Symmetric function

A symmetric function is a formal power series of bounded total degree that is invariant under every finite permutation of its variables. Symmetric functions form a graded ring with several useful bases.

The complete homogeneous symmetric polynomial $h_n$ is the sum of all monomials of total degree $n$. Its generating function is

The power-sum symmetric polynomial is $p_n={\sum }_i{x}_i^n$. For an integer partition $\mu$, write $p_{\mu }={\prod }_i{p}_{{\mu }_i}$.

The Schur polynomials form a basis of the ring of symmetric functions indexed by integer partitions.

The Jacobi–Trudi identity expresses a Schur polynomial asIn particular, $s_{(a,b)}=h_a{h}_b-h_{a+1}{h}_{b-1}$.

The Frobenius characteristic map sends class functions on $S_n$ to homogeneous symmetric functions of degree $n$. It satisfies