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.
199
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Pieri's formula over general rings
Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Pieri's formula gives an isomorphism
$S^{\lambda}V \otimes \Lambda^{m}V \cong \bigoplus_{\lambda(m)}S^{\lambda(m)}V$,
where $S^{\lambda}...
- 702
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0
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14
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Key Polynomials expansion in terms of Non-symmetric Hall-Littlewood functions
I know from Symmetric Functions and Hall Polynomials on Macdonald's book, that the following holds:
\begin{equation}
s_{\lambda} (x) = \sum_{T \in A} t^{charge(T)} HS_{cont(T)} (x;t)
\end{equation}
...
- 571
1
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0
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49
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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 $N=S_{r_1}wr S_{m_1} \times ...\times S_{r_k}wr S_{m_k}$....
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1
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Exercise 8.6 of Algebraic Combinatorics by Stanley
Problem 6 in Chapter 8 of Algebraic Combinatorics by Stanley: Show that the total number of standard Young tableaux (SYT) with $n$ entries and at most two rows is ${n \choose \lfloor n/2 \rfloor}$. ...
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12
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Conjugate of a Gel'fand pattern
Background:
A Gel'fand pattern is a set of numbers
$$
\left[\begin{array}{}
\lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
- 300
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48
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How to compute the character by "removing the hooks"?
I am reading a paper by McKay in 1971, the name of the paper is Irreducible Representations of Odd Degree. There is a theorem said: ${m}_{2}({S}_{n})=2^r$, where $n=\sum 2^{k_i}$, ${k}_{1}>{k}_{2}&...
- 61
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88
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Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients
I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
- 4,936
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1
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75
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An example of application of the Littlewood–Richardson rule [closed]
I am computing the Littlewood–Richardson coefficients (https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule) of the product $s_{[2,2]}s_{[1,1]}$ both by hand and a software tool (https://...
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0
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I want to count the multiplicity of specific peak sets occurring in a standard shifted tableau with some restrictions. Possibly using path counting?
Ok first some definitions:
Let a shifted diagram of some strict partition $\lambda$ be a Young tableau whose $i^{th}$ row is shifted $i-1$ spaces to the right, (I use french notation, and start ...
- 31
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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 ...
- 702
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Dimension of irreducible totally-symmetric tensor product of two su(n) representations
Consider the irreducible representation of $\mathfrak{su}$(n) given by the Young tableu
The dimension of the representation is easily obtained by the usual rules, and it is $d=n(n-1)/2$. One can also ...
- 133
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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$...
- 31
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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 ...
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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
$$\...
- 9,982
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0
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Idempotency of the Young symmetrizer
Let $t$ be a (not necessarily standard) tableau of shape $\lambda \vdash n$ consisting of the numbers $1, ..., n$, each used exactly once.
Notation:
(i) $\mathfrak{S}_{n}$ is the symmetric group on $n$...
- 245
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1
answer
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Permuting the rows in ascending order first and then the columns of any Young tableau gives a standard Young tableau
Show that if you take any Young tableau and permute the rows in ascending order first and then the columns in ascending order (or columns first and then row), then you get a standard Young tableau.
I ...
- 9,982
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Weighted sum over integer partitions involving hook lengths
I am trying to compute the following quantity:
$$ g_n(x) = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \exp\left[x\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right]...
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Constructing Young tableaux basic question
I'm trying to understand Young tableaux and was making some exercises.
I'm a bit confused with the following question
Given a tensor $B^{ijk}$ where $B^{ijk} = -B^{jik}$ and $B^{ijk} + B^{kij} + B^{...
- 806
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Bijection between noncrossing sets of arcs and row-strict Young tableaux
There is a well-known bijection between (i) noncrossing partitions of the set {1,...,2n} where all blocks have size 2, and (ii) standard Young tableaux with n rows and 2 columns. There exist a Catalan ...
- 67
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How many Young tableaux of size 6 are there? [duplicate]
How many Young tableaux of size 6 are there? I have come up with the number 76, by counting the number of every possible Young tableau of weight 6, namely
{6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {...
- 502
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0
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55
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Number of semi-standard Young tableaux of shape $\lambda$ with some entries fixed
Given a partition $\lambda$, the number of semi-standard Young tableaux (SSYT) of shape $\lambda$ with maximum entry $n$ is given by
\begin{equation}
\prod_{1\leq i<j\leq n} \frac{\lambda_i-\...
- 11
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vote
0
answers
36
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Linear involution for Specht modules
Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$.
Here the Specht modules are ...
- 3,082
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0
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127
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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
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Littlewood-Richardson coefficients and conjugation of Young diagrams
I am currently reading William Fulton's Young Tableaux and struggling to understand the proof of Corollary 2 in Section 5 of the book.
Suppose that $\lambda$ and $\mu$ are Young diagrams (or ...
- 4,322
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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$, ...
- 479
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1
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Schur functors for $\mathfrak{S}_3$
I have been trying to calculate the explicit images of the Schur functors for the action of $\mathfrak{S}_3$ on $V^{\otimes 3}$ where $V$ is some vector space, for the sake of concreteness of ...
