# 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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### For which $n$ can $S_n$ act transitively on $n+k$ elements?

It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$. Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Finding the decorated permutation of a non-reduced plabic graph

This is a question about Postnikov's theory of positroids and plabic graphs. The short version is If we have an non-reduced plabic graph $G$, how do we look at the alternating strands and read off ...
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### Faithful representation into $\operatorname{GL}(9,3)$

Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$. Does there exist an injective ...
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### I want to know the name of or any references for a matrix in the book “The representation theory of the symmetric groups” by Gordon James

$\DeclareMathOperator{\Ind}{\operatorname{Ind}}$I'm reading "The representation theory of the symmetric groups" written by Gordon James. I found the matrix $B$ in the chapter 6 ("The ...
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### A question related to Young symmetrizers

Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
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### Classification of posets that are quotient posets of the Boolean lattice

Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics. $B_n/G$ for a subgroup $G$ of the ...
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### A basic question about Young symmetrizers

This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I ...
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### On a certain expansion in term of Schur functions

This question is related to this other one A Schur positivity conjecture related to row and column permutations by Richard Stanley (thanks to Sam Hopkins for letting me know about it). Consider a ...
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### Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
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### Question about generalizing Cauchy identity

One of the Cauchy identities says that $$\prod_{i,j}(1+x_iy_j) = \sum_\lambda s_\lambda (x_1, \cdots,x_m) s_{\lambda'} (y_1, \cdots,y_n)$$ Where $\lambda$ is a Young diagram, $\lambda'$ is the ...
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### Number of paths in the Bruhat order in the symmetric group

Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
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### A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation

Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
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### Consequences of Littlewood-Richardson rule

I am trying to read Deligne's paper 'Categories Tensorielles', and in the first chapter Deligne states some results obtained from the Littlewood-Richardson rule that I do not understand. He states: '...
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### Schur multiplier of 2-Sylow subgroups of symmetric group

Is the Schur multiplier of 2-Sylow subgroups of symmetric groups on $n\geq 4$ symbols known? I couldn’t find much except that multiplier of $S_n$ is contained in the multiplier of 2-Sylow subgroup.
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### Combinatorics of merging sequences from multinomial coefficients

If you have $m$ sequences $a_{11},\dots,a_{1n_1}$ through $a_{m1},\dots,a_{mn_m}$ each sorted in ascending order (assume there are no duplicates) then there is an unique way to merge them. How many ...
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### Do highly symmetric cones have “small” supporting hyperplanes?

Let $C$ be a full-dimensional cone in $\mathbb{R}^{d}$, defined as the positive span of $c = {n \choose 3} \gg d$ vectors. $C$ is highly symmetric in the following sense: each such vector is labelled ...
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### Smith normal form of conjugacy class actions

This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group. Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the symmetric group $S_n$. Identify a ...
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### Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that $$H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y),$$ where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
### 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 $\tau(S_n)$, ...
### Probability of Words Summing to $1$ in $S_n$ or $PGL_2(n)$
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 abstract variables $x_1, x_2, \ldots,x_k$, i.e. $X$ is a product ...