All Questions
Tagged with symmetric-groups polynomials
9 questions
3
votes
1
answer
149
views
Drunken X-mas polynomials for graphs
Given a finite connected graph $\Gamma$ with vertices $\lbrace 1,\ldots,N\rbrace$,
we can consider the polynomial
$$\sum_{\pi\in\mathcal S_N}x^{\sum_{j=1}^Nd_\Gamma(j,\pi(j))}$$
where $\mathcal S_N$ ...
- 17.9k
4
votes
0
answers
217
views
Detecting symmetries in polynomials that lead to nice geometric properties
If we plot the single variable polynomial $p(x) = (x^2-1)^2$, it is easy to see that it has a nice property: all of its local minima are actually global minima.
In particular, it has precisely two ...
- 116
2
votes
0
answers
99
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 ...
- 34.3k
10
votes
1
answer
274
views
When are immanants irreducible?
For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$.
Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
- 26.5k
6
votes
1
answer
843
views
Symmetry Group of a Polynomial
Given a polynomial $P \in \mathbb{Z}[X_1,\ldots,X_n]$, is there a poly-time algorithm which computes the group of permutations of variables that leaves $P$ unchanged? (Clearly, the trivial $O(n!)$-...
- 61
6
votes
2
answers
630
views
Generalized cycle index polynomial for the symmetric group
The answer to a particular calculation in quantum information theory gives me the following expression:
Given $M$ specific elements of the symmetric group $S_n$, define the polynomial
$$Z_n(\pi_1, \...
- 163
1
vote
0
answers
147
views
$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$? [closed]
Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$.
My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
- 19
3
votes
3
answers
413
views
Polynomials of low degree that clone polynomials of higher degree
Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$.
Let $\mathcal{Z}$ be the zero set of $f$ in $\mathcal{C_{16}}=\{0,1\}^{...
- 13.9k
6
votes
4
answers
977
views
Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?
For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$
...
- 3,283