All Questions
Tagged with young-tableaux combinatorics
81
questions
4
votes
1
answer
57
views
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 ${n \choose \lfloor n/2 \rfloor}$. ...
0
votes
1
answer
74
views
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://...
0
votes
0
answers
38
views
Does this partial order on semistandard Young tableaux have a name?
Given a semistandard Young tableau $T$ of shape $\lambda$, let $c_i(T)$ is the number of $i$'s that appear in $T$. We can define a partial order on the set of all semistandard Young tableau of some ...
- 437
1
vote
0
answers
27
views
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
3
votes
0
answers
222
views
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
1
vote
0
answers
164
views
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
1
vote
1
answer
122
views
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 ...
- 9,592
2
votes
0
answers
79
views
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]...
- 484
2
votes
0
answers
59
views
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
1
vote
0
answers
34
views
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 ...
- 4,232
0
votes
0
answers
171
views
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
0
votes
0
answers
148
views
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 ...
- 12.1k
3
votes
2
answers
67
views
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
1
vote
0
answers
21
views
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
1
vote
0
answers
92
views
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: ...
- 567