All Questions
Tagged with young-tableaux abstract-algebra
15
questions
3
votes
0
answers
86
views
Schur functors = Weyl functors in characteristic zero?
In the paper `Schur functors and Schur complexes' by Akin et al., the notion of a Schur functor had been defined for the first time over an arbitrary commutative ring $R$.
To recall the definition (I ...
- 642
1
vote
0
answers
29
views
Embed U(5) to U(16) by specifying the 16-dimensional complex representation
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
My question concerns all the possible ways to embed SU(5) to U(16) by specifying the 16-dimensional complex ...
- 5,969
1
vote
0
answers
56
views
Young tableaux - column group
I am studying young tableaux and at one point in a demonstration the author states that
$$C_{\pi t} = \pi C_{t}\pi^{-1}$$
where $C_{t}$ is a subgroup of $S_{n}$ consisting of permutations which only ...
- 739
2
votes
1
answer
328
views
Hook-length formula (equivalent)
I would like to know if anyone knows where I can find a proof
for the equivalent hook-length formula
$$f^{\lambda}=\frac{n! \cdot \prod_{i<j}(l_i-l_j)}{l_1! \cdot l_2! \cdot ... \cdot l_k!}$$
...
- 1,438
2
votes
1
answer
208
views
Irreducible representations of symmetric group
Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes.
That bijection is proved showing that the ...
- 1,446
1
vote
1
answer
96
views
$\mu$, $\nu$ are compositions with the same parts then for any $\lambda$, $K_{\lambda\mu}=K_{\lambda\nu}$ ($K$ Kostka number)
I want to show the following.
If $\mu, \nu$ are compositions with the same parts (only rearranged) then for any $\lambda$ we have that $K_{\lambda\mu}=K_{\lambda\nu}$.
I know that the Kostka ...
- 379
0
votes
0
answers
93
views
Combinatorial identity
Let $a_1,\cdots,a_n$ be n positive consecutive integers. So I want to know if there exists a close combinatorial form for the coefficient of $x^k$ in
$$(x+a_1)(x+a_2)\ldots (x+a_n) .$$
In ...
- 1,065
2
votes
1
answer
384
views
Representation from Young tabloids
I am following the note Young Tableaux and the Representations
of the Symmetric Group to work out a representation from a Young tableau for $S_n$.
Here $\lambda$ is a partition of an integer $n$. In ...
- 559
2
votes
1
answer
93
views
Confusing partitions of $S_5$ in two different sources
I am trying to understand the partitions of $S_5$ created by it's conjugacy classes but two sources have two different partitions.
Source 1:
Source 2:
So, for example, in the first table, the ...
- 559
3
votes
2
answers
2k
views
Young diagram for $S_5$
I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$.
The order of the group is $120$.
The number of conjugacy classes and so partitions is $7$.
...
- 559
1
vote
0
answers
204
views
Product of standard and sign representation of $S_5$
I am able to work out the sign representation of $S_5$ and standard representation of $S_5$.
How do I compute the product of standard and sign representation of $S_5$?
What kind of product do I need ...
- 559
0
votes
2
answers
913
views
Standard representation of $S_5$
I am trying to determine the standard representation of $S_5$. I understand that it will be a map from group elements to $\mathbb{C}^4$. The character table is as follows.
I understand that the ...
- 559
4
votes
2
answers
3k
views
Young tableaux of $8\otimes 8$ in $SU(3)$
In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$:
I am unsure of all the cancellations. Let us number the canceled tableaus increasing ...
- 3,556
8
votes
1
answer
482
views
A Question on the Young Lattice and Young Tableaux
Let:
$\lambda \vdash n$ be a partition of $n$
$f^\lambda$ - number of standard Young Tableaux of shape $\lambda$
$\succ$ - be the covering in the Young Lattice (that is, $\mu \succ \lambda$ iff $\mu$ ...
- 733
3
votes
1
answer
329
views
Standard Young Tableaux and Bijection to saturated chains in Young Lattice
I'm reading Sagan's book The Symmetric Group and am quite confused.
I was under the assumption that any tableau with entries weakly increasing along a row and strictly increasing down a column would ...
- 733