The Chern classes for degeneracy loci of quivers are natural
generalizations of the Thom-Porteous-Giambelli formula. Suppose that
$E,F$ are vector bundles over a manifold $M$ and that $s : E\to F$ is
a vector bundle homomorphism. The question is, which cohomology class
is defined by the set $\Sigma\sb k(s)\subset M$ consisting of points
$m$ where the linear map $s(m)$ has corank $k$? The answer, due to
I. Porteous, is a determinant in terms of Chern classes of the bundles
$E,F$. We can generalize the question by giving more bundles over $M$
and bundle maps among them. The situation can be conveniently coded by
an oriented graph, called a quiver, assigning vertices for bundles and
arrows for maps.
¶ We give a new method for calculating Chern class formulae for
degeneracy loci of quivers. We show that for representation-finite
quivers this is a special case of the problem of calculating Thom
polynomials for group actions. This allows us to apply a method for
calculating Thom polynomials developed by the authors. The
method–reducing the calculations to solving a system of linear
equations–is quite different from the method of A. Buch and
W. Fulton developed for calculating Chern class formulae for
degeneracy loci of $A\sb n$-quivers, and it is more general (can be
applied to $A\sb n$-, $D\sb n$-, $E\sb 6$-, $E\sb 7$-, and $E\sb
8$-quivers). We provide sample calculations for $A\sb 3$- and $D\sb
4$-quivers.