On the head characters of the Monster simple group

Smith, Stephen

doi:10.1090/conm/045/822245

Cited by 25 publications

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“…It was verified by Atkin-Fong-Smith [212], using results of Thompson [220] (cf. also [207]), that a graded (possibly virtual) infinite-dimensional monster module V , such that the functions T g of (3.2) are exactly those predicted by Conway-Norton in [58], exists.…”

Section: Conjecture 2 (Monstrous Moonshine: Conway-norton)mentioning

confidence: 69%

Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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illustrate the case of J(τ ) = j(τ ) − 744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of M. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions.

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Section: Conjecture 2 (Monstrous Moonshine: Conway-norton)mentioning

confidence: 69%

Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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“…2.3 for a definition of this term) of the conjugacy class with respect to the module. An abstract proof of this conjecture (i.e., one whose construction of the module is done only implicitly in terms of the McKay-Thompson series) was announced by Atkin-Fong-Smith [20,36]. Their proof was based on an idea of Thompson.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…This gives an (probably 6 ) alternate proof the theorem of Atkin-Fong-Smith [20,36]. As in the case of Thompson moonshine, we have calculated a list of congruences for each prime dividing the order of the Monster, proven by means of Sturm's theorem.…”

Section: Proofmentioning

confidence: 99%

A proof of the Thompson moonshine conjecture

Griffin

Mertens

2016

Res Math Sci

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“…V G n such that for all g ∈ G, the McKay-Thompson series Tr(g|V G ) is a modular function strictly of level ord(g), and such that the homogeneous components V G n are virtual modules for only finitely many n. Remark 9. Theorem 4.1 may be thought of as a natural generalization of the work of Atkin-Fong-Smith [37], with more freedom in the choice of modular functions.…”

Section: Some Representation Theory For Virtual Moonshinementioning

confidence: 99%

Moonshine for all finite groups

et al. 2018

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In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional C[G]-modules where the McKay-Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on Γ 0 (ord(g)). As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level ord(g) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include (Z/5Z) 5 and (Z/7Z) 4 , which are not subgroups of the Monster.

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On the head characters of the Monster simple group

Cited by 25 publications

References 0 publications

Moonshine

Moonshine

A proof of the Thompson moonshine conjecture

Moonshine for all finite groups

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