All Questions
Tagged with young-tableaux lie-groups
14
questions
0
votes
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answers
122
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Negative powers of the determinant representation of $U(N)$
Consider the determinant representation of $U(N)$ defined by $\det:U(N)\ni U\mapsto\det U\in U(1)$. If I'm not mistaken, $\det^{n}$ for $n\geq 1$ are all irreducible representations. When classifying ...
- 113
1
vote
0
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29
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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
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
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- 3,697
3
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253
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Complex conjugated representation and Young tableaux
Imagine you have the Young tableu and the Dynkin numbers, $(q_1, q_2, ..., q_r)$, of the Lie algebra of $SU(n)$ which has $r$ simple roots. The way I assign Dynkin numbers is increasing its value from ...
- 549
1
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201
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$SU(N)$ irreps and Young-Diagrams
I have read (source not publicly available) that there is a one-to-one correspondence between young-diagrams with less then $n$ rows and the irreducible representations (irrep) of $SU(n)$. Is this ...
2
votes
1
answer
536
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Adjoint representation of SU(2) and Young Tableaux
Suppose we have the fundamental representation of $SU(N)$ (represented by a box) which acts on the vector space $V$. Then, the irreducible representation of $SU(N)$ can be found using the rules for ...
- 1,715
1
vote
1
answer
830
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Young Tableaux of $SU(N)$: Group or Algebra?
I am slightly confused about the use of Young Tableaux in the context of the Lie group and Lie algebras of $SU(N)$. A Young Tableaux has an associated representation which acts on a tensor e.g. $\psi_{...
1
vote
1
answer
71
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Basis of the representations of the B and C series Lie groups
As is well-known, the representations of $SU(n)$ are labelled by Young diagrams. Moreover, there exists a canonical basis of each representation labelled by all the possible tableaux of the diagram.
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2
votes
1
answer
96
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Tensoring Young Tableaus
As we all know very well, the finite dimensional irreducible representations of the compact Lie groups $SU(N)$ are labelled by Young tableaus. Now when we tensor two irreducible representations we get ...
- 338
5
votes
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answers
344
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Short pedagogical introduction to Young-tableaux and weight diagrams?
I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them. Hopefully one which would contain many detailed and worked out examples of ...
- 1,207
2
votes
1
answer
213
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Do we have $\mathbb{C}[\text{SL}_n] = \bigoplus_{\lambda, \,\text{ht}(\lambda)\leq n} V_{\lambda} $?
The coordinate algebra $$\mathbb{C}[\text{SL}_n]=\mathbb{C}\big[x_{ij}: i, j \in \{1, \ldots, n\}\big]/\big(\det(x_{ij}) - 1\big)$$ is a representation of $\text{SL}_n$: $$(g'.f)(g)=f(g'^T g)\,.$$
Let ...
- 14.6k
4
votes
2
answers
3k
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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
10
votes
1
answer
512
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Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?
Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$.
Is this because spinor representation are projective representations? If so, where does ...
- 221
6
votes
1
answer
2k
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Can Young tableaux determine all the irreducible representations of Lie groups?
Can Young tableaux, or generalisations thereof, determine and parametrise (uniquely) all the irreducible representations of each simple Lie group over the complex numbers, ignoring the 5 exceptions?
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- 6,173