Simplification of $\lambda$-ring expressions in the Grothendieck ring of Chow motives

TL;DR

This paper develops algorithms for simplifying complex $ ext{λ}$-ring expressions in the Grothendieck ring of Chow motives, enabling explicit computations of motives of moduli spaces and verification of conjectures.

Contribution

It introduces effective algorithms for symbolic simplification of $ ext{λ}$-ring and Adams operations expressions in the Grothendieck ring of Chow motives, facilitating computational applications.

Findings

Algorithms successfully simplify $ ext{λ}$-ring expressions

Explicit motives of certain moduli spaces are computed

Verification of conjectural formulas for motives is achieved

Abstract

The Grothendieck ring of Chow motives admits two natural opposite $\lambda$-ring structures, one of which is a special structure allowing the definition of Adams operations on the ring. In this work I present algorithms which allow an effective simplification of expressions that involve both $\lambda$-ring structures, as well as Adams operations. In particular, these algorithms allow the symbolic simplification of algebraic expressions in the sub-$\lambda$-ring of motives generated by a finite set of curves into polynomial expressions in a small set of motivic generators. As a consequence, the explicit computation of motives of some moduli spaces is performed, allowing the computational verification of some conjectural formulas for these spaces.