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.
436
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7
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What makes skew characters of the symmetric group special?
For integer partitions $\mu\subset\lambda$ we can define the skew character $\chi^{\lambda/\mu}$ (for example?) via the Littlewood-Richardson rule.
Many combinatorial gadgets and algorithms extend in ...
8
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0
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Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
2
votes
0
answers
71
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CAS for finite-dimensional complex representations of $S_n$
Does there exist a computer algebra system that can work with finite-dimensional complex representations of the symmetric groups on finitely many letters? It should have the following functionality: (...
2
votes
1
answer
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Minimum local permutation data needed to globally merge locally sorted sequences?
We have $k$ blocks of integer sequences $B_1,\dots,B_k$ where $B_i$ is a sequence $$a_{i,1},\dots,a_{i,n_i}$$ with $a_{i,j}\leq a_{i,j+1}$.
Denote the permutation matrix $M_{\ell,\ell'}$ that merges $...
0
votes
0
answers
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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 ...
11
votes
0
answers
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
5
votes
0
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Narayana numbers as character values?
The Catalan numbers show up as character values of the symmetric group: Let $\lambda = (n,n)$, a partition with two parts. Then $\chi^\lambda(1^{2n}) = \frac{1}{n+1}\binom{2n}{n}$, the $n$:th Catalan ...
2
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0
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Representations of the symmetric group from subgroups
Consider the finite symmetric group on $n$ letters $S_n$ and $\chi$ a representation on a finite dimensional complex vector space $V$. Further assume that you know the dimension of the invariant sub-...
3
votes
0
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Constructing a centrally primitive idempotent in the group algebra of the symmetric group
Consider the group algebra of the symmetric group $ \mathbb{C} S_k$.
Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...
12
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2
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How rare are unholey permutations?
For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$.
Given a permutation $\pi$ of $[n]$, we define the holeyness $D(\pi)$ of $\pi$ as being $...
11
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2
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A criterion for finite abelian group to embed into a symmetric group
Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p_1^{i_1}\mathbb{Z}\times\dots \mathbb{Z}/p_m^{i_m}\mathbb{Z}$, with $m\ge 0$, $p_1,\dots,p_m$ primes (not necessarily distinct) and $i_k\...
3
votes
0
answers
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Is there some sort of formula for $t(S_n)$?
Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$.
Is there some sort of formula for $t(S_n)...
2
votes
1
answer
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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 ...
15
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1
answer
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Characteristic classes of symmetric group $S_4$
For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
6
votes
1
answer
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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 $\tau(S_n)$, ...
11
votes
1
answer
538
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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 ...
2
votes
1
answer
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Enumeration and structure of abelian 2-subgroups of a symmetric group
I am struggling with a group theoretic problem arising in my research. Given a symmetric group $\Sigma_{n}$, let's consider all its abelian 2-subgroups up to conjugation. Is it possible to give a ...
7
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4
answers
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Maximum conjugacy class size in $S_n$ with fixed number of cycles
Context: It is well known that given a permutation in $S_n$ with $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by
$$ \frac{n!}{\prod_{j=1}^...
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0
answers
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Groups that can occur as graph automorphisms of a fixed size graph
From theorem $4$ and corollary $1$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...
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0
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A certain kind of permutations and transport of Bruhat chains under conjugation
Let $(W,S)$ be a finite Coxeter system. Let us consider the following situation:
Let $v_1,v_2,w\in W$ such that $v_1=wv_2w^{-1}$. Let $s_{\beta_r}\ldots s_{\beta_1}$ be a reduced expression of $v_2$. ...
1
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0
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Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups
First of of all I'm trying to find a general interpretation to the following facts below.
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
3
votes
1
answer
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Growth rate of $|{\rm cd}(S_n)|$
The question
What is the order of magnitude for the function $n\mapsto |{\rm cd}(S_n)|$?
The motivation
In my research on character degrees of finite groups, I have in recent years been focusing on ...
3
votes
1
answer
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Covering with Deck group $\mathfrak{S}_3$
I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...
1
vote
0
answers
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Two versions of the Murnaghan-Nakayama rule
I have always see the following Murnaghan-Nakayama rule for a partition $\lambda$ and a permutation $\sigma \in \mathfrak{S}_n$ of cycle structure $(\sigma_1, ..., \sigma_n)$:
$$
\chi_{\lambda}(\sigma)...
