A GAP package for braid orbit computation, and applications

Let G be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus many problems on algebraic curves require the computation of braid orbits. In this paper we describe an implementation of this computation. We discuss several applications, including the classification of irreducible families of indecomposable rational functions with exceptional monodromy group

Primitive axial algebras of Jordan type

An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.Comment: 41 pages; comments welcom

Split spin factor algebras

Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.Comment: 17 pages. The results in Section 5 have been simplified and strengthened. A new section has been added to deal with a family of exceptional algebras which arise for $\alpha=-1

Extended F_4-buildings and the Baby Monster

The Baby Monster group B acts naturally on a geometry E(B) with diagram c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2, and (b) three elements of type 1 are pairwise incident to common elements of type 2 iff they are incident to a common element of type 5. It is shown that E(B) is the only (non-necessary flag-transitive) c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first characterization of B in terms of an incidence geometry, similar in vein to one known for classical groups acting on buildings. Further, it is shown that E(B) contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2 with flag-transitive automorphism groups are constructed. A complete list of possibilities for the isomorphism type of the subgraph induced by the common neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t) satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Majorana representations of the symmetric group of degree 4

AbstractThe Monster group M acts on a real vector space VM of dimension 196,884 which is the sum of a trivial 1-dimensional module and a minimal faithful M-module. There is an M-invariant scalar product (,) on VM, an M-invariant bilinear commutative non-associative algebra product ⋅ on VM (commonly known as the Conway–Griess–Norton algebra), and a subset A of VM∖{0} indexed by the 2A-involutions in M. Certain properties of the quintetM=(M,VM,A,(,),⋅) have been axiomatized in Chapter 8 of Ivanov (2009) [Iv09] under the name of Majorana representation of M. The axiomatization enables one to study Majorana representations of an arbitrary group G (generated by its involutions). A representation might or might not exist, but it always exists whenever G is a subgroup in M generated by the 2A-involutions contained in G. We say that thus obtained representation is based on an embedding of G in the Monster. The essential motivation for introducing the Majorana terminology was the most remarkable result by S. Sakuma (2007) [Sak07] which gave a classification of the Majorana representations of the dihedral groups. There are nine such representations and every single one is based on an embedding in the Monster of the relevant dihedral group. It is a fundamental property of the Monster that its 2A-involutions form a class of 6-transpositions and that there are precisely nine M-orbits on the pairs of 2A-involutions (and also on the set of 2A-generated dihedral subgroups in M). In the present paper we are making a further step in building up the Majorana theory by classifying the Majorana representations of the symmetric group S4 of degree 4. We prove that S4 possesses precisely four Majorana representations. The Monster is known to contain four classes of 2A-generated S4-subgroups, so each of the four representations is based on an embedding of S4 in the Monster. The classification of 2A-generated S4-subgroups in the Monster relies on calculations with the character table of the Monster. Our elementary treatment shows that there are (at most) four isomorphism types of subalgebras in the Conway–Griess-Norton algebra of the Monster generated by six Majorana axial vectors canonically indexed by the transpositions of S4. Two of these subalgebras are 13-dimensional, the other two have dimensions 9 and 6. These dimensions, not to mention the isomorphism type of the subalgebras, were not known before

Lie algebras and 3-transpositions

We describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3-transpositions in a group. In particular, we determine which simple Lie algebras arise by this construction. Among other things, this construction yields a natural embedding of the sporadic simple group \Fi{22} in the group $^2E_6(2)$.Comment: 23 page