All Questions
Tagged with young-tableaux symmetric-groups
60
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Normalizer of a Young subgroup in symmetric group
In $S_n$, a Young subgroup $S_{r_1}^{m_1}\times S_{r_2}^{m_2}\times ... S_{r_k}^{m_k}$ where $m_1\times r_1+...+m_k\times r_k=n$ has normalizer $N=S_{r_1}wr S_{m_1} \times ...\times S_{r_k}wr S_{m_k}$....
- 1,303
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7
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Conjugate of a Gel'fand pattern
Background:
A Gel'fand pattern is a set of numbers
$$
\left[\begin{array}{}
\lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
- 248
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46
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How to compute the character by "removing the hooks"?
I am reading a paper by McKay in 1971, the name of the paper is Irreducible Representations of Odd Degree. There is a theorem said: ${m}_{2}({S}_{n})=2^r$, where $n=\sum 2^{k_i}$, ${k}_{1}>{k}_{2}&...
- 1
2
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81
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Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients
I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
- 4,886
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164
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Idempotency of the Young symmetrizer
Let $t$ be a (not necessarily standard) tableau of shape $\lambda \vdash n$ consisting of the numbers $1, ..., n$, each used exactly once.
Notation:
(i) $\mathfrak{S}_{n}$ is the symmetric group on $n$...
- 245
1
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32
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Linear involution for Specht modules
Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$.
Here the Specht modules are ...
- 3,052
3
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84
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Schur functors applied to irreducible representations of $S_n$
For a $d$-box Young diagram $\lambda$, the Schur functor is a functor $S_\lambda: \text{Vect}\rightarrow \text{Vect}$. If $\lambda = d$ then $S_\lambda V=S^d V$ the $d$-th symmetric power of $V$, ...
- 479
3
votes
1
answer
140
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Schur functors for $\mathfrak{S}_3$
I have been trying to calculate the explicit images of the Schur functors for the action of $\mathfrak{S}_3$ on $V^{\otimes 3}$ where $V$ is some vector space, for the sake of concreteness of ...
- 311
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2
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67
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Given n, do we have a formula for the greatest Hook number of an n-box Young diagram?
Obviously the least Hook number is 1, by considering boxes stacked vertically or horizontally. Is there a formula for the greatest possible Hook number ?
EDIT: The Hook number of a Young diagram is ...
- 443
1
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1
answer
199
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Are idempotents in the group algebra of $S_n$ equivalent to Specht modules?
I am studying the irreps of Sn. I will use the following example tableau:
$$|1|2|\\
|3|4|
$$
From what I understand, one approach to constructing the irreducible representations is as follows:
For ...
- 443
1
vote
0
answers
21
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Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
- 135
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92
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Sum involving ${\frak{S}}_n$-character values and Kostka numbers
Let $\lambda$, $\mu$, and $\rho$ be partitions of $n$ and let
$\chi^\lambda_\rho$ and $K_{\lambda \mu}$ denote the associated ${\frak{S}}_n$-character value and Kostka number respectively.
Question: ...
- 567
2
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0
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106
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A question about Fomin's local rules for growth diagrams
Let $w\in S_n$. Define the growth diagram of $w$ as follows: Start by an array of $n\times n$ squares, with an $X$ in the i'th column and row $w(i)$ from bottom. Then we obtain $(n+1)^2$ vertices (the ...
- 3,052
2
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223
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
- 549
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1
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63
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Computing permutation character associated to a Young subgroup.
If $\lambda = (\lambda_1,\lambda_2,\ldots)$ is a partition of $n$, then there is a permutation character of $S_n$ associated to the Young subgroup $S_\lambda$:
$$
\pi_\lambda = \mathrm{Ind}_{S_\lambda}...
- 712