Standard Young tableau: fill n boxes of shape λ with 1..n, increasing in rows and columns. Count f^λ = n!/Π_{(i,j)∈λ}h(i,j). For λ=(3,2): 5!/(4·3·1·2·1)=5. The 5 SYT are the basis vectors of the Specht module S^λ.
How to Use
Enter shape λ:
- Count: f^λ via hook formula
- Hooks: h(i,j) display
- List: All SYT (small shapes)
Representation Theory
Irreducible representations of S_n are indexed by partitions λ⊢n. Dimension = f^λ = #SYT of shape λ. The identity Σ(f^λ)² = n! (sum over λ⊢n) reflects that regular representation decomposes into all irreducibles.
RSK Correspondence
Robinson-Schensted-Knuth: bijection between permutations of [n] and pairs (P,Q) of SYT of the same shape λ⊢n. This proves Σ(f^λ)²=n! bijectively. The shape λ encodes permutation statistics.
Step-by-Step Instructions
- 1Enter shape.
- 2Compute hook lengths.
- 3Count f^λ.
- 4List SYT.
- 5Explore RSK.