Questions tagged [symmetric-groups]
The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.
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Reference request : Point of view for representations of the symmetric group
This question was posted on MSE a few days ago, where it received some upvotes, but no answer.
I am doing my thesis studying representations of the symmetric group $\mathfrak{S}_n$. More precisely, I ...
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Sum over all elements in conjugacy class of $S_n$ has all integer eigenvalues in any representation?
Sorry, for the question being obvious or well-known for some, just want to reconfirm not to mislead myself and colleagues. It seems the answer might follow from previous posts by N.Elkies and B....
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Subgroups of the symmetric group $S_n$ without $2^k$-type of cycles for $1\leq k\leq t$ for some given $t\leq \lfloor \frac{n}{2}\rfloor$
Let $S_n$ be a symmetric group of degree $n\in \mathbb{N}$,
and $t$ a fixed positive integer with $t\leq \lfloor \frac{n}{2}\rfloor$.
Question: Can we characterize the maximal subgroups of $S_n$ ...
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The Peel exact sequence for hook Specht modules: conceptual proof
This is a question I will answer myself, as the answer took me long enough to
figure out that I found it worth explaining, but it is too advanced and
off-topic for my notes and I don't have a blog.
...
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Does the Gelfand--Tsetlin subalgebra have a characteristic-free basis?
This is so easy to ask that I'm surprised I've never seen it asked before.
Let $n\geq0$ be an integer.
Let $\left[ n\right] :=\left\{ 1,2,\ldots,n\right\} $.
Consider the group algebra $\mathbf{k}\...
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Equations in the permutation group
Let $n \in \mathbb N$ and let $\sigma,\tau \in {\rm Sym}(n)$. I am looking for a permutation $x \in {\rm Sym}(n)$ that minimizes the Hamming distance between $x^2 \sigma$ and $\tau x$. Here, the ...
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Similar triangles and line in the plane
The construction of figures to produce special results in plane geometry is an interesting problem, attracting the attention not only of students but also of professional mathematicians. Pizza's ...
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Sign of the permutation mapping a row-major tableau to a column-major tableau
Consider the Young diagram of an integer partition $\lambda \vdash n$. I can fill the boxes of the Young diagram with the integers $1,2,\ldots,n$ in row-major order (i.e., in increasing order row-by-...
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Real Schur indices for spin characters of symmetric groups
Let $\widetilde{S}_n^\pm,\widetilde{A}_n$ denote the double covers of the symmetric and alternating groups for $n\geq 4$. I would like to know the Schur indices over the reals (or equivalently, the ...
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Embedding the symmetric group algebra inside the Schur algebra in terms of PBW Basis
I posted a question on MSE here. There was not much response from users. So I am posting it here. The question asks to write down the simple reflections in the symmetric group in terms of the PBW ...
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Integral vs. vector representations of symmetric group
Let $A$, $B$ be two integral representations of $S_n$. Moreover, we assume that $A$, $B$ are sufficiently nice. In my question we can assume something like
$A$, $B$ are free and finitely generated as ...
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Embeddings of the finitary symmetric group in Polish groups
Let $\mathrm{Sym}(\mathbb{N})$ be the symmetric group on the set of natural numbers, endowed with the Polish tolopogy of pointwise convergence, and $\mathrm{FSym}(\mathbb{N})<\mathrm{Sym}(\mathbb{N}...
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Finite quotients of the braid group that remember the winding number modulo $d$
Let $B_n$ be the braid group on $n$ points, and let $S_n$ be the symmetric group on $n$ letters. There is a homomorphism $B_n\rightarrow S_n$, given by forgetting the paths of the strings and only ...
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Asymptotic growth rates of orbits of symmetric subgroups
Let $\mathfrak{S}_N$ be the symmetric group acting on the set $X_N = \{1,\ldots,N\}$, and let $X_N^{[n]} \subseteq X_N^n$ be the collection of ordered $n$-tuples from $X_N$ with distinct elements. $\...
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A question about binary partitions and permutations of the elements of their sets
Suppose I have a set whose cardinality is a power of 2, i.e. made of $N=2^n$ elements. If we consider the group of all the permutations of this set (the symmetric group $S_N$), we can say that it has $...
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