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Questions tagged [young-tableaux]

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.

173 questions
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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.
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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
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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
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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 this caveat of projective representations enter…
DJBunk
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Young diagram for standard representation of $S_d$

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
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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
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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
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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
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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.
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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
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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 symmetric group on $d$ letters: $$ P = P_\lambda…
Alex Ortiz
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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 these rules for the orthogonal group SO(d).…
user264317
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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? There are four families of lie groups: The…
Mozibur Ullah
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Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape $\lambda$, define the "flattened tableaux" by deleting…
Alex R.
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Given a Ferrers diagram, prove that $\det(M)=1$

Let $\lambda$ be a Ferrers diagram corresponding to some integer partition of $k$. We number the rows and the columns, so that the j'th leftmost box in the i'th upmost row is denoted as $(i,j)$. Let $n$ be the largest number, such that the box…
Ido
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