Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
1 answer
378 views

Where to begin in Computational Group Theory?

I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
G. Fougeron's user avatar
16 votes
3 answers
1k views

Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group? I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
Mare's user avatar
  • 26.5k
9 votes
0 answers
254 views

An identity for characters of the symmetric group

I am looking for a reference for the identity $$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$ for the irreducible characters of the ...
Hjalmar Rosengren's user avatar
3 votes
2 answers
451 views

Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?

I'm interested in the representation theory of symmetric groups. I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
gualterio's user avatar
  • 1,013
3 votes
0 answers
115 views

Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius

I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius. There he mentioned some theorems of Netto. I'm depending on the Google translator. and the translation ...
gualterio's user avatar
  • 1,013
1 vote
0 answers
213 views

Is there any research on the action of a subgroup on the whole finite group by conjugation?

I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.) I'm especially ...
gualterio's user avatar
  • 1,013
10 votes
1 answer
269 views

Edge-transitive Cayley graphs of $S_n$

I came across the following question which I haven't seen before: Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...
Igor Pak's user avatar
  • 17k
9 votes
2 answers
2k views

alternating and symmetric powers of the standard representation of the symmetric group

Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$ Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
Libli's user avatar
  • 7,300
9 votes
2 answers
762 views

Solutions of $x^d=1$ in the symmetric group

L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations: $$f_d(n):=\#\{\pi\in\...
T. Amdeberhan's user avatar
16 votes
2 answers
818 views

Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
Benjamin Steinberg's user avatar