Lecture notes for Introduction to Representation Theory

4.20 Schur polynomials
Let ∂ = (∂ 1 , ..., ∂ p ) be a partition of n, and N ⊂ p. Let
Define the polynomials
N
j +N−j j +N−j
s(j)
i
sS N j=1
D (x) = (−1) s x = det(x ).
D (x)
S (x) :=
D0 (x)
(clearly D 0 (x) is just (x)). It is easy to see that these are indeed polynomials, as D is antisymmetric
and therefore must be divisible by . The polynomials S are called the Schur
polynomials.
Proposition 4.61.
m
m
(x1 + ... + x ) im = ν N (C i )S (x).
m :p→N
Proof. The identity follows from the Frobenius character formula and the antisymmetry of
m
m
(x) (x 1 + ... + x N ) im .
m
Certain special values of Schur polynomials are of importance. Namely, we have
Proposition 4.62.
Therefore,
S (1, z, z 2 , ..., z N−1 ) =
S (1, ..., 1) =
1→i