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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_{...
Mare's user avatar
  • 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 $O(n!)$-...
Pax Kaufman's user avatar
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: $$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$ ...
Mike Spivey's user avatar
  • 3,283
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, \...
Mark's user avatar
  • 163
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 ...
谁家的鸡's user avatar
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\}^{...
Turbo's user avatar
  • 13.9k
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 $\mathcal S_N$ ...
Roland Bacher's user avatar
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 ...
Max Alekseyev's user avatar
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 $\text{Harm}(\mathbb{R}^n, S_n)$?
Ron Donalds's user avatar