
TL;DR
This paper extends the concept of weak moonshine to arbitrary width by using Frobenius' r-characters, enabling the distinction between non-isomorphic groups with identical McKay-Thompson series through modular functions.
Contribution
It introduces width r McKay-Thompson series using Frobenius r-characters, providing a new framework to differentiate groups beyond character table invariants.
Findings
Defined width r McKay-Thompson series for each r-tuple in G^{(r)}
Established orthogonality relations for Frobenius r-characters
Extended weak moonshine to arbitrary width using these new series
Abstract
\textit{Weak moonshine} for a finite group is the phenomenon where an infinite dimensional graded -module has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width . For each and each irreducible character , we employ Frobenius' -character extension to define \textit{width McKay-Thompson series} for ( copies) for each…
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