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.
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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 ...
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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 $...
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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 ...
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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 ...
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Can essentially primitive idempotents be defined both as $e_\lambda=s_\lambda a_\lambda$ and $e_\lambda=a_\lambda s_\lambda$?
I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
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Non-Intersecting up-right lattice paths and standard Young Tableaux
Consider the Lattice $\mathbb{Z}^2$ and an initial set of points with coordinates $(0,u_1)$, $(0,u_2)$, $\cdots$ $(0,u_n)$, final set of points $(m,v_1),(m,v_2),\cdots,(m,v_n)$, where $v_i,u_i$ are ...
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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 ...
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Inequality regarding kostka numbers in representation theory
Before I post my question, let me set up some notation.
Notation. For $k\geq 1$, let $\lambda \vdash k$ be a partition of $[k]$. Let $C(k,m)$ be the set of all partitions $\lambda \vdash k$ of size $m$...
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Schur functors applied to irreducible representations of $S_n$
For a $d$-box Young diagram $\lambda$, the Schur functor is a functor $S_\lambda: \text{Vect}\rightarrow \text{Vect}$. If $\lambda = d$ then $S_\lambda V=S^d V$ the $d$-th symmetric power of $V$, ...
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Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?
I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
user279515
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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 ...
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Reading off tensor index symmetries from a Young Tableau
I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form
$[\mu][\nu]$
$[\rho][\sigma]$
I understand that if ...
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Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$
What I am trying to do?
Work out the irreducible representation of the group element $(1,3,5) \in S_5$ for the partition $2+2+1$ .
Motivation:
Learn how to calculate irreducible representation from ...
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How do we know if a Young Tableau represents $3$ or $\bar{3}$?
Dimension three in the title was just an example...In general I want to know for any dimension $d$ and any $\mathsf{SU}(n)$.
I am aware of the rule for tensors; if there are more lower indices than ...
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On Schur map and tableaux
My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37.
Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider ...
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