All Questions
Tagged with young-tableaux group-theory
23
questions
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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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42
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Tableau which corresponds to alternating square representation
Recall that $S_5$ acts on $\mathbb C^5$ by permuting its coordinates and that $\mathbb C^5$ decomposes as $V\oplus W$ where $W$ is the trivial representation and $V$ has dimension $4$. Show that the ...
- 9,592
1
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1
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140
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The irreducible representation of $S_n$ corresponding to the partition $n = (n − 1) + 1$
Let a finite group $G$ act doubly transitively on a finite set $X$, i.e., given $x,y,z,w\in X$ such that $x\neq y$ and $z\neq w$, there is a $g\in G$ such that $g.x=z$ and $g.y=w$. So, we can write
$$\...
- 9,592
1
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0
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Irreducible representations of $S_5$ and their Young diagrams
Given a Young tableau, we can construct its Young symmetrizer $c_\lambda$. Then, the ideal $\mathbb{C} S_n \cdot c_\lambda$ is an irreducible representation of $S_n$. Exercise 4.5 in Fulton and Harris ...
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44
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Does the product of two Schur functions always have a lattice structure with respect to the dominance order of partitions?
The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in ...
- 59
2
votes
1
answer
80
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Weyl constructions for finite groups
Let $G$ be a finite group.
Is there a complex finite dimensional irreducible representation $V$ such that all irreducible ones are submodules of $V^{\otimes n}$ for some $n \in \mathbb{N}$?
If not, ...
- 1,822
4
votes
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314
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Operators commuting with tensor product representations of SU(2)
I am currently investigating $SU(2)$ symmetric qubit systems. In the course of this work I proved the following theorem:
Let $S_n$ denote the permutation group of $n$ elements. For $\sigma\in S_n$ ...
- 125
1
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1
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1k
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$8 \otimes 8$ in $SU(3)$, dimension of the Young-tableau corresponding to the $\bar{10}$
In Georgi's Lie Algebras in Particle Physics, he calculates the decomposition of $8\otimes 8$ in $SU(3)$, and obtains
$$8\otimes 8 = 27 \oplus 10 \oplus \bar{10} \oplus 8 \oplus 8 \oplus 1,$$
...
- 157
3
votes
1
answer
155
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Explicit construction of a representation of Young diagram/tableaux from fundamental representations
Given $SU(N)$ fundamental representation say $U^i$ in the fundamental $N$ of $SU(N)$, with indices $i=1,2,3,\dots,N$.
We can construct a representation whose Young diagram/tableaux look like
Given
...
- 3,697
1
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0
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56
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Young tableaux - column group
I am studying young tableaux and at one point in a demonstration the author states that
$$C_{\pi t} = \pi C_{t}\pi^{-1}$$
where $C_{t}$ is a subgroup of $S_{n}$ consisting of permutations which only ...
- 739
4
votes
1
answer
281
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GAP: how to obtain the Young Symmetrizer?
Given a partition $\lambda$ of $n$ and a standard Young Tableaux filled with numbers from $1$ to $n$ (e.g. increasing row by row), how does one obtain the corresponding Young Symmetrizer using GAP?
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- 292
1
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1
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672
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Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram
I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$:
First definition
Given $\sigma \in S_n$ ...
2
votes
1
answer
208
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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
4
votes
1
answer
717
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What is the mistake in finding the irreps of $SU(3)$ multiplets $6 \otimes 8$, or $15 \otimes \bar{15}$?
I wrote a Mathematica paclet that can be used to find irreducible representations of $SU(n)$.
Specifically, given a product of $SU(n)$ multiplets, it can compute the corresponding sum.
To test the ...
- 153
6
votes
2
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992
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Littlewood Richardson rules for the orthogonal group SO(d)
I have a question related to tensor products of Young diagrams. More precisely, I know the Littlewood Richardson rules for the general linear group GL(d) and would like to know the restriction of ...
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