Young tableau

In mathematics, a Young tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.

Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.

The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10.

In their original application to representations of the symmetric group, Young tableaux have n distinct entries, arbitrarily assigned to boxes of the diagram.

This definition incorporates the partitions λ and μ in the data comprising the skew tableau.

Young tableaux have numerous applications in combinatorics, representation theory, and algebraic geometry.

Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions.

Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson–Schensted–Knuth correspondence.

Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique).

The standard monomial basis in a finite-dimensional irreducible representation of the general linear group GLn are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ..., n}.

This has important consequences for invariant theory, starting from the work of Hodge on the homogeneous coordinate ring of the Grassmannian and further explored by Gian-Carlo Rota with collaborators, de Concini and Procesi, and Eisenbud.

Applications to algebraic geometry center around Schubert calculus on Grassmannians and flag varieties.

Certain important cohomology classes can be represented by Schubert polynomials and described in terms of Young tableaux.

Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers.

Young tableaux are involved in the use of the symmetric group in quantum chemistry studies of atoms, molecules and solids.

[6][7] Young diagrams also parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SUn (again when they have at most n − 1 nonempty rows).