Related papers: Symmetric Generation of $M_{22}$

If an $m+2$-manifold $M$ is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity…

Suppose that M is countable, binary, primitive, homogeneous, and simple, and hence 1-based. We prove that the SU-rank of the complete theory of M is~1. It follows that M is a random structure. The conclusion that M is a random structure…

We further discuss the relation between the elliptic genus of K3 surface and the Mathieu group M24. We find that some of the twisted elliptic genera for K3 surface, defined for conjugacy classes of the Mathieu group M24, can be represented…

The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology…

We give a relatively self-contained proof that if a group $G$ fibres algebraically and is part of a $\mathrm{PD}^3$-pair, then $G$ is the fundamental group of a fibred compact aspherical 3-manifold. This yields a homological proof of a…

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the…

In this article we give an self contained existence proof for J. Conway's sporadic simple group Co_1 [4] using the second author's algorithm [14] constructing finite simple groups from irreducible subgroups of GL_n(2). Here n = 11 and the…

Generalised moonshine is reviewed from the point of view of holomorphic orbifolds, putting special emphasis on the role of the third cohomology group H^3(G, U(1)) in characterising consistent constructions. These ideas are then applied to…

For all Frobenius groups and a large class of finite multiply transitive permutation groups, we show that the corresponding group-subgroup subfactors are completely characterized by their principal graphs. The class includes all the sharply…

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…

We identify the sporadic simple group $\mathrm{M}_{12}$ and the simple group $\mathrm{SL}_3(3)$ from some part of their 3-local structure and give a graph theoretic analogue of the resulting theorem.

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M_24. The reason is yet a mystery.

There is a `Mathieu moonshine' relating the elliptic genus of K3 to the sporadic group M_{24}. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3 \times T^2 to…

Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $\mathbb{C}\ell(8)$ by incorporating a $U(1)_{em}$ gauge symmetry. The algebra $\mathbb{C}\ell(8)$ corresponds to…

For an in invertible quasihomogeneous singularity $w$ we prove an all-genus mirror theorem establishing an isomorphism between two cohomological field theories. On the $B$-side it is the Saito-Givental theory given by a certain choice of a…

We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.

Let $SB_n$ be the singular braid group generated by braid generators $\sigma_i$ and singular braid generators $\tau_i$, $1 \leq i \leq n-1$. Let $ST_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$,…

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the…

For each simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple provide a reductive pair related to a semi-Riemannian homogeneous manifold. It is proved…

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two…