Thom polynomials and Schur functions: towards the singularities $A_i(-)$

TL;DR

This paper introduces combinatorial tools using Schur functions to compute Thom polynomials for Morin singularities, providing explicit formulas and exploring their structure in specific cases.

Contribution

It develops a new algebro-combinatorial approach to explicitly compute Thom polynomials for $A_i$ singularities using Schur functions.

Findings

Thom polynomial ${ m extbf{ extit{T}}}^{A_i}$ is given by $F^{(i)}_{k+1}$ under certain conditions.

The 1-part of ${ m extbf{ extit{T}}}^{A_i}$ is explicitly described by the function $F^{(i)}_{k+1}$.

In specific examples, the 2-part of the Thom polynomial is a single Schur function with multiplicity.

Abstract

We develop algebro-combinatorial tools for computing the Thom polynomials for the Morin singularities $A_i(-)$ ($i\ge 0$). The main tool is the function $F^{(i)}_r$ defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial ${\cal T}^{A_i}$ for the singularity $A_i$ (any $i$) associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$, with any parameter $k\ge 0$, under the assumption that $\Sigma^j=\emptyset$ for all $j\ge 2$, is given by $F^{(i)}_{k+1}$. Equivalently, this says that "the 1-part" of ${\cal T}^{A_i}$ equals $F^{(i)}_{k+1}$. We investigate 2 examples when ${\cal T}^{A_i}$ apart from its 1-part consists also of the 2-part being a single Schur function with some multiplicity. Our computations combine the characterization of Thom polynomials via the "method of restriction equations" of Rim\'anyi et al. with the techniques of Schur functions.