All Questions
Tagged with gr.group-theory symmetric-groups
23
questions
4
votes
0
answers
234
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 22.9k
14
votes
5
answers
2k
views
A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...
- 1,000
29
votes
3
answers
4k
views
Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
- 99k
11
votes
5
answers
2k
views
Structure of the adjoint representation of a (finite) group (Hopf algebra) ?
Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...
- 22.9k
3
votes
0
answers
911
views
Necklaces and the generating function for inversions
The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated:
http://mathworld.wolfram.com/Necklace.html
Let us ...
24
votes
5
answers
3k
views
Why are Jucys-Murphy elements' eigenvalues whole numbers?
The Jucys-Murphy elements of the group algebra of a finite symmetric group (here's the definition in Wikipedia) are known to correspond to operators diagonal in the Young basis of an irreducible ...
- 3,435
22
votes
6
answers
2k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,312
17
votes
8
answers
13k
views
"Natural" generating sets for symmetric groups
The symmetric group on $n$ letters has
many sets of generators. Some of them are more natural than others, eg the
set $(i,i+1)$ of adjacent transpositions (natural with respect to the type A Weyl ...
- 17k
16
votes
0
answers
324
views
Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 21.7k
15
votes
2
answers
803
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,455
15
votes
2
answers
776
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^...
- 36.8k
11
votes
1
answer
537
views
Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the ...
- 9,282
11
votes
2
answers
740
views
Characters of permutation groups
Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the ...
- 4,639
10
votes
5
answers
1k
views
Number of Permutations?
Edit: This is a modest rephrasing of the question as originally stated below the fold: for $n \geq 3$, let $\sigma \in S_n$ be a fixed-point-free permutation. How many fixed-point-free permutations $\...
- 101
10
votes
2
answers
542
views
Arbitrarily large finite irreducible matrix groups in odd dimension?
I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an ...
- 11.9k
10
votes
3
answers
641
views
Low-dimensional irreducible 2-modular representations of the symmetric group
I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
- 1,308
9
votes
1
answer
340
views
Diameter of the modified bubble-sort graph
The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
- 406
6
votes
1
answer
259
views
Is there some sort of formula for $\tau(S_n)$?
Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S_n)$, ...
- 2,618
6
votes
2
answers
311
views
Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$?
I have recently encountered a triangular array $(a_{i,j})_{0\le i\le j}$, each line of which might (should?) have a combinatorial interpretation in terms of $S_{2n+1}$. Here it is (the first entry of ...
- 13.1k
5
votes
1
answer
393
views
Which subgroup order of the symmetric group is the most frequent?
Question: What is the most frequent order of subgroups of $S_n$?
More precisely: Let $a_k$ be the number of subgroups of $S_n$ with order $k$. What is the maximum of $a_k$?
This question came up ...
- 2,994
3
votes
1
answer
731
views
Number of double cosets of a Young subgroup
Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$.
Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
- 33
2
votes
1
answer
621
views
Combinatorial problem in $\mathsf{S}_4$
I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below.
I am looking for ...
- 64.8k
1
vote
2
answers
486
views
What are all the transitive extensions of cyclic groups?
"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the ...
- 349