Young tableau - Wikipedia

Abstraction: Combinatorial diagrams encoding symmetric and linear group representations

Key points:

• Introduced by Alfred Young in 1900; a Young diagram encodes a partition λ of integer n as left-justified rows of boxes
• Standard Young tableaux (entries strictly increasing in rows and columns) describe bases in irreducible representations of the symmetric group S_n
• Hook length formula gives dimension of irreducible representation: n! divided by the product of all hook lengths
• Semistandard Young tableaux parametrize irreducible polynomial representations of GL_n; the number of such tableaux determines dimension
• Littlewood-Richardson rule (via skew semistandard tableaux) governs decomposition of tensor products of GL_n irreducibles
• Applications span algebraic geometry (Schubert calculus on Grassmannians), invariant theory, and quantum chemistry

Connections: Representation Theory · Combinatorics · Group Theory

Source: http://en.wikipedia.org/wiki/Young_tableau