37 votes
Accepted

What is the standard 2-generating set of the symmetric group good for?

Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots. You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a ...
24 votes

What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

Write $S = \bigsqcup_n BS_n$ for the symmetric monoidal category of finite sets and bijections under disjoint union, and write $\mathbb{S}$ for the sphere spectrum, thought of as a symmetric monoidal $...
24 votes
Accepted

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

By Schur–Weyl duality there is an isomorphism of $\mathrm{GL}(V) \times S_m$-representations $$V^{\otimes m} \cong \bigoplus_\lambda \Delta^\lambda(V) \boxtimes S^\lambda$$ where the sum is ...
  • 10.4k
23 votes
Accepted

A cancellation property for permutations?

Let $n$ be some integer greater than 2. Since the number of even and odd permutations in $S_n$ is the same we have $\sum_{\sigma\in S_{n}}(-1)^{\ell(\sigma)}=0$ therefore the contribution of $\sum_{\...
21 votes
Accepted

Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result?

This turns out to be a completely general phenomenon for configuration spaces on any open manifold, though we did not know this at the time; it came in the later paper "FI-modules and stability for ...
  • 7,946
21 votes

The character table of the symmetric group modulo m

This is true when $m$ is prime and false in general. Counterexample. Take $S_8$ with $m=6$. Computer calculations show that the $\mathbb{Z}$-rank of the character table of $S_8$ with entries taken ...
  • 10.4k
21 votes

The character table of the symmetric group modulo m

When $m$ is prime there is a simpler proof. The Smith normal form of the character table of $S_n$ is computed at Problem 14 here (solution here). From this it follows that the rank of the character ...
20 votes
Accepted

trace and involution permutations: Part I

Let us identify an involution $\sigma$ in $\mathfrak{S}_n$ with a set partition $\Pi_\sigma$ of $[n] := \{1,2,...,n\}$ into nonempty blocks of size at most two in the obvious way: we have $\{a,b\} \in ...
  • 19.6k
20 votes
Accepted

Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

Here are a few comments and a slightly different approach, though we take advantage of some of the earlier comments. We first note the well-known (at least to people who work with factorizations) ...
20 votes

What is the standard 2-generating set of the symmetric group good for?

It might be interesting (to some) to see that every possible shuffle of a pack of $n$ cards can be achieved by a sequence of operations in which you either swap the first two cards or move the bottom ...
20 votes

Bijective proof for a partition identity

The answer is yes, there is a combinatorial proof, and both Sam's and Fёdor's proofs work. However, this is a really old result and a combinatorial proof is old. Here is the reference: H. Gupta, ...
  • 15.8k
19 votes
Accepted

Simplicity of alternating group $A_n$

Just turning comments into an answer. Iwasawa's Criterion. This will do $A_5$ (using the natural action), $A_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}...
  • 10.9k
19 votes
Accepted

Is the Normal centralizer problem in P?

Yes. This is Proposition 7.3 of Eugene M. Luks. Permutation groups and polynomial-time computation. Pages 139-175 of: Larry Finkelstein and William M. Kantor, editors. Groups and Computation, Volume ...
  • 34.4k
19 votes

Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

As has been pointed out in comments, the only subgroups $G$ and $H$ of $S_{10}$ isomorphic to $S_7$ and $S_6$ that could possibly have trivial intersections are the copy of $S_7$ that lies in $A_{10}$ ...
  • 34.4k
18 votes
Accepted

Cohomology of configuration space as a representation of the symmetric group

One can indeed modify the formulas in the paper of Getzler to get an answer for any $\mathbf R^d$, by judiciously inserting minus signs and making substitutions $x \mapsto x^{d-1}$ in various places, ...
  • 37.5k
16 votes
Accepted

On an asymptotic formula of Keating and Snaith involving the Riemann zeta function

There is a connection! (Though see the edit below.) Keating and Snaith make their conjecture by modeling the distribution of $\zeta(s)$ by the distribution of the characteristic polynomial of a random ...
  • 2,016
16 votes
Accepted

Is this sum of cycles invertible in $\mathbb QS_n$?

$\newcommand{\cyc}{\operatorname{cyc}} \newcommand{\id}{\operatorname{id}} \newcommand{\BB}{\mathbf{B}} \newcommand{\AA}{\mathbf{A}} \newcommand{\kk}{\mathbf{k}} \newcommand{\ww}{\mathbf{w}} $ PART 1 ...
15 votes
Accepted

Minimal maximal subgroup of the symmetric group

Maximal intransitive groups will be $S_a\times S_b$ and have comparatively small index. I think once $d>6$ (otherwise there is a small number effect) the maximal subgroup of maximal index will ...
  • 1,090
15 votes
Accepted

Okounkov-Vershik approach to representation theory of $S_n$

First, I'll note that they provide a pretty clear explanation of their motivation on page two of that paper. So it would strengthen your question if you indicated you had read that, and what about ...
  • 41.8k
15 votes

How to constructively/combinatorially prove Schur-Weyl duality?

This is a quick answer to explain the statement that the hard direction of Schur-Weyl duality is the same thing the First Fundamental Theorem of invariant theory. Let $V$ be a finite dimensional ...
15 votes

A finite dimensional algebra associated to the symmetric group

For starters one can think about an algebraically closed field of characteristic zero. I will only sketch, so there is no need to accept it as an anwer. One can think in terms of representation theory....
  • 5,352
15 votes

What is the standard 2-generating set of the symmetric group good for?

As has been more or less said in comments, I think the important and useful thing to know is that $S_n$ can be generated by two elements. It is less important which two you choose, but $(1,2)$ and $(1,...
14 votes

universality of Macdonald polynomials

There are plenty of polynomials, and only a few are specializations of Macdonald polynomials. As a polynomial botanist, the following is a very incomplete family tree. A more comprehensive list of ...
14 votes
Accepted

The number of subgroups of ${\frak S}_n$

As Francesco Polizzi mentions, the answer is no alredy for ${\frak S}_4$: there are $30$ subgroups, but $4!=24$. Here are some more (very small) calculations: \begin{array}{|c|c|c|c|} \hline \mathrm{...
  • 17.1k
14 votes

Number of squares in a finite group

I answer here positively the second question (it's completely independent of the first one so it could have been 2 distinct posts). Let $p\le n$ be prime. Let $C\subset S_n$ be the group generated by ...
  • 52.9k
14 votes

Symmetry Group of a Polynomial

If the input has fully-expanded polynomials, then this is equivalent to graph isomorphism. In one direction, given a graph, create a variable for each vertex and consider the polynomial $\prod_{vw\in ...
14 votes
Accepted

Schur-Weyl duality and q-symmetric functions

As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those ...
14 votes
Accepted

Bijective proof for a partition identity

Lemma. For $n>1$, the number of partitions of $n$ onto an even number of powers of 2 (here powers of 2 are 1,2,4,...) and the number of partitions of $n$ onto an odd number of powers of 2 are equal....
  • 90.4k
13 votes

Problems which use S₄ → S₃

I use this or something equivalent in teaching projective geometry to show that the cross-ratio has at most 6 distinct values (when the points are permuted) as opposed to the 24 naively expected. This ...
  • 12.5k
13 votes
Accepted

hooks and contents: Part I

Specialize the Cycle Index Formula to get $$\prod_{n=1}^\infty \exp \bigl( \frac{a_i}{i}z^i \bigr) = \sum_{n=0}^\infty \frac{z^n}{n!} \sum_{\pi \in \mathfrak{S}_n} a_1^{\mathrm{cyc}_1(\pi)} a_2^{\...
  • 10.4k

Only top scored, non community-wiki answers of a minimum length are eligible