Semistandard Young Tableaux

1. Definition

A semistandard (Young) tableau of a partition $\lambda \vdash n $ is map from the cells (also known as boxes) of the Young diagram of $\lambda$, to the natural numbers, such that rows are weakly increasing and the columns are increasing. A semistandard tableau of size $n$ is a semistandard tableau of a partition of size $n$.

All standard (Young) tableaux are also semistandard. The set of semistandard tableaux are in bijection with Gelfand-Tsetlin patterns.

2. Examples

Here is an example from $\mathcal{SST}_7$.

$\begin{matrix}1 & 1 & 2 & 6\\2 & 5 \\3\\\end{matrix}$ which in Sage input form is $[[1,1,2,6], [2,5],[3]]$.

3. Properties

4. Remarks

5. Statistics

We have the following 8 statistics in the database:

The cocharge of a semistandard tableau.
The charge of a semistandard tableau.
The sum of the entries of a semistandard tableau.
The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....
The major index of a semistandard tableau obtained by standardizing.
The trace of a semistandard tableau.
The segment statistic of a semistandard tableau.
The flush statistic of a semistandard tableau.

6. Maps

We have the following 5 maps in the database:

reading word permutation
to Gelfand-Tsetlin pattern
shape
Schuetzenberger involution
catabolism

7. References

8. Sage examples

SemistandardTableaux (last edited 2015-12-18 04:49:46 by ZachariahNeville)