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 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 k describe bases in irreducible representations of the symmetric group on k letters. 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 Grassmanian and further explored by Gian-Carlo Rota with collaborators, de Concini and Procesi, and Eisenbud. The Littlewood–Richardson rule describing (among other things) the decomposition of tensor products of irreducible representations of GLn into irreducible components is formulated in terms of certain skew semistandard tableaux.
Applications to algebraic geometry center around Schubert calculus on Grassmanians and flag varieties. Certain important cohomology classes can be represented by Schubert polynomials and described in terms of Young tableaux.
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