Related papers: Arithmetic groups and the affine E8 Dynkin diagram

We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic…

In our previous paper math/0502157 we classified a large class of finite-dimensional pointed Hopf algebras up to isomorphism. However the following problem was left open for Hopf algebras of of type $A,D$ or $E_6$, that is whose Cartan…

This paper gives an elementary proof of the result that the conjugacy classes of pairs (X,Y) of unimodular 2x2 complex matrices is an affine 3-space, parametrized by the traces of X, Y and XY. Identities for triples of elements of SL(2,C)…

We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In…

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

The set of supercharacter theories of a fixed group $G$ forms a natural lattice. An open question in the study of supercharacter theories is to classify this lattice, and to date, this has only been done for the cyclic groups…

We show that any two homomorphisms of the alternating group A_5 into E_8 (over the complex numbers) whose images has finite centralizers are conjugate under E_8. This, in conjunction with earlier work of Frey, completes the classification…

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group.…

We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the…

For a simple algebraic group $G$ over an algebraically closed field, we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of $G$. We use two types of labels, a binary marking and a labeling with non-negative…

The simplest string theory compactifications to 3D with 16 supercharges -- the heterotic string on $T^7$, and type II strings on $K3 \times T^3$ -- are related by U-duality, and share a moduli space of vacua parametrized by $O(8,24;…

Let $V$ be a rational, selfdual, $C_2$-cofinite vertex operator algebra of CFT type, and $G$ a finite automorphism group of $V.$ It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated…

In $26+1$ space-time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to $25+1$ dimensions. This theory generalizes M-theory in many respects and we name it Monstrous…

E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and…

For a finite group G and a finite-dimensional G-module V, we prove a general result on the Poincar\'e series for the G-invariants in the tensor algebra T(V). We apply this result to the finite subgroups G of the 2-by-2 special unitary…

We describe the subracks of the conjugacy classes of $\mathrm{PSL}(2,q)$ based on Dickson's theorem on subgroups of $\mathrm{PSL}(2,q)$. All minimal non-abelian subracks of $\mathrm{PSL}(2,q)$ are determined. Further, we provide a general…

We show the details of certain computations that are used in the paper "Verification of the conjugacy classes and ordinary character table of the Monster".

We establish a surprisingly close relationship between universal Horn classes of directed graphs and varieties generated by so-called adjacency semigroups which are Rees matrix semigroups over the trivial group with the unary operation of…

We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional $\mathfrak{sl}_3$-modules which arise when acting by the latter monoidal category on arbitrary simple $\mathfrak{sl}_3$-modules.…