Moonshine for all finite groups

Abstract

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.

Higher width moonshine

“…In 2017, Dehority, Gonzalez, Vafa, and Van Peski [6] examined the question of the extent to which dimensions of irreducible representations of finite groups and Fourier coefficients of modular functions are related. They proved (see Theorem 1.1 of [6]) that the seemingly rare phenomenon of moonshine holds for every single finite group if we relax certain requirements. Namely, for every finite group G there is an infinite-dimensional graded G-module…”

“…Theorem 1.1 of [6] guarantees that there is a G-module V G = n V G (n), q-graded traces T (1, g; τ ) which are modular functions for all g ∈ G, and non-negative integer multiplicities m i (n) for the representation spaces of each ρ i in V G (n). Moreover, the results in Section 5 of [6] guarantee that V G can be chosen to be asymptotically regular. Therefore, it suffices to construct the McKay-Thompson series for V (r) G for r > 1.…”

p-Adic Properties of Hauptmoduln with Applications to Moonshine

“…This proves all of the cases with r = 0 above. For the remaining cases with r > 0, we follow the method given in [12]. Let J, T 1 , .…”

“…. , a n such that J + a 1 T 1 + · · · + a n T n ≡ 0 mod q r+1 , then we have shown that v q (|G|) ≤ r. As in [12], this computation was carried out using Sage [36] by computing the kernel of the matrix of coefficients of J, T 1 , . .…”

“…In order to show that this gives a valid moonshine module, we must show that these multiplicities are both positive and integral. For positivity, we may use inequality (4.16) in [12]. Indeed, this inequality holds for n = 2, and hence for all n ≥ 2 since the left-hand side is monotonically increasing.…”