Bases
Each of the following families is indexed by partitions \(\lambda\) of \(n\) and is a basis of the space of symmetric functions of degree \(n\).
| Basis | Notation | Description |
|---|---|---|
| monomial | \(m_\lambda\) | the sum of the distinct monomials \(x^\alpha\) with \(\alpha\) a rearrangement of \(\lambda\) |
| elementary | \(e_\lambda\) | \(e_\lambda=\prod_i e_{\lambda_i}\), where \(e_k\) is the coefficient of \(t^k\) in \(\prod_i(1+x_it)\) |
| complete homogeneous | \(h_\lambda\) | \(h_\lambda=\prod_i h_{\lambda_i}\), where \(h_k\) is the coefficient of \(t^k\) in \(\prod_i\inverseof{(1-x_it)}\) |
| power sum | \(p_\lambda\) | \(p_\lambda=\prod_i p_{\lambda_i}\) with \(p_k = \sum_i x_i^k\); related to \(e\) by Newton’s identities, and to the characters of \(S_n\) |
| Schur | \(s_\lambda\) | the sum over semistandard Young tableaux of shape \(\lambda\); the characters of the polynomial irreducible representations of \(\GL_n\) |
By the fundamental theorem of symmetric polynomials, the ring of symmetric polynomials in \(x_1,\ldots,x_n\) over \(\ZZ\) is \(\ZZ[e_1,\ldots,e_n]\), and \(e_1,\ldots,e_n\) are algebraically independent.
Computations
Change of basis. Newton’s identities relate the power sums and the elementary symmetric functions: \[ p_k - e_1p_{k-1} + \cdots + (-1)^{k-1}e_{k-1}p_1 + (-1)^k k e_k = 0. \]
Expansion of Schur functions. \(s_\lambda\) is the sum over semistandard tableaux of shape \(\lambda\) of the corresponding monomials, and by the Jacobi–Trudi identity \(s_\lambda = \det(h_{\lambda_i - i + j})\).
Products of Schur functions. The coefficients of \(s_\mu s_\nu = \sum_\lambda c^\lambda_{\mu\nu}s_\lambda\) are given by the Littlewood–Richardson rule; they are also the multiplicities in tensor products of polynomial representations of \(\GL_n\) and in induction products of representations of symmetric groups.
The characteristic map
The Frobenius characteristic map is an isometry from the class functions on \(S_n\) to the symmetric functions of degree \(n\), sending the irreducible character \(\chi^\lambda\) to \(s_\lambda\). For partitions \(\mu\) of \(k\) and \(\nu\) of \(n-k\), it sends \(\Ind_{S_k\times S_{n-k}}^{S_n}(\chi^\mu\times\chi^\nu)\) to \(s_\mu s_\nu\), so the multiplicity of \(\chi^\lambda\) in that induced character is the Littlewood–Richardson coefficient \(c^\lambda_{\mu\nu}\). It also gives \[ p_\rho = \sum_\lambda \chi^\lambda(\rho)\, s_\lambda, \] where \(\chi^\lambda(\rho)\) is the value of \(\chi^\lambda\) on permutations of cycle type \(\rho\); the Murnaghan–Nakayama rule computes these values. For \(\rho = (1^n)\), the coefficient \(\chi^\lambda(1) = \dim\chi^\lambda\) of \(s_\lambda\) in \(p_1^n\) is the number of standard Young tableaux of shape \(\lambda\), which the hook length formula evaluates.