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Higher width moonshine. (English) Zbl 1481.11045

This paper establishes a ‘higher width’ version of weak moonshine which distinguishes between non-isomorphic groups. For a finite group \(G\), weak moonshine asserts there is an infinite-dimensional graded \(G\)-module \[ V_G = \bigoplus_{n\in \{-d\}\cup \mathbb{Z}^+} V_G(n) \] for \(d>0\) sufficiently large, such that the McKay-Thompson series \[ T_g(\tau) = \sum_{n\gg -\infty} \operatorname{Tr} \left(g\vert V_G(n)\right)q^n =\sum_{n \gg -\infty} \sum_{1\leq i \leq t} m_i(n)\chi_i (g)q^n \] (\(g\in G, q=e^{2\pi i \tau}, \tau \in \operatorname{H}\)) are modular functions. Here, for \(1\leq i \leq t\) let \(m_i(n)\) denote the multiplicity of the irreducible representation \(\rho_i\) of \(G\) in \(V_G(n)\) and \(\chi_i\) be the corresponding irreducible character. It is proved in [S.DeHority et al., Res.Math.Sci.5, No.1, Paper No.14, 34 p.(2018; Zbl 1440.11059)] that weak moonshine holds for every finite group, however in this setting non-isomorphic groups might produce the same weak moonshine data.
To develop a higher width generalization that differentiates between non-isomorphic groups, the authors replace \(\chi_i(g)\) for \(g\in G\) with Frobenius \(r\)-characters \(\chi_i^{(r)}(\underline{g})\) defined for \(\underline{g} \in G^{(r)} = G\times G\times \cdots \times G\) (\(r\)-copies). Then a group \(G\) is said to have width \(s\) weak moonshine for \(s\in \mathbb{Z}^+\) if the width \(r\) McKay-Thompson series \[ T(r,\underline{g},\tau ) =\sum_{n \gg -\infty} \sum_{1\leq i \leq t} m_i(n)\chi_i^{(r)} (\underline{g})q^n \] is a weakly holomorphic modular function for each \(1\leq r\leq s\) and \(\underline{g} \in G^{(r)}\). In this way, higher width weak moonshine theory incorporates the original weak moonshine theory. However, the presence of the Frobenius \(r\)-characters means the higher width theory can distinguish between non-isomorphic groups.
Theorem 1.1, which is the main result of the paper, establishes that for any \(s\in \mathbb{Z}^+\) a finite group has width \(s\) weak moonshine, and that it is asymptotically regular. To establish this result, the authors also develop general results pertaining to the orthogonality of the Frobenius \(r\)-characters. As pointed out in the paper, these results are of interest in character theory and complete work began by Frobenius, Hoehnke, and Johnson. To illustrate the usefulness of their results, examples are provided of non-isomorphic groups sharing the same weak moonshine but emitting different higher width moonshines.

MSC:

11F22 Relationship to Lie algebras and finite simple groups
11F11 Holomorphic modular forms of integral weight
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F50 Jacobi forms
20C34 Representations of sporadic groups
20C35 Applications of group representations to physics and other areas of science

Citations:

Zbl 1440.11059
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References:

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