm studying Symmetric Functions and I came across a doubt that could be considered stupid but I need clarifications.
In the course I'm following we introduced symmetric functions as formal series of ...
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Write the polynomial $(2X_1^2-X_1X_2+2X_2^2)(2X_2^2-X_2X_3+2X_3^2)(2X_3^2-X_3X_1+2X_1^2)$ as a polynomial of fundamental symmetric polynomials [duplicate]
Write the polynomial $F(X_1,X_2,X_3)=(2X_1^2-X_1X_2+2X_2^2)(2X_2^2-X_2X_3+2X_3^2)(2X_3^2-X_3X_1+2X_1^2)\in \mathbb{Z}[X_1,X_2,X_3]$ as a polynomial of fundamental symmetric polynomials, i. e. Find a ...
- 1,144
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Number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$?
What is the number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$?
(Here I define $p \sim q$ iff $p(x) = q(x)$ for all $x$.)
What about the case if we allow permutations? ...
- 1,397
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a Sum Involving Elementary Symmetric Polynomials and Quadratic Exponents
I'm trying to compute the sum
$
S(y)= \sum_{s=0}^n e_s(x_1,…,x_n) y^{s^2}
$
where $e_s$ is the elementary symmetric polynomial of order $s$, and $x_1, ..., x_n, y$ are all positive real numbers. One ...
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How to decompose the symmetric polynomical into elementary symmetric polynomials.
I was trying to decompose into elementary symmetric polynomials: $$(x_1^2 - x_2 - x_3 + 1)(x_2^2 - x_1 - x_3 + 1)(x_3^2 - x_1 - x_2 + 1)$$
But it didn't work out for me.
I tried to add one of the ...
- 123
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Different bases of vector spaces of symmetric polynomials
Fix some positive integer $N$ and consider the vector space $V$ of symmetric polynomials of weight $N$, for simplicity in $N$ variables. We have different bases for $V$:
(1) - the $\sigma_{j_1}\dots\...
- 169
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Proof of Newton's Formulas.
This is question 22 in section 14.6 in Dummit and Foote, I am trying to understand its solution:
(Newton's Formulas)Let $f(x)$ be a monic polynomial of degree $n$ with roots $\alpha_1, \dots, \alpha_n$...
- 3,127
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Relation between Fourier series and Schur polynomials
I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the variables $x_j=e^{i\theta_j}$.
...
- 1,847
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Galois group of a quartic, determine all intermediate subfields explicitly
Let $F$ be the splitting field of an irreducible quartic polynomial $f \in \Bbb Q[x]$.
If Galois group of $F/\Bbb Q$ is $D_4$, I try to determine all intermediate subfields explicitly.
$D_4=⟨σ,τ⟩$, $σ=...
- 3,047
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Find the dimension and basis of the vector of the symmetric polynomials
Let $V$ be the vector space of polynomials in two variables $x$ and $y$ over $\mathbb{R}$ with degree at most two. That is, $V = \{a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 : a_i \in \mathbb{R}\}$.
...
