αn), deﬁne aα(x1, . . . , xn) = w∈Sn sgn(w)w(xα). This is skew-symmetric, i.e., w(aα) = sgn(w)aα. In particular, aα = 0 unless all parts of α are distinct. So may as well assume that α1 > α2 > · · · > αn ≥ 0 and thus write α = λ + δ where δ = (n − 1, n − 2, . . . , 1, 0) and λ is a partition. Now aδ = det(xn−j i ) = i<j (xi − xj ) is a Vandermonde determinant. It divides aα = aλ+δ, and we deﬁne the Schur function sλ to be the quotient sλ(x1, . . . , xn) = aλ+δ(x1, . . . , xn) aδ(x1, . . . , xn) ∈ Symn . The function sλ is symmetric. If λ is not a partition, then either sλ = 0 or sλ = ±sµ where µ is a partition. So: Theorem {sλ(x1, . . . , xn) | λ a partition, (λ) ≤ n} is a Z-basis of Symn .