All Questions
Tagged with young-tableaux algebraic-combinatorics
12
questions
4
votes
1
answer
57
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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 ${n \choose \lfloor n/2 \rfloor}$. ...
2
votes
0
answers
81
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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^{\...
- 4,886
1
vote
0
answers
49
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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
1
vote
1
answer
87
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RSK and Matrices
It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square ...
- 593
2
votes
2
answers
264
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Proof that the Lascoux-Schützenberger involutions satisfies the braid-relations
I am interested in the Lascoux-Schützenbereger involutions $\theta_i$, defined on semistandard Young tableaux. See this paper for definitions, page 4. These involutions satisfy the braid relations:
(...
- 63
5
votes
2
answers
384
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$\eta$-value of a partition and its meaning
The $\eta$-value of an integer partition $\lambda = \big( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0 \big)$ is defined as
\begin{equation}
\eta \big( \lambda \big) \ := \ \sum_{i=1}^k ...
- 567
9
votes
1
answer
746
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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 ...
- 25.2k
1
vote
0
answers
96
views
Murnaghan-Nakayama rule for general dimension of a hook
Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from
$(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
- 1,065
2
votes
1
answer
1k
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Is there a explicit formula for the number of Semi-standard Young Tableaux over $\{1,\dots,n\}$ for a given partition $\lambda$ and a given type $\mu$
I was given an exercise to give all SSYT over $\{1,\dots,12\}$ of shape $\lambda=(4,4,3,1)$ and type $\mu=(4,2,2,2,2,0,\dots,0)$. Now I was wondering if there is an formula to say something about the ...
- 987
4
votes
2
answers
736
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Reference request: Representation theory over fields of characteristic zero
Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook ...
- 2,584
9
votes
0
answers
183
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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 $...
- 1,366
-1
votes
2
answers
609
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Young tableaux of shape lambda. [closed]
Consider the partition $\lambda=(m,n-m)$ of $n$ (thus $2m \ge n$).
The number of standard Young
tableaux of shape $\lambda$ is given by
$$f_{(m,n-m)} = \binom nm - \binom{n}{m+1}$$
a) Prove this ...
- 1,302