Thom polynomials of invariant cones, Schur functions, and positivity

TL;DR

This paper extends Thom polynomials to invariant cones in representations and vector bundles, demonstrating their Schur function expansions have nonnegative coefficients, thus generalizing previous stability results.

Contribution

It introduces a generalized framework for Thom polynomials in invariant cones and proves their Schur function expansions are nonnegative, extending stability results to broader cases.

Findings

Thom polynomials expanded in Schur functions have nonnegative coefficients.

Extension of stability results to non-stable singularities.

Provides an expository discussion on related aspects of Thom polynomials.

Abstract

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.