Questions tagged [symmetric-polynomials]
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126 questions
41
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The sum of squared logarithms conjecture
I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
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23
votes
2
answers
3k
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 ...
- 493
23
votes
1
answer
1k
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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, ...
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17
votes
2
answers
3k
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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 $k$, ...
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14
votes
2
answers
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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 ...
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13
votes
3
answers
894
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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*}
e_{h}(\...
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13
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4
answers
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$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial
Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and $y=(...
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3
answers
651
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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:
$$\sum_{cyc}\...
- 1,054
11
votes
2
answers
2k
views
Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
- 2,356
11
votes
2
answers
670
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A particular specialization of symmetric polynomials: is it bijective?
Let $\Lambda^d_n$ the space of symmetric polynomials
in $n$ variables, with maximum 'partial degree' of each variable $d$.
A basis for this space is the set of symmetrized monomials $m_\lambda$,
where ...
- 451
11
votes
0
answers
615
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
11
votes
0
answers
387
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
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10
votes
1
answer
478
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Symmetric polynomials that detect positivity
Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you ...
- 1,513
10
votes
1
answer
522
views
Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y),
$$
where $s_\lambda(x)$ is the Schur function.
Is there a known decomposition of the ...
- 273
10
votes
1
answer
492
views
Generalization of symmetric functions
A $n$-variable function $f$ is a symmetric function if
$$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$
for every permutation $\sigma \in S_n$.
In particular, if $f$...
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9
votes
1
answer
454
views
Combinatorics and symmetric functions
No answers from stackexchange, so I'll try this here:
(The actual questions in this posting are at the bottom.)
Occasionally someone asks on stackexchange how to show that every nonempty finite set ...
9
votes
1
answer
253
views
Hyper-symmetric polynomials (reference request)
Let $M_n$ be the linear space of $n\times n$ matrices. The product of symmetric groups $S_n\times S_n$ acts naturally on $M_n$, and thus induces an action on the coordinate algebra $k[M_n]$. Is there ...
- 417
8
votes
1
answer
586
views
When the Littlewood-Richardson rule gives only irreducibles?
Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:
$$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$
is there a classification of the cases where the LR ...
8
votes
0
answers
145
views
Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
- 413
8
votes
0
answers
88
views
Generalization of Lucas sequences to order 3 (and above)
For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...
- 34.3k
7
votes
2
answers
533
views
Schur polynomial, change of variable
Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...
- 3,804
7
votes
2
answers
480
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Free $k[x_1, \dots, x_n]^{S_n}$-module?
Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{...
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7
votes
1
answer
354
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:\...
- 960
7
votes
1
answer
320
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PDE characterisation of elementary symmetric functions?
For $k\leq{}n$ the elementary symmetric polynomials are defined by:
$$e_k(x_1,\ldots,x_n)=\sum_{1\leq{}i_1<...<i_k\leq{}n}x_{i_1}\cdots x_{i_k}$$
I believe I can prove (by a complex brute force ...
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7
votes
2
answers
1k
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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
$$ \lambda_1^2 +...
- 9,853
7
votes
2
answers
406
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Proving an identity for flagged Schur without use of determinants?
In proposition 3 of Determinantal transition kernels for some interacting particles on the line, Dieker and Warren prove the following identity: consider vector $a:=(a_1,\dotsc,a_N)$ and kernels
$$\...
- 5,474
7
votes
1
answer
271
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Existence of a linear map resulting in the determinant being an elementary symmetric polynomial
Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
- 6,351
7
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0
answers
239
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Characterizing $n$-exceptions of the ring of symmetric polynomials
(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials)
We say that an homogeneous symmetric polynomial ...
- 307
6
votes
3
answers
515
views
Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials
The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials.
Is it known how to write the result of the application of $L_0$, ...
- 1,549
6
votes
1
answer
533
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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 ...
6
votes
1
answer
320
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Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
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6
votes
1
answer
303
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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 $x_1,x_2,x_3,x_4$:
$$m_{(4, 3, 2, 1)} + m_{(4, ...
- 34.3k
6
votes
0
answers
308
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Coefficients in expansion of a classical symmetric polynomial
If we expand \begin{equation} P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k), \end{equation} then \begin{equation} P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\...
- 61
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0
answers
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Find a symmetric polynomial with a projection divisible by a known polynomial
Consider the polynomial $Q$, a homogeneous quartic in seven variables:
$$ Q(R, s_1, s_2, s_3, s_4, d_1, d_2) = \\
(d_1^2-(R+s_1+s_2-s_3-s_4)(R+s_1-s_2+s_3+s_4))\\(d_1^2-(R-s_1-s_2-s_3-s_4)(R-s_1+s_2+...
- 2,902
5
votes
5
answers
873
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Green polynomials
Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in ...
- 1,590
5
votes
3
answers
510
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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
5
votes
1
answer
852
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 ...
- 818
5
votes
2
answers
1k
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Isotypic components of the action of the symmetric group on polynomials
The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...
- 2,537
5
votes
1
answer
327
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The coefficients of the Jack polynomials are polynomials in the Jack parameter
I implemented the Jack polynomials with a symbolic Jack parameter $\alpha$ in their coefficients ($\alpha=1$ for Schur polynomials, $\alpha=2$ for zonal polynomials). From my implementation (and also ...
- 2,560
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votes
2
answers
275
views
Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
I am looking at a polynomial of the entries of a matrix, and this polynomial is invariant under permutation of the rows or columns of the matrix. Is there a similar characterization as in the case of ...
- 244
5
votes
1
answer
189
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Normalization of Jack polynomial integral-scalar product?
In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are ...
- 331
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votes
1
answer
189
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Symmetric polynomial separating points
I've been looking for references/answers to this problem for several days and I couldn't find anything.
If we consider the closed unit ball $B$ in $\mathbb C^2$ then for any point $(z_1,z_2)\notin B$ ...
5
votes
1
answer
226
views
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 ...
- 26.5k
5
votes
1
answer
354
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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
5
votes
1
answer
439
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 $S^n(A)$ ...
- 618
5
votes
1
answer
453
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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 $i\in\{1,2,...,n-1,n\}$, ...
- 149
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votes
1
answer
202
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Alternating elements in free graded-commutative algebras
It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...
- 17.4k
5
votes
2
answers
284
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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 ...
5
votes
2
answers
242
views
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
- 161
5
votes
2
answers
330
views
sum of squares of Schur polynomials indexed over partition valued functions on a set
Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...
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