Related papers: Modular invariance groups and defect McKay-Thompso…
We study aspects of the theory of generalized Kac-Moody Lie algebras (or Borcherds algebras) and their standard modules. It is shown how such an algebra with no mutually orthogonal imaginary simple roots, including Borcherds' Monster Lie…
In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally…
Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an…
For the moonshine module $V^{\natural},$ whose automorphism is the Monster ${\Bbb M},$ We show how to give a uniform existence proof for irreducible $g$-twisted modules for elements of type $2A,$ $2B$ and $4A$ in ${\Bbb M}.$ The most…
V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula…
We give a summary of R. Borcherds' solution (with some modifications) to the following part of the Conway-Norton conjectures: Given the Monster simple group and Frenkel-Lepowsky-Meurman's moonshine module for the group, prove the equality…
In a recent paper, Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we…
The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular…
The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a…
The Conway--Norton conjectures unexpectedly related the Monster with certain special modular functions (Hauptmoduls). Their proof by Borcherds et al was remarkable for demonstrating the rich mathematics implicit there. Unfortunately…
If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called \textsl{strictly invariant}. For the case of $%…
A conjecture in [Ish20] states that for a finite subgroup $G$ of $GL(2; \mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability parameter…
In this short note, by modifying the results and proofs given in the author's recent work, we simply show that the difference of a Hauptmodul for any genus zero group $\Gamma_{0}(N)$ is a Borcherds lift. This work extends Scheithauer's…
The theory of topological modular forms (TMF) predicts that elliptic genera of physical theories satisfy a certain divisibility property, determined by the theory's gravitational anomaly. In this note we verify this prediction in Duncan's…
We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory,…
We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that…
We investigate the relation between the generating functions of the highest weight vectors of the Monster module and the McKay-Thompson series.
Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e}…
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant…
The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex…