The symmetric-functions tag has no wiki summary.
1
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0answers
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decomposition of products of monomial symmeric polynomials into sums of them
I'm trying to make sense of the answer given in: this question
I am stuck at the phrase 'where the partitions γ result from adding, respectively, from α all distinct partitions obtained by permuting ...
1
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0answers
23 views
Branching rule restriction to $\mathrm{O}_9 \Bbb C$ from $\mathrm{GL}_9 \Bbb C$
On page 427 of Fulton and Harris's Representation Theory, the authors give the branching rule for the above restriction as
$$
\mathrm{Res}_{\mathrm O_m \Bbb C}^{\mathrm{GL}_m \Bbb C} (\Gamma_\lambda) ...
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0answers
29 views
Forgotten symmetric function
Let $f_\lambda$ be the forgotten symmetric functions, given by $f_\lambda=\omega (m_\lambda )$. Expand $f_{(13)},f_{(21)},f_{(3)}$ in monomial symmetric functions.
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votes
1answer
32 views
Anti-Symmetric Complex Polynomial
Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have ...
1
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0answers
52 views
Reference request on symmetric polynomials
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$).
I know very little about these polynomials. I've just noticed this odd ...
1
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0answers
76 views
Probabilistic results on the elementary symmetric polynomials
The elementary symmetric polynomials of degree $k$ in $N$ variables are defined as
$$e_k(x_1, \ldots, x_N) = \sum_{(i_1,\ldots,i_N) \in I_k^N}{x_1^{i_1}\ldots x_N^{i_N}}, \quad 0 \le k \le N$$
with ...
5
votes
2answers
111 views
Symmetry of a Plücker function
Let $d \in \mathbb{N}$ and let $I$ be a set. Let $\omega : I^d \times I^d \to \mathbb{R}$ be a function, denoted by $(a_1,\dotsc,a_d,b_1,\dotsc,b_d) \mapsto a_1 \cdots a_d | b_1 \cdots b_d$, with the ...
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1answer
54 views
Convergence of series of elementary symmetric functions
Let $x_1,x_2,x_3,\ldots$ be an infinite sequence of real numbers (or assume they're complex numbers if you find that convenient).
Let $e_0,e_1,e_2,e_3,\ldots$ be the elementary symmetric functions of ...
-2
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3answers
98 views
If $f(x,y) = f(y,x)$, does it follow that $x=y$? [closed]
If $f(x,y) = f(y,x)$, does it follow that $x=y$?
If yes, please show a proof. If no, please demonstrate a counter-example.
Thank you.
1
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2answers
29 views
Finding correct symmetry axis
before I ask for anything I must admit I'm working hard to understand this beautiful subject. Thanks in advance.
$$
f(x)= 2(x)^2+8x+5
$$
Acoording to the graph of this function, there is a x-axis ...
3
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0answers
39 views
Symmetrizing a sequence of vectors
Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
2
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1answer
103 views
Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions
A rational function $f$ in $n$ variables is a ratio of $2$ polynomials,
$$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$
where $q$ is not identically $0$. The function is called symmetric if ...
0
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0answers
52 views
what exactly does symmetric game and symmetric equilibrium mean?
I am confused about the ideas of a symmetric game and symmetric equilibrium of a game under the following conditions.
1) pure strategy Nash equilibrium
2) Nash bargaining game where players set a ...
5
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0answers
116 views
Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.
(this is an improved version of What about other symmetric functions of the eigenvalues? )
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
7
votes
1answer
123 views
What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?
The product of monomial symmetric polynomials can be expressed as
$m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$
for some constants $c_{\lambda\mu}^{\nu}$.
In the case of Schur ...
1
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1answer
185 views
Doubt with parametric and symmetric equations
In the line through $P(0, 0, 0)$ and is perpendicular to $x=y-5$, $z=2y-3$, when we solve the equations and get the symmetric equations in order to find the vectors $V_1$ and $V_2$, why the normal ...
3
votes
2answers
502 views
Scalar Product for Vector Space of Monomial Symmetric Functions
Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$:
$$
P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} .
$$
Since ...
2
votes
1answer
122 views
Symmetric polynomials have unique expressions as polynomials in symmetric elementary functions
Let $s_i$ be the symmetric elementary functions. For example, $s_1=x_1+\cdots+x_n$.
Suppose a polynomial $p(z_1,\ldots,z_n)\in R[z_1,\ldots,z_n]$ satisfies $p(s_1,\ldots,s_n)=0$ in ...
2
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95 views
Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.
Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
6
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2answers
503 views
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via
$$
\prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N.
$$
How can one get the monomial symmetric functions ...
0
votes
1answer
104 views
Roots of elementary monomials
Let $m_\lambda(X_1,X_2,...X_N)$ be a monomial symmetric function with partition $\lambda$.
For example:
$$
m_{(3,1,1)}(X_1,X_2,X_3) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3
$$
Is there a general ...
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3answers
388 views
How do I find out the symmetry of a function?
For example, how do I know that with:
$$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$
$f$ has the property:
...
5
votes
0answers
96 views
Schur skew functions
Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If
$s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$
...
4
votes
2answers
417 views
Sum of cubed roots
I need to calculate the sums
$$x_1^3 + x_2^3 + x_3^3$$
and
$$x_1^4 + x_2^4 + x_3^4$$
where $x_1, x_2, x_3$ are the roots of
$$x^3+2x^2+3x+4=0$$
using Viete's formulas.
I know that ...
