All Questions
Tagged with integer-partitions symmetric-groups
9
questions with no upvoted or accepted answers
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
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
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
0
answers
7
views
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
0
votes
0
answers
53
views
Show $S^\lambda \otimes sgn$ is a simple representation of $S^n$.
Let $\lambda \vdash n$. Identify $S^\lambda \otimes sgn$ as a simple representation of $S^n$.
I know that that $S^\lambda$ is the Specht module (over $\mathbb{C}$) with a set of polytabloids as a ...
- 591
0
votes
0
answers
47
views
Which of partitions of 5 correspond exclusively to even permutations?
I am ultimately want to prove that $A_{5}$ is simple and the first step in doing so is to:
$(a)$ Write out all partitions of $5.$ Which of these correspond exclusively to even permutations?
I was able ...
user777833
0
votes
0
answers
88
views
How can we map a partition of $n$ to some permutation of [1,2, ... , n]?
Here is the question I was reading:
Does every partition of n correspond to some permutation of [1,2, ... n]?
And here is a statement in the answer given there that I want to use:
If the partition is $...
user777833