All Questions
Tagged with symmetric-polynomials co.combinatorics
15 questions with no upvoted or accepted answers
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
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
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
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...
- 149
5
votes
0
answers
250
views
A question on symmetric functions
Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
- 7,249
5
votes
0
answers
186
views
algebra of endomorphisms over the diagonal invariants
Let $k$ be a field of characteristic 0 (say $\mathbb{C}$).
Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$.
It is known that the ...
- 887
3
votes
0
answers
43
views
Kostka-Jack numbers with the zero Jack parameter
Define the Kostka-Jack number $K_{\lambda,\mu}(\alpha)$ as the coefficient of the monomial symmetric polynomial $m_\mu$ in the expression of the Jack $P$-polynomial $P_\lambda(\alpha)$ as a linear ...
- 2,560
3
votes
0
answers
206
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
2
votes
0
answers
187
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$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
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