Questions tagged [symmetric-polynomials]
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102
questions
0
votes
1
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219
views
Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?
The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
- 2,715
4
votes
1
answer
303
views
Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?
I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials ...
3
votes
1
answer
55
views
Alternative bases of symmetric polynomials in cohomology ring of flag varieties and coinvariant algebras
$\DeclareMathOperator\Fl{Fl}$It is known that $H^*(\Fl(m)) \cong R^{\mathbb Z}(m)$, where $\Fl(m)$ denotes the variety of complete flags in $\mathbb C^m$, and $R^{\mathbb Z}(m)$ is the coinvariant ...
- 185
6
votes
1
answer
291
views
Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?
For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...
- 940
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votes
0
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82
views
Applications of Jack polynomials
I developed four libraries (Julia, R, Python, Haskell) for the computation of Jack polynomials. I developed them for fun because I found this was programmatically interesting. But now I'd like them to ...
- 2,293
7
votes
0
answers
77
views
Generalization of Lucas sequences to order 3 (and above)
For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...
- 29.1k
6
votes
1
answer
408
views
Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers
Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$.
Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...
- 151
3
votes
1
answer
235
views
Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?
Consider a set $N$ with elements $n_1, n_2, \dots, n_k$ which are distinct integers. Introduce the notation $N_{i=1,2,\dots,s}$ for the $s$ blocks of a set partition of $N$. Consider a supplementary ...
- 71
6
votes
1
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465
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Construction of a symmetric polynomial in the roots that acts like the discriminant
The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
8
votes
1
answer
556
views
When the Littlewood-Richardson rule gives only irreducibles?
Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:
$$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$
is there a classification of the cases where the LR ...
1
vote
0
answers
106
views
Schur polynomial with integer values
There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...
2
votes
0
answers
87
views
Symmetric polynomial constructed from symmetric group
Let $n$ be a positive integer, $S_n$ be the symmetric group. For a permutation $p=[p_1,\dots,p_n]\in S_n$, define $x^p := x_1^{p_1}\cdots x_n^{p_n}$. It can be seen that the following polynomial is ...
- 29.1k
3
votes
1
answer
126
views
Bounds on symmetric polynomials in power-sum form with bounded coefficients
Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$.
For a partition $\lambda= (j_1,\...
- 36.1k
3
votes
0
answers
36
views
Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
- 213
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votes
0
answers
95
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Positive values of Schur polynomials
Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials
$$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...
- 718
4
votes
1
answer
264
views
Nonnegativity locus of Schur polynomials
Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
- 5,270
10
votes
1
answer
332
views
Symmetric polynomials that detect positivity
Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you ...
- 1,513
23
votes
1
answer
922
views
Symmetric polynomial inequality arising from the fixed-point measure of a random permutation
A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows.
Given $n$ non-negative reals $a_1, ...
- 543
5
votes
0
answers
862
views
A generalization of the difference of two squares identity
This question deals with finding explicit integer functions for the coefficients of the monomial expansion of the polynomial
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\kappa_1, \ldots , \kappa_{n-...
- 129
3
votes
1
answer
298
views
Polynomial function defined recursively by a resultant - is it well defined?
Preliminaries
Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| ...
- 129
1
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1
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89
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A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$
If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\...
- 133
3
votes
1
answer
186
views
Conjecture on some combinatorial constant
In the process of computing Shapley values, I observed an interesting combinatorial constant. I am not exactly sure where such behavior comes. And here is the conjecture.
Notations
For any finite non-...
- 183
3
votes
0
answers
119
views
Multiplicities of irreducible $U(n)$-modules in the tensor product $V_{\lambda}\otimes V_{\mu}$
It is known that every irreducible representation of the unitary group $U(n)$ can be uniquely described by the non-increasing sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of integers (denote the ...
- 333
6
votes
1
answer
285
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Representing a symmetric polynomial as a conical sum of squares
This question in inspired by the recent solution to another question.
The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$:
$$m_{(4, 3, 2, 1)} + m_{(4, ...
- 29.1k
2
votes
0
answers
299
views
The Galois resolvent in Lagrange
In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...
- 407
4
votes
2
answers
226
views
Schur positivity of a polynomial
Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...
