Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
4.20 Schur polynomials<br />
Let ∂ = (∂ 1 , ..., ∂ p ) be a partition of n, and N ⊂ p. Let<br />
Define the polynomials<br />
N<br />
j +N−j j +N−j<br />
s(j)<br />
i<br />
sS N j=1<br />
D (x) = (−1) s x = det(x ).<br />
D (x)<br />
S (x) :=<br />
D0 (x)<br />
(clearly D 0 (x) is just (x)). It is easy <strong>to</strong> see that these are indeed polynomials, as D is antisymmetric<br />
and there<strong>for</strong>e must be divisible by . The polynomials S are called the Schur<br />
polynomials.<br />
Proposition 4.61.<br />
m<br />
m<br />
(x1 + ... + x ) im = ν N (C i )S (x).<br />
m :p→N<br />
Proof. The identity follows from the Frobenius character <strong>for</strong>mula and the antisymmetry of<br />
m<br />
m<br />
(x) (x 1 + ... + x N ) im .<br />
m<br />
Certain special values of Schur polynomials are of importance. Namely, we have<br />
Proposition 4.62.<br />
There<strong>for</strong>e,<br />
S (1, z, z 2 , ..., z N−1 ) =<br />
S (1, ..., 1) =<br />
<br />
1→i