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

Gordon introduced a class of matroids M ( n ) , for prime n ≥ 2 , such that M ( n ) is algebraically representable, but only in characteristic n . Lindström proved that M ( n ) for general n ≥ 2 is not algebraically representable if n > 2 is an even number, and he conjectured that if n 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.