Symmetric group representations via Young diagrams and tableaux
Young diagrams are arrangements of boxes into left-justified rows of decreasing length. Each Young diagram corresponds to a partition of n, and (remarkably) the partitions of n are in bijection with both the conjugacy classes and the irreducible representations of the symmetric group Sn.
A standard Young tableau is a filling of a Young diagram with the numbers 1,...,n such that entries increase along each row and down each column. The number of standard tableaux equals the dimension of the corresponding irreducible.
Click on a Young diagram to see its hook lengths. The hook length formulagives the dimension of the corresponding irreducible representation: dim = n! divided by the product of all hook lengths.
Click a Young diagram to see its hook lengths and the hook length formula for computing the dimension. Each partition of n corresponds to an irreducible representation of Sn.
Key insight: The hook length at a box is the number of boxes directly to its right plus the number directly below it, plus 1 (for the box itself). This simple combinatorial quantity determines representation dimensions.
For each partition shape, enumerate all possible standard fillings. Each filling corresponds to a basis vector in the irreducible representation. Hover over a tableau to see the increasing condition highlighted.
Hover over a tableau to highlight it. The count of standard tableaux for a given shape equals the dimension of the corresponding irreducible representation.
Key insight: The partition [n] (one row) has exactly 1 standard tableau and corresponds to the trivial representation. The partition [1,...,1] (one column) also has 1 standard tableau and corresponds to the sign representation.
The partition function p(n) counts the number of ways to write n as a sum of positive integers. It grows rapidly and has deep connections to number theory via Ramanujan's congruences and the Hardy-Ramanujan formula.
The number of partitions p(n) equals the number of conjugacy classes of Sn, which equals the number of irreducible representations.
Key insight: p(n) = number of conjugacy classes of Sn = number of irreducible representations of Sn. This triple equality is one of the most beautiful facts in algebra.