Symmetric function

Summary

A symmetric function is a formal power series of bounded degree in infinitely many variables $x_1, x_2, \ldots$, see the definition. It does not change under any permutation of the variables. The symmetric functions form the ring $\Lambda$, which collects the symmetric polynomials in all numbers of variables at once. The ring has several bases indexed by integer partitions: monomial, elementary, complete homogeneous and Schur functions. Over $\Q$, the power sum symmetric functions give one more basis. It is a central object of algebraic combinatorics and of the representation theory of symmetric and general linear groups.