For questions on the Young tableau, a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties.
Questions tagged [young-tableaux]
201 questions
25
votes
1 answer
Involutions, RSK and Young Tableaux
Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes $\mbox{sh}(P)=\mbox{sh}(Q)=\lambda$, where…
Alex R.
- 33,289
15
votes
2 answers
Young diagram for exterior powers of standard representation of $S_{n}$
I'm trying to solve ex. 4.6 in Fulton and Harris' book "Representation Theory". It asks about the Young diagram associated to the standard representation of $S_{n}$ and of its exterior powers. The one of the standard representation $V$ is the…
Stefano
- 4,544
12
votes
3 answers
Show via Young diagrams that the standard representation of $S_d$ corresponds to the partition $d=(d-1)+1$
I'm working through Fulton-Harris and I'm kind of "stuck" at the following question. I'm looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is:
"Show that for general $d$, the standard…
Shaf_math
- 275
10
votes
1 answer
Theorem 1 chapter 8 of Fulton's Young Tableaux
I am reading Theorem 1 on page 110 of Fulton's Young Tableaux and have several questions on it. Let $E$ be a free module on $e_1,\ldots,e_m$ (for our purposes $E$ being a finite dimensional complex vector space will do) and consider the module…
user38268
10
votes
1 answer
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 this caveat of projective representations enter…
DJBunk
- 221
10
votes
0 answers
Inequality for hook numbers in Young diagrams
Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $q_{ij} = i + j -1$. Let
$$H(\lambda) =…
Igor Pak
- 1,376
10
votes
3 answers
Direct proof of Gelfand-Zetlin identity
Denote by $D(a_1,\dots,a_n)$ the product $\prod_{j>i}(a_j-a_i)$. Assuming that $a_i$ are integers s.t. $a_1\le a_2\le\dots\le a_n$, prove that $D(a_1,...,a_n)/D(1,...,n)$ is the number of Gelfand-Zetlin triangles (that is, triangles consisting of…
Grigory M
- 18,082
9
votes
5 answers
What are the end and coend of Hom in Set?
A functor $F$ of the form $C^{op} \times C \to D$ may have an end $\int_c F(c, c)$ or a coend $\int^c F(c, c)$, as described for example in nLab or Categories for Programmers. I'm trying to get an intuition for this using concrete examples, and the…
Hew Wolff
- 4,550
9
votes
1 answer
Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the symmetric group on $d$ letters:
$$
P = P_\lambda…
Alex Ortiz
- 26,211
8
votes
1 answer
Historical reference request: Young tableaux
I am writing up an article on the RSK correspondence. To this end, I want to understand the history behind the invention of the Young tableaux and how it was introduced into the study of the symmetry group by Frobenius.
Could someone point me…
historybuff
- 81
8
votes
1 answer
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 $\mu$ is obtained by adding a single box to…
gone
- 763
8
votes
0 answers
Young Tableaux as Matrices
These questions are motivated only by curiosity.
Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any physical meaning or importance to the eigenvalues of…
Alex R.
- 33,289
8
votes
2 answers
the number of Young tableaux in general
From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ …
Qiang Li
- 4,307
7
votes
1 answer
How to apply the Schur-Weyl duality to a three-qubit system?
I am interested in applying Schur-Weyl duality to quantum information theory, specifically "qubits". But I have been stuck for some time on understanding how the Young symmetrizers work in this situation.
Let $V$ be a 2 dimensional complex vector…
Simon Burton
- 263
7
votes
2 answers
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 paclet, I compare its output with various examples…
JEM_Mosig
- 183