Lindström’s conjecture on a class of algebraically non-representable matroids

Gordon introduced a class of matroids M(n)M(n), for prime n≥2n≥2, such that M(n)M(n) is algebraically representable, but only in characteristic nn. Lindström proved that M(n)M(n) for general n≥2n≥2 is not algebraically representable if n>2n>2 is an even number, and he conjectured that if nn is a composite number it is not algebraically representable. We introduce a new kind of matroid called a harmonic matroid of which the full algebraic matroid is an example. We prove the conjecture in this more general case.