Power sums

See [Mac95].

Power sums is a natural basis on the space of symmetric polynomials. They are constructed using elemetary symmetric sums

over which product is taken for the Young diagram

For example, for and Young diagrams of size

Sym = SymmetricFunctions(QQ)
p = Sym.power()
diagrams = [
    lam
    for n in range(1, 3)
    for lam in Partitions(n).list()
]
arr = [[lam, p[lam].expand(3, alphabet=['x1','x2', 'x3'])]
       for lam in diagrams];

table(arr, frame=True)

References