Questions tagged [symmetric-polynomials]
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131 questions
12
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0
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Existence of odd integer solutions to a system of elementary symmetric equations
Consider the following system of equations:
$$
\sigma_{2k}(x_1,x_2,\dots,x_{2n+1}) = (-1)^k \binom{n}{k}, \quad 1 \le k \le n,
$$
where $\sigma_k(x_1,x_2,\dots,x_{2n+1})$ denotes the $k$-th elementary ...
- 476
2
votes
0
answers
100
views
Regularity of a variant of elementary symmetric polynomials
Let $Y_1, Y_2,\dots, Y_n$ be $n$ bases of linear forms in the polynomial ring $k[x_1,\dots, x_n]$, where $k$ is a field or $\mathbb{Z}$. Examples of bases $Y_i$'s are $A_i\cdot[x_1,\dots, x_n]^T$, ...
- 21
4
votes
1
answer
508
views
Maximal ideals in rings with polynomial relations
Let $F$ be a perfect field, and let $p(t) \in F[t]$ be an irreducible monic polynomial of degree $n$ such that: $$p(t) = t^n+s_1 t^{n-1}+s_2 t^{n-2}+\dots+s_n$$ Let $\theta_1, \theta_2, \dots, \...
12
votes
2
answers
456
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Is $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ Schur positive for large enough $p$?
I would like to know if the symmetric polynomial $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ is Schur positive for large enough $p$. Here, how large the $p$ is can depend on $n$ and $...
- 121
1
vote
1
answer
190
views
Expressing factorial elementary symmetric polynomials in terms of factorial complete symmetric polynomials
There are well known identities relating elementary symmetric polynomials in $k$ variables and complete homogeneous symmetric polynomials in the same set of variables. For example,
$$e_{2,k}(x) = h_{1,...
- 2,448
2
votes
2
answers
250
views
Inequality of elementary symmetric polynomials written in terms of derivatives of $(s+ta_1)\dotsm(s+ta_n)$
For positive integers $k,n$ with $k\le n{-}1$ and $a\in\mathbb{R}^{n}$, let $S_{k}(a):=\sum_{i_{1}<\dots<i_{k}}a_{i_{1}}\dotsm a_{i_{k}}$ be the $k$-th elementary symmetric polynomial. If the ...
11
votes
3
answers
429
views
Positivity of elementary symmetric polynomials under linear fractional transformations
The general question
For $1\leq k\leq n$, let $$e_k(a_1,\dots,a_n):=\sum_{j_1<\dots<j_k}a_{j_1}\cdots a_{j_k}$$ be the $k$-th elementary symmetric polynomial.
Let $a_1,\dots,a_n<1$ and $e_1(...
5
votes
2
answers
361
views
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
- 161
8
votes
0
answers
178
views
Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
- 513
2
votes
0
answers
101
views
Hall-Littlewood polynomials for $n$-tuples that are not partitions
For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
- 6,459
6
votes
1
answer
460
views
Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
- 99
2
votes
0
answers
228
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
7
votes
2
answers
608
views
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
$$\...
- 5,599
0
votes
0
answers
180
views
Irrational elements can always be moved
Let $x_1,x_2,x_3,\ldots,x_n$ be the roots of a polynomial $P_n(x)$. Let $F$ be the field $\mathbb{Q}[x_1,x_2,x_3,\ldots,x_n]$, i. e. all the possible combinations of rational numbers with $x$'s.
It's ...
- 125
3
votes
0
answers
65
views
Kostka-Jack numbers with the zero Jack parameter
Define the Kostka-Jack number $K_{\lambda,\mu}(\alpha)$ as the coefficient of the monomial symmetric polynomial $m_\mu$ in the expression of the Jack $P$-polynomial $P_\lambda(\alpha)$ as a linear ...
- 2,590
2
votes
0
answers
80
views
Skew Jack polynomial when the Jack parameter is zero
According to Macdonald's book, when the Jack parameter $\alpha$ is $0$,then the Jack $P$-polynomial $P_\lambda(\alpha)$ is the elementary symmetric polynomial $e_{\lambda'}$ where $\lambda'$ is the ...
- 2,590
7
votes
1
answer
357
views
Existence of a linear map resulting in the determinant being an elementary symmetric polynomial
Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
- 6,882
0
votes
0
answers
132
views
"Degenerate" Schur polynomials
Let's say that a Schur polynomial is degenerate if its number of variables is less than the weight of the partition it is associated to. For example, according to Sage, the Schur polynomial of the ...
- 2,590
3
votes
1
answer
236
views
Eigenvalues of the Jack polynomials for the Calogero-Sutherland operator
The Calogero-Sutherland operator on the space of homogeneous symmetric polynomials in $n$ variables is defined by
$$
\frac{\alpha}{2}\sum_{i=1}^n x_i^2\frac{\partial^2}{\partial x_i^2} + \frac{1}{2}\...
- 2,590
3
votes
0
answers
256
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
5
votes
1
answer
428
views
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 ...
- 2,590
12
votes
0
answers
688
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} \...
1
vote
1
answer
175
views
Multidimensional power series with coefficients equal to an order of stabilizer of a set of powers
I have encountered a necessity to work with a series of the following form.
There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They ...
- 159
6
votes
3
answers
658
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
0
votes
0
answers
121
views
Quick calculation of a symmetric product with two indices
Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
- 397
1
vote
0
answers
198
views
Hall-Littlewood polynomials with sage
I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...
- 6,459
2
votes
0
answers
125
views
Lower & upper bound on the maximal component given the system of power sums
Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like)
$$
\begin{cases}
x_1+x_2+\...
- 21
-1
votes
1
answer
325
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 ...
- 2,883
4
votes
1
answer
463
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
115
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
8
votes
1
answer
450
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:\...
- 970
0
votes
0
answers
134
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 ...
- 2,590
8
votes
0
answers
200
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 ...
- 38.5k
17
votes
1
answer
3k
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$, ...
- 381
11
votes
0
answers
447
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}$).
...
- 605
3
votes
1
answer
448
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
answer
587
views
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
633
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,$$
is there a classification of the cases where the LR ...
1
vote
0
answers
144
views
Schur polynomial with integer values
There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...
2
votes
0
answers
118
views
Symmetric polynomial constructed from symmetric group
Let $n$ be a positive integer, $S_n$ be the symmetric group. For a permutation $p=[p_1,\dots,p_n]\in S_n$, define $x^p := x_1^{p_1}\cdots x_n^{p_n}$. It can be seen that the following polynomial is ...
- 38.5k
3
votes
1
answer
201
views
Bounds on symmetric polynomials in power-sum form with bounded coefficients
Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$.
For a partition $\lambda= (j_1,\...
- 38.4k
3
votes
0
answers
76
views
Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
- 213
2
votes
0
answers
154
views
Positive values of Schur polynomials
Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials
$$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...
- 973
4
votes
1
answer
421
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 ...
- 6,886
10
votes
1
answer
691
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 ...
- 1,523
25
votes
2
answers
1k
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, ...
- 563
6
votes
0
answers
1k
views
A generalization of the difference of squares identity
Let us find explicit integer functions for the coefficients of the monomial expansion of
$$
\prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}}{\left( x_n+\sum_{i=1}^{n-1}{\...
- 219
3
votes
1
answer
381
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| ...
- 219
1
vote
1
answer
147
views
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
223
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