4
votes
1
answer
228
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proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"
I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
- 41
1
vote
0
answers
91
views
sum of squares of the coefficients of a monic polynomial [closed]
Consider the monic polynomial $$f: = X^n + a_1 X^{n-1} + \cdots + a_n$$ over $\mathbb Z$. Let $p_i$ be the $i$_th power sum ($1 \le i \le n$) of the roots of $f$ (in some extension of $\mathbb Z$). ...
- 311
10
votes
1
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415
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Generalization of symmetric functions
A $n$-variable function $f$ is a symmetric function if
$$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$
for every permutation $\sigma \in S_n$.
In particular, if $f$...
- 245
20
votes
2
answers
2k
views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 463
5
votes
1
answer
195
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Frobenius algebras from symmetric polynomials
Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
- 25k
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votes
0
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149
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Generalization of elementary symmetric polynomials
The elementary symmetric polynomials (ESPs) are defined as -
\begin{align*}
E_{1}^{1}
&= X_1,
\\
E_{1}^{2}
&= X_1 + X_2,
\\
E_{2}^{2}
&= X_1 X_2,
\\
E_{2}^{3}
&= X_1 X_2 + X_1 X_3 + ...
- 47
5
votes
0
answers
324
views
Expressing the elementary symmetric polynomials in the $(x_i-x_j)^2$ variables in term of the elementary polynomials in the $x_k$ variables
Let $n>1$ be an integer and let $P$ be a multivariate symmetric polynomial in $n(n-1)/2$ variables.
Let us define the multivariate polynomial $Q_P$ in n variables as:
$Q_P(x_1,...,x_n)=P\left(\{(...
- 51
4
votes
1
answer
479
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Derivatives of Riemann $\xi$ and traces of zeros
Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
- 9,377
1
vote
0
answers
132
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Symmetric functions for multidimensional variables
I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to make permutation invariant.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. ...
- 11
11
votes
3
answers
586
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Polynomial inequality of sixth degree
There is the following problem.
Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$
Prove that:
$$\sum_{cyc}\...
- 1,054
5
votes
3
answers
464
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Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$
Let $n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $ and $\sum\limits_{1\leq i<j<k\leq n}...
- 1,054
3
votes
0
answers
132
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Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
- 131
4
votes
1
answer
177
views
Applying a simple involution to Hall-Littlewood polynomials
Consider the Hall-Littlewood polynomial
$$
P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-...
- 2,219
5
votes
1
answer
421
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Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
- 129
1
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0
answers
388
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An explicit formula for characteristic polynomial of matrix tensor product [closed]
Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...
- 119
4
votes
1
answer
366
views
Possible values of symmetric functions evaluated on quaternions
$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be the units of quaternions, in particular $i^2=j^2=k^2=-1$, $ijk=-1$.
We will use non commutative variables $x$, $y$, $z$. Define $\sym_{a,b,c}$ to be ...
- 43
0
votes
1
answer
472
views
Symmetric polynomials in two sets of variables
Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
user133644
7
votes
2
answers
830
views
Maximize $L^p$ norm over sphere
For $p \in \mathbb{R}$, consider the function
$$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda_1^2 +...
- 9,297
2
votes
0
answers
57
views
Classes of curves with "determinant-like operation"
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
- 743
1
vote
1
answer
66
views
Does stability of equilibrium point preserved by permutation matrix (symmetry)?
Given the following differential equations:
\begin{equation}
\begin{aligned}
\dot{x}_1 &= f_1(x_1,\ldots,x_n) \\
\vdots \\
\dot{x}_n &= f_n(x_1,\ldots,x_n)
\end{aligned}
\end{equation}
In ...
- 345
3
votes
0
answers
64
views
Bounds related to Monomial Symmetric Functions
Two of the well studied bases for symmetric functions over $\mathbb{Q}$ are the monomial symmetric functions $\{ m_{\lambda} \}_{\lambda \text{ a partition}}$ and the power sum symmetric functions $\{ ...
- 12.9k
3
votes
0
answers
316
views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
- 131
5
votes
1
answer
303
views
When is a linear combination of the elementary symmetric polynomials reducible?
Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
user111524
3
votes
1
answer
640
views
Symmetry in the triangular distribution
A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$
The mean of ...
- 10.7k