# Tag Info

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 ...
• 1,857

• 82.5k
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### 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 ...
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### 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

### 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 ...
• 45.6k
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• 10.9k
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### 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

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

• 31.4k
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### 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 ...
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### 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

### 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 ...
• 141k

### 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....
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• 34.4k
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### 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 ...
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### 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....
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### 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
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^{\...