The Schur polynomial 
associated with a partition 
of length at most 
is the symmetric polynomial
 |
Here 
is padded with zeros to length 
. The quotient is a polynomial
because the numerator is alternating and hence divisible
by the Vandermonde determinant in the
denominator. Equivalently, 
is the generating
function for semistandard Young tableaux of
shape 
with entries in 1, 2, ..., 
, with a tableau contributing the monomial
recording the multiplicities of its entries. The Schur polynomials form a basis
for the ring of symmetric
polynomials and are the characters of polynomial
irreducible representations of the
general linear group 
.
Schur polynomials are the 
special case of the Jack
polynomials and also occur as specializations of the Macdonald
polynomials.
