Eigenweights for arithmetic Hirzebruch Proportionality

TL;DR

This paper introduces a novel AI-based method to compute eigenweights in the arithmetic Hirzebruch proportionality principle, linking algebraic combinatorics and representation theory to generalize previous results.

Contribution

It develops an AI-driven approach that connects eigenweights to symmetric group representations, enabling calculations for all classical groups.

Findings

Successfully computed eigenweights for all classical groups.

Established a connection between eigenweights and symmetric group representation theory.

Demonstrated the effectiveness of AI tools in advanced algebraic computations.

Abstract

Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to $L$-functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.