# 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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### 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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139 views

### Generalized Young symmetrizers, why not?

For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$.
We will denote by $G(\...

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### Permutation representation of a finite $p$-group

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...

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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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162 views

### 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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### Character table of the symmetric group $S_n$ according to James

In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...

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### Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...

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### Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753.
Consider the following PDE: ...

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**1**answer

202 views

### Induction step in Bóna and Ehrenborg's proof that the generating function of the alternating runs has -1 as a root of a certain multiplicity

This is a crosspost of a question I asked on Mathematics SE four months ago. Periodically bumping it and placing a bounty on it to attract more attention were to no avail. There are some comments ...

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### Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...

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### action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$.
Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via
$$\pi(e_i \wedge e_j) ...

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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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### Transposition Cayley graphs are planar

Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...

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### Recurrence relation for number of reduced words of longest element in $S_n$

Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)?
Edit: Sorry for unaccepting the answer, but I realized that I really ...

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### Carter Payne homomorphisms and reduced expressions

Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...

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### Expansion of polytabloids in the standard basis

would like to know the most efficient way to write a polytabloid in terms of standard ones.
I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...

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### On bounding a certain discrepancy between probability distributions on the symmetric group

Disclaimer. This is a follow up to a question I asked and answered on SE https://math.stackexchange.com/q/3579311/168758. The question was about upper-bounds. Here I'm interested in lower bounds, and ...

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### Sum of products of irreducible characters of the symmetric group over a subgroup

When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind
$$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \...

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### Combinatorial bijection on monotone sequences

Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions
$$ (1,2,\...

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### Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...

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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 ...

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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 $...

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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: (...

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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 $...

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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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### 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|\...

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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 ...

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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-...

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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 ...

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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 $...

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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\...

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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)...

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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 \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 ...

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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 $\tau(S_n)$, ...

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### 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 ...

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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 ...

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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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### 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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### 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 $v_2$. ...