All Questions
Tagged with integer-partitions symmetric-groups
21
questions
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5
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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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1
answer
29
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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 ...
- 1,072
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53
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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
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47
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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
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0
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88
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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
5
votes
1
answer
574
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
3
votes
0
answers
73
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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
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2
answers
77
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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
2
votes
1
answer
87
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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 \...
- 4,125
9
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1
answer
739
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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 ...
- 25.1k
1
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274
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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
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70
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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
0
votes
1
answer
117
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Interpreting the table of classification of the partitions of $n$
I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
- 549
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1
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58
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Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$
I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
- 549
1
vote
1
answer
125
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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