[Paper Review] Thom polynomials and Schur funcions: the singularities $A_3(-)$

Alain Lascoux, Piotr Pragacz|ArXiv.org|Oct 14, 2008

Nonlinear Waves and Solitons22 references3 citations

TL;DR

This paper computes the Schur function expansions of Thom polynomials for the Morin singularities $A_3$ in maps $({f C}^{ullet},0)\to({f C}^{ullet+k},0)$ for any $k\geq 0$ using the method of restriction equations and symmetric functions. The key result is a complete description of the $2$-part of the Thom polynomial, with all coefficients nonnegative and partitions containing the row $(k+1)$, achieved via supersymmetric Schur functions and Pascal staircase constructions.

ABSTRACT

Combining the "method of restriction equations" of Rimányi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\bf C}^{\bullet},0) o ({\bf C}^{\bullet + k},0)$ for any nonnegative integer $k$.

Motivation & Objective

• To determine the Schur function expansion of the Thom polynomial for the $A_3$ singularity in maps $({\bf C}^{\bullet},0)\to({\bf C}^{\bullet+k},0)$ for arbitrary $k\geq 0$.
• To establish a conceptual framework for computing families of Thom polynomials parameterized by $k$, avoiding brute-force computation.
• To reveal structural properties—such as nonnegative coefficients and the presence of the partition $(k+1)$—in the Schur basis.
• To extend the method of restriction equations using symmetric function techniques, particularly supersymmetric Schur functions and their properties.

Proposed method

• Applying the method of restriction equations from Rimányi et al. to derive a system of linear equations for the Thom polynomial coefficients.
• Using supersymmetric Schur functions (Schur functions in difference of alphabets) as the basis for expressing the Thom polynomial, leveraging their vanishing, cancellation, and factorization properties.
• Employing the functorial $\lambda$-ring approach to handle symmetric functions in $2x_1, 2x_2, x_1+x_2$ alongside $x_1, x_2$, enabling algebraic simplification.
• Constructing a Pascal staircase $P$ from generating functions to encode the coefficients of the Schur function expansion.
• Using the function $W(n, \mathbb{A})$ to compute contributions to Schur functions, with $W(n, \mathbb{A})$ expressed via resultants and generating functions.
• Specializing generating functions such as $f = \frac{5-6z}{(1-z)(1-2z)(1-3z)}$ to compute the $2$-part of the Thom polynomial via $y$-specialization at $1, 2, 3$.

Experimental results

Research questions

• RQ1 What is the Schur function expansion of the Thom polynomial for the $A_3$ singularity across all values of $k\geq 0$?
• RQ2 How can the method of restriction equations be systematically applied to compute families of Thom polynomials parameterized by $k$?
• RQ3 Why do the coefficients in the Schur basis of the $A_3$ Thom polynomial remain nonnegative, and what structural constraints govern the partitions that appear?
• RQ4 What role do supersymmetric Schur functions and their properties—vanishing, cancellation, factorization—play in simplifying the computation of Thom polynomials?
• RQ5 How does the presence of the partition $(k+1)$ in all terms of the Schur expansion reflect the geometric closure of $A_3$ in the orbit of $\Sigma^1$?

Key findings

• The $2$-part of the Thom polynomial for $A_3$ is given by $3^{n+1}S_n(\mathbb{X}_2) - 2\cdot 3^n S_{1,n-1}(\mathbb{X}_2)$ for $n\geq 3$, with initial conditions $W(0)=1$, $W(1)=(y-3)S_1(\mathbb{X}_2)$, and $W(2)=(y-1)(y-2)S_2(\mathbb{X}_2)-2(y-3)S_{11}(\mathbb{X}_2)$.
• All coefficients in the Schur function expansion of the $A_3$ Thom polynomial are nonnegative, confirming a general property established by Weber and Pragacz.
• All partitions in the Schur expansion contain the single row $(k+1)$, reflecting that $A_3$ lies in the closure of the $\Sigma^1$ orbit.
• The $h$-parts of the $A_3$ Thom polynomial vanish for $h\geq 3$, so only the $1$-part and $2$-part are nontrivial.
• The function $W(n, \mathbb{A})$ encodes contributions to Schur functions and is computed via $W(n, \mathbb{A}) = (y-1)y^{n-1}S_n(\mathbb{X}_2 - y^{-1}\mathbb{B})$ for $\mathbb{A} = \mathbb{X}_2 + \mathbb{B}$, with $\mathbb{B} = 2x_1 + 2x_2$.
• The Pascal staircase construction from the generating function $f = \frac{5-6z}{(1-z)(1-2z)(1-3z)}$ enables the computation of the $2$-part of the Thom polynomial through $y$-specialization at $y=3$.