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2
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Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?
By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
5
votes
1
answer
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Toggles for non-broken-circuit sets in matroids
Let $M$ be a matroid with ground set $E$. If $t$ is a total order on $E$, and if $S$ is a nonempty subset of $E$, then $\max_t S$ will mean the $t$-largest element of $S$ (that is, the maximum of $S$ ...
2
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0
answers
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How to prove this identity on summations and partitions?
Let $f$ be a symmetric function of $s$ variables. The identity is
$$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\...
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A cancellation property for permutations?
Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$.
QUESTION. Assume $n>2$. Does this cancellation property hold true?
$$\sum_{\...
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0
answers
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A symmetric function that appears in the coefficients of a power expansion
Let's say we have the expression
$$\sum_{k_1=1}^\infty\sum_{k_2=1}^\infty\sum_{k_3=1}^\infty\sum_{k_4=1}^\infty\sum_{k_5=1}^\infty x^{k_1+k_2+k_3+k_4+k_5} f(k_1,k_2+k_3,k_4+k_5)$$
where $f(a,b,c)$ is ...
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0
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Has this result about the number of permutations of a given cycle type (or centralizers) been proved?
I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...
1
vote
0
answers
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Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
2
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0
answers
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Explicit symmetry adapted basis for the symetric square of the standard representation
I already posted a related question here, which is more detailed: https://math.stackexchange.com/posts/2786382/edit
The permutation group $S_n$ has standard representation $S^{(n-1,1)}$ (irreducible)....
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0
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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 ...
1
vote
0
answers
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To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$
Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...
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1
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Schur-Weyl duality and q-symmetric functions
Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
4
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1
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Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $S_d, S_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
3
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Induced Homomorphism on Cohomology of Symmetric Group 3
For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
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votes
1
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Young's natural representation of the symmetric group
The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.
One of the standard ways to ...
10
votes
2
answers
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Closed formulas for the character of the symmetric group
I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:
$$\chi_{n}(\sigma) = 1$$
$$\chi_{11...1}(\sigma) ...
1
vote
1
answer
134
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The number of permutations with a special condition
Suppose we are considering $S_n$. For any permutation, let $h$ be the number of derangement and $N$ be the number of cycles with length no less than 2.
I'm interested in the number of permutations ...
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3
answers
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The number of permutations with specified number of cycles and fixed points
I'm interested in the number of permutations for a specified number of fixed points and cycles.
Suppose we are in $S_n$. For any permutation in $S_n$, let $h$ be the number of changed points (the ...
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3
answers
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Is this sum of cycles invertible in $\mathbb QS_n$?
I am interested the following element of the group algebra $\mathbb{Q}S_n$:
\begin{align}
\phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n)
\end{align}
where $e$ is the identity permutation. My question ...
13
votes
2
answers
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Cycle generating function of permutations with only odd cycles
Let $\mathrm{ODD}(n)$ be the set of permutations in $\mathfrak{S}_n$ whose cycle lengths are all odd. It is known that
$$ \#\mathrm{ODD}(n) = \begin{cases} ((n-1)!!)^2 &\textrm{ if $n$ is even}; \\...
4
votes
0
answers
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Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
6
votes
0
answers
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Combinatorics of $p$-Kazhdan--lusztig polynomials
When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion?
Let's assume that I'm going to try to do this using the theory ...
6
votes
1
answer
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Can $S_n$ be partitioned into subsets containing an involution and satisfying $∀σ≠τ, ∃j$ s.t. $σ(j)≠τ(j),σ^{−1}(j)=τ^{−1}(j)$?
Background
Let $\sigma, \tau \in S_n$. We will say that $\sigma$ and $\tau$ are locally orthogonal and write $\sigma \perp \tau$ if there exists $j \in \{1, 2, \ldots, n\}$ such that $\sigma(j) \neq \...
3
votes
1
answer
247
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Cycle Structure of a Permutation Based on the Binary Representation
This is a question I posted on math.stackexchange.com before but never got an answer. I am cross-posting it here.
Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number ...
8
votes
1
answer
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Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\...
15
votes
1
answer
710
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Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...