All Questions
Tagged with co.combinatorics symmetric-polynomials
36 questions
23
votes
1
answer
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, ...
- 543
13
votes
3
answers
894
views
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}(\...
- 231
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. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
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 $f$...
- 325
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 ...
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
406
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
$$\...
- 5,474
7
votes
0
answers
239
views
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
5
votes
1
answer
327
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 ...
- 2,560
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
453
views
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
5
votes
2
answers
243
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$. ...
- 725
5
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
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...
- 149
5
votes
0
answers
250
views
A question on symmetric functions
Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
- 7,249
5
votes
0
answers
186
views
algebra of endomorphisms over the diagonal invariants
Let $k$ be a field of characteristic 0 (say $\mathbb{C}$).
Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$.
It is known that the ...
- 887
4
votes
1
answer
245
views
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
4
votes
1
answer
208
views
Applying a simple involution to Hall-Littlewood polynomials
Consider the Hall-Littlewood polynomial
$$
P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-...
- 2,516
3
votes
1
answer
336
views
Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?
Consider a set $N$ with elements $n_1, n_2, \dots, n_k$ which are distinct integers. Introduce the notation $N_{i=1,2,\dots,s}$ for the $s$ blocks of a set partition of $N$. Consider a supplementary ...
- 71
3
votes
1
answer
170
views
Eigenvalues of the Jack polynomials for the Calogero-Sutherland operator
The Calogero-Sutherland operator on the space of homogeneous symmetric polynomials in $n$ variables is defined by
$$
\frac{\alpha}{2}\sum_{i=1}^n x_i^2\frac{\partial^2}{\partial x_i^2} + \frac{1}{2}\...
- 2,560
3
votes
2
answers
824
views
On a positivity property of Hall-Littlewood polynomials
Here's the new, more thought through version.
Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
- 3,513
3
votes
1
answer
193
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
43
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 ...
- 2,560
3
votes
0
answers
206
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
2
votes
0
answers
187
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 $$M_{ij}=(-1)^i e_{i-1}(...
2
votes
0
answers
49
views
Skew Jack polynomial when the Jack parameter is zero
According to Macdonald's book, when the Jack parameter $\alpha$ is $0$,then the Jack $P$-polynomial $P_\lambda(\alpha)$ is the elementary symmetric polynomial $e_{\lambda'}$ where $\lambda'$ is the ...
- 2,560
2
votes
0
answers
169
views
counting how many boxes from a given Young tableau contribute to hook length made out of two YTs
Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...
- 533
1
vote
1
answer
129
views
Multidimensional power series with coefficients equal to an order of stabilizer of a set of powers
I have encountered a necessity to work with a series of the following form.
There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They ...
- 61
1
vote
0
answers
40
views
Matrix transform of the bivariate Narayana polynomials into the arithmetic and geometric means of the two indeterminates
The matrix identity presented below is a specialization of the more general result displayed in the MSE-Q "Lah and associahedra partition polynomials and symmetric functions (reference request)&...
- 10.5k
1
vote
0
answers
118
views
Schur polynomial with integer values
There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...
0
votes
0
answers
96
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 ...
- 2,560
0
votes
0
answers
109
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 ...
- 2,560
0
votes
0
answers
186
views
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