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

Daniel Sanchez; David Alfaya; Jaime Pizarroso; Daniel Sanchez; David Alfaya; Jaime Pizarroso

doi:10.3934/era.2025093

Electronic Research Archive

2025, Volume 33, Issue 4: 2118-2147. doi: 10.3934/era.2025093

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

1.
Institute for Research in Technology, ICAI School of Engineering, Comillas Pontifical University, Calle del Rey Francisco 4, Madrid 28015, Spain
2.
Department of Applied Mathematics and Institute for Research in Technology, ICAI School of Engineering, Comillas Pontifical University, C/Alberto Aguilera 25, Madrid 28015, Spain
3.
Department of Telematics and Computation and Institute for Research in Technology, ICAI School of Engineering, Comillas Pontifical University, C/Alberto Aguilera 25, Madrid 28015, Spain

Received: 31 December 2024 Revised: 28 March 2025 Accepted: 01 April 2025 Published: 15 April 2025

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.
- lambda-rings,
- symbolic computations of motives,
- Chow motives,
- moduli spaces,
- Higgs bundles moduli space
Citation: Daniel Sanchez, David Alfaya, Jaime Pizarroso. Motives meet SymPy: studying $ \lambda $-ring expressions in Python[J]. Electronic Research Archive, 2025, 33(4): 2118-2147. doi: 10.3934/era.2025093

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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.

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