Editing Young tableau (section)

== Overview of applications ==

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 polynomial|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'').

In representation theory, standard Young tableaux of size {{mvar|''k''}} describe bases in irreducible representations of the [[symmetric group]] on {{mvar|''k''}} letters. The [[standard monomial basis]] in a finite-dimensional [[irreducible representation]] of the [[general linear group]] {{math|''GL''<sub>''n''</sub>}} are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ..., {{mvar|''n''}}}. This has important consequences for [[invariant theory]], starting from the work of [[W. V. D. Hodge|Hodge]] on the [[homogeneous coordinate ring]] of the [[Grassmannian]] and further explored by [[Gian-Carlo Rota]] with collaborators, [[Corrado de Concini|de Concini]] and [[Claudio Procesi|Procesi]], and [[David Eisenbud|Eisenbud]]. The [[Littlewood–Richardson rule]] describing (among other things) the decomposition of [[tensor product]]s of irreducible representations of {{math|''GL''<sub>''n''</sub>}} into irreducible components is formulated in terms of certain skew semistandard tableaux.

Applications to algebraic geometry center around [[Schubert calculus]] on Grassmannians and [[flag varieties]]. Certain important [[cohomology class]]es can be represented by [[Schubert polynomial]]s and described in terms of Young tableaux.