Motives meet SymPy: studying $\lambda$-ring expressions in Python

TL;DR

The paper introduces 'motives', a Python package based on SymPy, for symbolic manipulation of motivic expressions in $\lambda$-rings, enabling advanced algebraic computations and verification of conjectural formulas.

Contribution

It provides a novel computational tool for manipulating motivic expressions in $\lambda$-rings, including explicit handling of motives of moduli schemes and stacks.

Findings

Verified Mozgovoy's conjecture for rank 2 and 3 cases.

Proved the conjecture for curves of genus up to 18.

Demonstrated the package's capability in complex motivic computations.

Abstract

We present a new Python package called "motives", a symbolic manipulation package based on SymPy capable of handling and simplifying motivic expressions in the Grothendieck ring of Chow motives and other types of $\lambda$-rings. The package is able to manipulate and compare arbitrary expressions in $\lambda$-rings and, in particular, it contains explicit tools for manipulating motives of several types of commonly used moduli schemes and moduli stacks of decorated bundles on curves. We have applied this new tool to advance in the verification of Mozgovoy's conjectural formula for the motive of the moduli space of twisted Higgs bundles, proving that it holds in rank 2 and 3 for any curve of genus up to 18 and any twisting bundle of small degree.