No abstract

We show that all 2A-Majorana representations of the Harada-Norton group F 5 have the same shape. If R is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span U of the Majorana axes. Finally, we prove that, if R is based on the (unique) embedding of F 5 in the Monster, U is closed under the algebra product. (Clara Franchi), a.ivanov@imperial.ac.uk (Alexander A. Ivanov), mario.mainardis@uniud.it (Mario Mainardis) c b This work is licensed under http://creativecommons.org/licenses/by/3.0/

Let G be a finite group, W be a ℝ[G]-module equipped with a G-invariant positive definite bilinear form (,)_W, and X a finite generating set of W such that X is transitively permuted by G. We show a new method for computing the dimensions of the irreducible constituents of W. Further, we apply this method to Majorana representations of the symmetric groups and prove that the symmetric group S_n has a Majorana representation in which every permutation of type (2, 2) of S_n corresponds to a Majorana axis if and only if n≤12

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 $$(\alpha ,\beta )$$ ( α , β ) , over a field of characteristic other than 2, has dimension at most 8 if $$\alpha \notin \{2\beta ,4\beta \}$$ α ∉ { 2 β , 4 β } . In this note, we show that Rehren’s bound does not hold in the case $$\alpha =4\beta $$ α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type $$(2,\frac{1}{2})$$ ( 2 , 1 2 ) over an arbitrary field $${{\mathbb {F}}}$$ F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

Rehren proved in [13,14] that a primitive 2-generated axial algebra of Monster type (α, β), over a field of characteristic other than 2, has dimension at most eight if α / ∈ {2β, 4β}. In this note we construct an infinitedimensional 2-generated primitive axial algebra of Monster type (2, 1 2 ) over an arbitrary field F with char(F) = 2, 3. This shows that the second special case, α = 4β, is a true exception to Rehren's bound.