Search

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (α , β) , over a field of characteristic other than 2, has dimension at most 8 if α ∉ { 2 β , 4 β } . In this note, we show that Rehren's bound does not hold in the case α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2 , 1 2) over an arbitrary field F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features. [ABSTRACT FROM AUTHOR]

We prove that the near hexagon associated with the extended ternary Golay code has, up to isomorphism, 25 hyperplanes, and give an explicit construction for each of them. As a main tool in the proof, we show that the classification of these hyperplanes is equivalent to the determination of the orbits on vectors of certain modules for the group 2 · M 12 . [ABSTRACT FROM AUTHOR]

We present a new non-existence proof for the strongly regular graph G with parameters (76, 21, 2, 7), using the unit vector representation of the graph. [ABSTRACT FROM AUTHOR]

We classify all non-collapsing Curtis–Tits and Phan amalgams with 3- spherical diagram over all fields. In particular, we show that amalgams with spherical diagram are unique, a result required by the classification of finite simple groups. We give a simple condition on the amalgam which is necessary and sufficient for it to arise from a group of Kac–Moody type. This also yields a definition of a large class of groups of Kac–Moody type in terms of a finite presentation. [ABSTRACT FROM AUTHOR]

Nonassociative commutative algebras A, generated by idempotents e whose adjoint operators ade: A → A, given by x → xe, are diagonalizable and have few eigenvalues, are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example, vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms. Axial algebras of Jordan type η are commutative algebras generated by idempotents whose adjoint operators have a minimal polynomial dividing (x-1)x(x-η), where η/∈ {0, 1} is fixed, with well-defined and restrictive fusion rules. The case of η = 1 2 was thoroughly analyzed by Hall, Rehren and Shpectorov in a recent paper, in which axial algebras were introduced. Here we focus on the case where η = 1/2, which is less understood and is of a different nature. [ABSTRACT FROM AUTHOR]

In Pralle and Shpectorov (Adv Geom 7(1):1-17, ) the class of ovoidal hyperplanes in dual polar spaces of rank 4 is described. In this paper we observe that by removing such a hyperplane and a related second hyperplane one obtains a nice geometry for the group stabilising the ovoidal hyperplane. We show that this group acts flag-transitively and that the geometry is simply connected. [ABSTRACT FROM AUTHOR]

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative. [ABSTRACT FROM AUTHOR]