- 311
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0
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Compute Schutzenberger involution of a Young tableau without using Jeu de Taquin
How does one compute Schutezenberger involution $T'$ of a Young tableau $T$ without using Jeu de Taquin.
Can we use Viennot's construction or some other technique and apply it on the contents of $T$ ...
- 99
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Question on definition of Schur polynomial from Fulton's Young Tableaux.
In the following from Young Tableaux by Fulton, what happens if our Tableaux has numbers greater than $m$? Fulton gives an example with $m=6$, but according to his definition, we also have a monomial ...
- 12.2k
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Given n, do we have a formula for the greatest Hook number of an n-box Young diagram?
Obviously the least Hook number is 1, by considering boxes stacked vertically or horizontally. Is there a formula for the greatest possible Hook number ?
EDIT: The Hook number of a Young diagram is ...
- 443
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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 ...
- 4,217
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1
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Are idempotents in the group algebra of $S_n$ equivalent to Specht modules?
I am studying the irreps of Sn. I will use the following example tableau:
$$|1|2|\\
|3|4|
$$
From what I understand, one approach to constructing the irreducible representations is as follows:
For ...
- 443
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0
answers
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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 ...
- 172
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About Young symmetrizer $c_{\lambda}$
I'm reading the Fulton and Harris's book "Representation Theory". I want to ask about the proof of lemma 4.25.
Let $c_{\lambda}$ be the young symmetrizer, and let $V_{\lambda} = {\mathbb C}...
- 51
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0
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Young-Tableau Exercise Solution Help
i have the following definition of a young tableau:
A Young tableau is an m × n matrix ($t_{i,j}$) with entries from N ∪ {∞}, for which it holds that in each row and each column the values ascend from ...
- 41
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0
answers
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Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
- 135
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Sum involving ${\frak{S}}_n$-character values and Kostka numbers
Let $\lambda$, $\mu$, and $\rho$ be partitions of $n$ and let
$\chi^\lambda_\rho$ and $K_{\lambda \mu}$ denote the associated ${\frak{S}}_n$-character value and Kostka number respectively.
Question: ...
- 577
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0
answers
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A question about Fomin's local rules for growth diagrams
Let $w\in S_n$. Define the growth diagram of $w$ as follows: Start by an array of $n\times n$ squares, with an $X$ in the i'th column and row $w(i)$ from bottom. Then we obtain $(n+1)^2$ vertices (the ...
- 3,082
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1
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Parity of hooklengths in a partition diagram
Main Question
Let $\lambda\vdash n$ be a partition, with hooklengths $\{h_1,\dots,h_n\}$ in its partition diagram. Is there a formula for determining
$$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$
...
- 405
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0
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
- 549
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0
answers
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Character of the irreducible representation $ψ^λ$ : $S_4$ → $Aut_C(S^λ)$
I am struggling with these exercise from group representations and would really appreciate some steps to take or sources with similar exercises.
The task is to compute the character of the irreducible ...
- 2,550
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Promotion on semistandard Young tableaux.
I searched on google and found that algorithms describe promotion operator on the set of standard Young tableaux. For example, the article. But I didn't find algorithms describe promotion operator on ...
- 14.7k
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Computing permutation character associated to a Young subgroup.
If $\lambda = (\lambda_1,\lambda_2,\ldots)$ is a partition of $n$, then there is a permutation character of $S_n$ associated to the Young subgroup $S_\lambda$:
$$
\pi_\lambda = \mathrm{Ind}_{S_\lambda}...
- 712
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0
answers
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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
1
vote
1
answer
182
views
Is a Standard Tableau determined by its descent set?
Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers ...
- 405
1
vote
1
answer
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A question on Young tableau.
I am reading Fulton's book representation theory. My question occurs in the proof of Lemma 4.23. I will introduce my question concisely without letting you read that book.
The book introduces an order:...
- 1,468
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0
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How to evaluate $s_\lambda(q,q^2,\cdots,q^m)$? (principal specialisation of the schur function)
It is required to show that
$$
s_\lambda(q,q^2,\cdots,q^m) = q^{m(\lambda)}\prod_{i,j \in \lambda}\frac{1-q^{c_{i,j}+m}}{1-q^{h_{i,j}}}
$$
where $c_{i,j}=j-i$ is the content of cell $(i,j)$, and $h_{i,...
user614535
0
votes
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Partition of integer and its conjugate
For the partition $(6,4,4,2)$ of integer $16$, if we draw its Young diagram with four rows of boxes, one below the other, of size $6$, $4$, $4$, and $2$, then flipping the resulting Young diagram ...
- 3,137
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49
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Combinatorics and Catalan Numbers [duplicate]
I was asked to investigate this question and to present my findings and I would like some sense of help and direction, I am very lost:-(
2n people, all of different heights
How many ways are there ...
1
vote
0
answers
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 ...
- 5,989
1
vote
0
answers
61
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Proof of conjecture about orthogonalized Specht polynomials.
Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a $r$-part partition of integer $N$ ($\lambda\vdash N$), i.e.,
$$\sum_{i=1}^r{\lambda_i}=N,$$
such that $\lambda_i\leq\lambda_j$ for $r\geq j>...
- 165