Splitting Principle - Symmetric Polynomial

Symmetric Polynomial

Under the splitting principle, characteristic classes correspond to symmetric polynomials (and for the Euler class, alternating polynomials) in the class of line bundles.

The Chern classes and Pontryagin classes correspond to symmetric polynomials: they are symmetric polynomials in the corresponding classes of line bundles ( is the kth symmetric polynomial in the of the line bundles, and so forth).

The Euler class is an invariant of an oriented vector bundle, and thus the line bundles are ordered up to sign; the corresponding polynomial is the Vandermonde polynomial, the basic alternating polynomial. Further, for an even dimensional manifold, its square is the top Pontryagin class, which corresponds to the square of the Vandermonde polynomial being the discriminant.