All Questions
30 questions
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
15
votes
2
answers
1k
views
Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
- 10.7k
15
votes
2
answers
762
views
Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\...
- 3,513
10
votes
7
answers
2k
views
Representations of products of symmetric groups
I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...
- 22.3k
10
votes
1
answer
2k
views
Permutation character of the symmetric group on subsets of certain size
The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...
- 238
9
votes
2
answers
2k
views
alternating and symmetric powers of the standard representation of the symmetric group
Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
- 7,300
9
votes
0
answers
254
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
- 2,306
8
votes
1
answer
1k
views
Irreducible decomposition of tensor product of irreducible $S_n$ representations
Are there well known results on the irreducibles in the decomposition of tensor products of irreducible $S_n$ representations? I would also like to know of some references where I can find formulas (...
- 596
8
votes
1
answer
455
views
Is there any good survey on the hook length formula and related topics?
I am recently doing some research related to the hook length formula.
The hook formula counts the number of Young tableaux of certain type.
I find there are plenty of research already been done and ...
- 393
8
votes
1
answer
203
views
Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\...
- 809
8
votes
1
answer
400
views
Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module
Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
- 4,902
7
votes
2
answers
376
views
Basis parametrized by the symmetric group elements for the coinvariant algebra
Let $A_n$ be the coinvariant algebra of the symmetric group $S_n$.
This algebra has vector space dimension $n!$.
$A_n$ is the quotient algebra of the polynomial ring $K[x_1,...,x_n]$ by the elementary ...
- 26.5k
7
votes
1
answer
2k
views
Tensor products of permutation representations of symmetric groups.
I am looking for a reference for the following fact which must be classical (which makes it harder, for me, to track a reference down). I am interested because there are similar (more complicated) ...
- 4,990
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
7
votes
0
answers
543
views
Representation theory of symmetric group for dummies
I have to grade a master project on representations of symmetric groups (char $0$) third time in my life and finally I came to a conclusion that I have to get a grasp of the matter. I am aware of ...
- 14.3k
5
votes
1
answer
299
views
Sum of skew characters over hooks and "odd" partitions
Let us call a partition odd if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$.
Let $H(n)$ denote the set of all hook ...
- 2,552
5
votes
1
answer
198
views
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, (...
- 15.8k
4
votes
1
answer
499
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 ...
- 1,013
4
votes
1
answer
392
views
Expression of basis vectors of permutation modules in different bases.
This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...
- 4,902
4
votes
0
answers
128
views
Filtrations of the irreducible representations of the symmetric groups
For a partition $\lambda$ of $n$, write $S^{\lambda}$ for the irreducible $S_n$-representation that corresponds to $\lambda$ (a.k.a. the Specht module).
For two integers $d<n$ write $Par_d(n) = \{\...
- 5,039
3
votes
2
answers
449
views
Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?
I'm interested in the representation theory of symmetric groups.
I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
- 1,013
3
votes
0
answers
311
views
What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
- 571
3
votes
0
answers
115
views
Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius
I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius.
There he mentioned some theorems of Netto.
I'm depending on the Google translator. and the translation ...
- 1,013
2
votes
1
answer
212
views
Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials
The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known.
The zonal spherical functions $\omega_\lambda(g)=\frac{...
- 1,549
2
votes
1
answer
736
views
Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
- 596
2
votes
1
answer
75
views
Reference for the action of the Mullineux involution on a partition with an added good node
Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...
2
votes
0
answers
352
views
On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
- 43.2k
1
vote
1
answer
423
views
$\lambda$-rings and hopf-rings
The direct sum of complex representation rings $R_*\oplus R\Sigma_n$, for $\Sigma_n$ the $n$th symmetric group is also the free $\lambda$-ring on one generator. Here, we take a product obtained from ...
- 822
1
vote
0
answers
213
views
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 ...
- 1,013