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.

Filter by
Sorted by
Tagged with
2
votes
0answers
89 views

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 ...
3
votes
0answers
135 views

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 ...
4
votes
1answer
396 views

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 ...
5
votes
0answers
83 views

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 ...
6
votes
0answers
113 views

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 ...
3
votes
1answer
180 views

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 ...
5
votes
1answer
133 views

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 ...
8
votes
0answers
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(\...
3
votes
1answer
84 views

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^{}}}) ...
4
votes
0answers
93 views

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 ...
4
votes
0answers
81 views

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 ...
4
votes
1answer
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 ...
7
votes
0answers
80 views

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 ...
3
votes
0answers
106 views

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: '...
3
votes
0answers
271 views

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) $$\...
3
votes
0answers
47 views

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 ...
4
votes
1answer
127 views

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: ...
8
votes
1answer
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 ...
9
votes
2answers
282 views

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 $x_1,x_2,...,x_n$ ...
3
votes
1answer
130 views

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) ...
0
votes
0answers
50 views

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.
5
votes
2answers
297 views

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)...
2
votes
1answer
214 views

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 ...
2
votes
0answers
33 views

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 ...
2
votes
0answers
37 views

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 ...
0
votes
1answer
79 views

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 ...
3
votes
0answers
171 views

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 \...
1
vote
0answers
171 views

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,\...
1
vote
0answers
52 views

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) \...
7
votes
2answers
283 views

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 ...
7
votes
0answers
172 views

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
0answers
68 views

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
1answer
114 views

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
0answers
107 views

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 ...
10
votes
0answers
168 views

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
0answers
220 views

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
votes
0answers
70 views

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
0answers
99 views

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
votes
2answers
904 views

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 $...
10
votes
2answers
549 views

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
0answers
128 views

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
1answer
84 views

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 ...
9
votes
0answers
97 views

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
votes
1answer
513 views

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
1answer
238 views

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 $\tau(S_n)$, ...
10
votes
0answers
367 views

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 ...
2
votes
1answer
122 views

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
votes
4answers
432 views

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}^...
3
votes
0answers
48 views

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 ...
1
vote
0answers
140 views

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
2 3 4 5
7