The image of the Burnside ring in the Representation ring for binary Platonic groups
We describe an efficient algorithm that computes, for any finite group G, the linear span of its virtual permutation representations inside all its linear representations, hence the image of the canonical morphism $\beta$ from the Burnside ring to the representation ring. We use this to determine the image and cokernel of $\beta$ for binary Platonic groups, hence for finite subgroups of SU(2), over $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$. We find explicitly that for the three exceptional subgroups and for the first seven binary dihedral subgroups, $\beta$ surjects onto the sub-lattice of the real representation ring spanned by the integer-valued characters. We conjecture that, generally, this holds true for all the binary dihedral groups.