Schur functions can be defined via Jacobi-Trudi (special case of Weyl character) formula
(1)
where is a Young diagram and are complete homogeneous symmetric functions. Then in power sum basis for all such that are given as
Sym = SymmetricFunctions(QQ)
s = Sym.s()
p = Sym.p()
diagrams = [
lam
for n in range(1, 4)
for lam in Partitions(n).list()
]
arr = [[lam, p(s[lam])]
for lam in diagrams];
table(arr, frame=True)
(2)
Notation conventions
In this notes conventions may be not so standard, though I did my best to make them as close to standard as possible. Schur function in power sum basis is denoted as . On occasion when power sum basis is specialized like that I write . Examples: for the corresponding Schur function is , for — . For the matrix argument I write and mean by that evaluated at power sum basis .
Properties
- Schur functions are characters of in a literal sense that , where is an irreducible representation of indexed by Young diagram .
- Schur functions are connected with characters of symmetric group via Frobenius characteristic map.