All Questions
Tagged with integer-partitions symmetric-groups
21
questions
15
votes
2
answers
373
views
A question on partitions of n
Let $P$ be the set of partitions of $n$. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : f_\...
- 228
9
votes
1
answer
742
views
Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
- 25.2k
8
votes
1
answer
742
views
Characters of the symmetric group corresponding to partitions into two parts
Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character $\...
user38451
6
votes
1
answer
196
views
Flattening Young Tableaux
Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape $\...
- 32.9k
6
votes
0
answers
445
views
Invariant element in the tensor product of rectangular Specht modules?
Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
- 14.9k
5
votes
1
answer
577
views
Why do partitions correspond to irreps in $S_n$?
As stated for example in these notes (Link to pdf), top of page 8, irreps of the symmetric group $S_n$ correspond to partitions of $n$. This is justified with the following statement:
Irreps of $S_n$ ...
- 7,095
5
votes
0
answers
284
views
On Applications of the Murnaghan-Nakayama rule
The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup.
Let $Y$ be a standard ...
- 32.9k
3
votes
1
answer
111
views
Symmetry of Plancherel measure (for $S_n$)
For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$:
$$
\begin{pmatrix}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 ...
- 2,052
3
votes
0
answers
73
views
What are some invariants/properties of the conjugacy classes of the symmetric group $S_n$? [closed]
I've recently realized that there is an isomorphism between the integer partitions of $n$ and the symmetric group $S_n$. There are many documented properties of integer partitions of $n$ but I can't ...
- 131
2
votes
1
answer
87
views
A question on an identity involving partition
Let $n$ be a natural number. Let $\lambda \mapsto n$ , be a partition of n. So $\lambda=(\lambda_1, \lambda_2, \ldots ,\lambda_k)$, with $\lambda_1\leq \ldots \leq \lambda_k$, and $\sum_{i=1}^k \...
- 4,135
1
vote
0
answers
275
views
Decompose the permutation module $M^{(2, 2)}$ into irreducible representations.
My current approach is to take some elements of $M^{(2, 2)}$ and examine the submodules generated by them, in the hopes of finding a basis for them. Each submodule will correspond to an irreducible ...
- 353
1
vote
0
answers
71
views
Why is the ladder tableau of an $e$-restricted partition $e$-restricted?
Lemma 3.40 on page 46 in Mathas's "Iwahori-Hecke Algebras and the Symmetric Group" states
Suppose that $\lambda$ is an $e$-restricted partition of $n$. Then the ladder tableau $\mathfrak{l}_e^\...
- 997
1
vote
1
answer
125
views
How many solutions of equation
How many solutions of equation
$x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$?
I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way :
...
- 153
0
votes
2
answers
77
views
Partitioning evens as sum of evens
Take the set $\{a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8\}$.
We can partition according to rules.
Every member in the partition has even number of elements.
Every member in partition have to be consecutive.
...
- 123
0
votes
1
answer
29
views
Summation indices suspiciously don’t align
In Diaconis’ book Group representations in Probability and Statistics, freely online there is the following formula on p. 40 labeled (D-2):
Let $\tau$ be any transposition in $S_n$, $\lambda$ any ...
- 1,092