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

• The number of semistandard tableaux of a partition $\lambda$ with all entries in $[r]$ is given by the well known hook-content formula $$\big|\mathcal{SST}_\lambda\big| = \prod_{(i,j)}\frac{r+i-j}{hook(i,j)},$$ where the product ranges over all boxes $(i,j)$ in the Young diagram of $\lambda$, and where $hook(i,j)$ is the hook length of the box$(i,j)$ in $\lambda$.

• Another similar formula is given by
• $$\prod_{1\leq i < j \leq r} \frac{\lambda_j-\lambda_i + j-i}{j-i}$$ where we implicity pad $\lambda$ with 0 if the index exceeds the number of parts of $\lambda$.

# 4. Remarks

• Semistandard tableaux can be used to generate a Schur Polynomial with $(x_n)^k$ where $n$ is the number and $k$ is the number of times $n$ appears in the SST

• Ex:
• $\begin{matrix}1 & 1 & 2 & 6\\2 & 5 \\3\\\end{matrix}$ = $(x_1)^2(x_2)^2(x_3)^1(x_4)^0(x_5)^1(x_6)^1$ = $(x_1)^2(x_2)^2(x_3)(x_5)(x_6)$

• Skew Tableaux come from taking the set difference of two partitions of different sizes. If the resulting boxes are filled with numbers such that the columns are increasing and the rows are weakly increasing, then it is called a semistandard skew tableaux. If the columns are increasing and the rows are increasing, it is called a standard skew tableaux.

• Ex:
• $\lambda_1 = (6,3,2,2)$ and $\lambda_2 = (4,1,1)$. Then $\lambda_1$ / $\lambda_2$ =

$\begin{matrix}* & * & * & * & 2 & 3 \\* & 1 & 5 \\* & 3\\6 & 6\\\end{matrix}$ where the * are the removed boxes

• The number of semistandard tableaux of shape $\lambda$ and content $\mu$ is equal to the Kostka number $K_{\lambda\mu}$. The hook length formula is a special case when $\mu = (1,1,1,...,1)$, since that corresponds to a standard tableau. However, there is no general formula for the Kostka numbers.

# 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: