What is the hook length formula?

The Hook Length Formula is a powerful combinatorial tool used to determine the exact number of standard Young tableaux (SYT) that can be formed from a given Young diagram.

What is the Hook Length Formula?

The Hook Length Formula provides a straightforward method to calculate the number of standard Young tableaux for a Young diagram with n cells. A standard Young tableau is a filling of a Young diagram with the numbers 1 to n such that entries increase from left to right in rows and from top to bottom in columns.

The formula is expressed as:

$$
\text{Number of SYT} = \frac{n!}{\prod_{u \in \lambda} h(u)}
$$

Where:

• n is the total number of cells in the Young diagram.
• n! is the factorial of n.
• λ represents the Young diagram.
• u is a specific cell within the Young diagram λ.
• h(u) is the hook length of the cell u.
• The product ∏ u ∈ [ λ ] h ( u ) is the product of the hook lengths of all cells u in the Young diagram λ.

Understanding Hook Length

The hook length of a cell u in a Young diagram is defined as the number of cells in the "hook" extending from that cell. This includes:

1. The cell u itself.
2. All cells to the right of u in the same row.
3. All cells below u in the same column.

Essentially, for a cell at position (i, j) (row i, column j), its hook length is (number of cells in row i to the right of (i,j)) + (number of cells in column j below (i,j)) + 1.

Example Application

Let's illustrate the Hook Length Formula with a concrete example. Consider a Young diagram of shape (3,2), which has n=5 cells.

The diagram looks like this:

■ ■ ■
■ ■

Now, we calculate the hook length for each cell. We'll denote cells by (row, column):

1. Cell (1,1):

   • Right: (1,2), (1,3) (2 cells)
   • Down: (2,1) (1 cell)
   • Itself: (1 cell)
   • Hook length h(1,1) = 2 + 1 + 1 = 4

2. Cell (1,2):

   • Right: (1,3) (1 cell)
   • Down: (2,2) (1 cell)
   • Itself: (1 cell)
   • Hook length h(1,2) = 1 + 1 + 1 = 3

3. Cell (1,3):

   • Right: None (0 cells)
   • Down: None (0 cells)
   • Itself: (1 cell)
   • Hook length h(1,3) = 0 + 0 + 1 = 1

4. Cell (2,1):

   • Right: (2,2) (1 cell)
   • Down: None (0 cells)
   • Itself: (1 cell)
   • Hook length h(2,1) = 1 + 0 + 1 = 2

5. Cell (2,2):

   • Right: None (0 cells)
   • Down: None (0 cells)
   • Itself: (1 cell)
   • Hook length h(2,2) = 0 + 0 + 1 = 1

The Young diagram with hook lengths filled in:

4	3	1
2	1

The hook lengths for this diagram are 4, 3, 1, 2, and 1.
The product of all hook lengths is: 4 × 3 × 1 × 2 × 1 = 24.

Since n = 5, n! is 5! = 5 × 4 × 3 × 2 × 1 = 120.

Now, applying the Hook Length Formula:

$$
\text{Number of SYT} = \frac{120}{24} = 5
$$

Therefore, there are 5 standard Young tableaux for a Young diagram of shape (3,2). This demonstrates how a product of hook lengths (like 4 ⋅ 3 ⋅ 1 ⋅ 2 ⋅ 1) leads to the number of tableaux (5) when incorporated into the full formula.

For further exploration of Young diagrams and tableaux, you can refer to resources like Wikipedia's article on Young Tableaux.