Singular

D.15.4.50 SchurS

Usage:

SchurS(I, S); I list of integers representing a partition, S list of polynomials

Return:

poly

Purpose:

computes the Schur polynomial in the Segre classes S (of the dual vector bundle), i.e., in the complete homogeneous symmetric polynomials, with respect to the partition I

Note:

if S are the Segre classes of the tautological bundle on a grassmanian, this gives the cohomology class of a Schubert cycle

LIB "chern.lib";
// The Schur polynomial corresponding to the partition 1,2,4
// and the Segre classes 1, s(1), s(2),..., s(6)
ring r=0,(s(1..6)), dp;
list I=1,2,4;
list S=s(1..6);
print( SchurS(I, S) );
==> s(1)*s(2)*s(4)-s(1)^2*s(5)-s(3)*s(4)+s(1)*s(6)
// compare this with the Schur polynomial computed using Chern classes
list C=chDual(chern(S));
print( SchurCh(I, C) );
==> s(1)*s(2)*s(4)-s(1)^2*s(5)-s(3)*s(4)+s(1)*s(6)