# 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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### 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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### 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 ...

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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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### 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 ...

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### $m-$cycles in $S_n$ modulo an equivalence relation

Let $A$ be the set of all $m-$cycles in $S_n$. Define an equivalence relation $i$ in $A$ by $\sigma_1$ is related to $\sigma_2$ by $i$ if $\sigma_1$ is a power of $\sigma_2$ or viz., then the number ...

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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 ...

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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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### 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 ...

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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) ...

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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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### 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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### 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 ...

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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}; \\...

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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 ...

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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 ...

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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 \...

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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 ...

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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) = |\...

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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 ...

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### Combinatorial model for twisted involutions in $S_n$

Let $(W,S)$ be a Coxeter group and $*:S \to S$ be an automorphism of the Dynkin diagram of $W$ so that $*^2$ is the identity.
This induces a bijection $*:W \to W$ mapping $w = s_1 \dots s_n$ to $w^* = ...

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### Large subgroups of $S_n$ without large symmetric or alternating subgroups

I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form.
$G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their ...

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### working with symmetric groups presented via nonstandard generators

This is follow-up to my earlier question.
Suppose that we have elements $\sigma_1,\ldots,\sigma_k\in S_n$, and that we established that these elements actually generate $S_n$.
Since that previous ...

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### A dimension formula for generalised symmetric powers of the natural module

I need a reference for the following well-known statement - does anyone know one?
Let $\mu$ a partition of $n$ into at most $d$ parts.
We let
$${\rm Sym}^\mu(\Bbbk^d)={\rm Sym}^{\mu_1}(\Bbbk^d) \...

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### Permutation factorizations according to number of generated orbits

Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$.
How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...

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### Homotopy type of the semi-simplicial set of symmetric groups

Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...

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### What are all the transitive extensions of cyclic groups?

"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the combined ...

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### Edge-transitive Cayley graphs of $S_n$

I came across the following question which I haven't seen before:
Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...

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### Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...

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### Is the Normal centralizer problem in P?

Notation
$\le$ is used for the subgroup relation;
$P$ means polynomial time in input size;
$\Omega = \{1,2,3,\cdots,n\}$ is a input domain;
$\mathrm{Sym}(\Omega)$ means the symmetric group on $\...

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### Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...

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### Symmetric functions of eigenvalues

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables).
Let $M_{n\times n}$...

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### Number of permutations in a set in an algorithm

Previously I asked a question about the space usage in an algorithm of mine: Upper bound on the number of permutations in a set during an algorithm. This question does not depend on the previous one, ...

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### Murnaghan-Nakayama rule when all cycles have same size

Let $\lambda \vdash nk$. Let $n^k$ denote the partiton with $k$ parts of size $n$. We can compute $\chi^\lambda(n^k)$ by using the Murnaghan-Nakayama rule, as a signed sum over border-strip tableaux, (...

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### Dimension of irreducible representation associated to a Young tableau

This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be ...

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### Upper bound on the number of permutations in a set during an algorithm

Fix $n\geq 2$ and let $S_n$ be the symmetric group on $n$ letters with identity $e$. We consider elements of $S_n$ to be bijections $[n]\to [n]$ as well as sequences (one line notation). For $1\leq i&...

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### Generating symmetric groups with small cycles

This was asked but never answered at MSE.
Let $S_n$ denote the symmetric group and let $H$ be a subgroup which contains
an $n$-cycle. If $n$ is prime, and if $H$ also contains a 2-cycle, then ...

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### Natural maps between Schur functors: understanding the image

Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...

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### alternating and symmetric powers of the standard representation of the symmetric group

Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...

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### Question on a reduction in Kirillov's paper on positivity of divided difference operators

As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...

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### Characterization of permutations which have at most one successor in the covering relation of the weak Bruhat order

Let $W$ be the symmetric group on $n+1$ letters. Let $\ell$ be the length function on $W$.
As the title says, can we characterize all $v\in W$ such that there exists a $w\in W$ such that for all ...

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### Generating sets of the symmetric group that yield isomorphic Cayley graphs

Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...

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### A question about set of inversion

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a ...

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### Permanent of the symmetric group

Let $A$ be the algebra corresponding to a representation-finite block of a Schur algebra.
See for example 6.1. of https://arxiv.org/pdf/1607.05965.pdf for quiver and relations and some relevance of ...

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### The number of permutations of a given cycle type that fix a string with a given histogram

Let $\lambda$ and $\mu$ be partitions of some integer $n \geq 1$. Let $d$ be the number of parts in $\mu$ and let $\bar{\mu} \in \{1,\dotsc,d\}^n$ denote the string $1^{\mu_1} 2^{\mu_2} \dotsb d^{\...

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### Induced representation of a Young subgroup

This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $n=k+l+r$ where $k\geq l\geq r\geq 0$. Let $G$ be the symmetric ...

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### Classification of transitive subgroups of finite symmetric groups generated by double transpositions

I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...

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### Character values at a cyclic permutation of a symmetric group

Let $S_n$ be the symmetric group of degree $n$ and $\sigma\in S_n$ be
a cyclic permutation of order $p$, where $p$ is a prime and $p>n/2$. Consider
ordinary irreducible characters of $S_n$. Are ...

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### Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux

Question. Can you find a bijective proof of the identity
$$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m}
= \dim \Lambda^p (\mathbb{C}^m \...

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### Reference request: Reduced reflection length in Coxeter groups

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...

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### Provoking involutions further

Let $\mathfrak{S}_n$ denote the permutation group, and $I_0(n)=\sum_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions see A000085 for more interpretations. There is also these numbers $...