Representable orientations of the free spikes

All orientations of binary and ternary matroids are representable [R.G. Bland, M. Las Vergnas, Orientability of matroids, J. Combinatorial Theory Ser. B 24 (1) (1978) 94–123; J. Lee, M. Scobee, A characterization of the orientations of ternary matroids, J. Combin. Theory Ser. B 77 (2) (1999) 263–291]. In this paper we show that this is not the case for matroids that are representable over GF(pk)GF(pk) where k⩾2k⩾2. Specifically, we show that there are orientations of the rank-k   free spike that are not representable for all k⩾4k⩾4. The proof uses threshold functions to obtain an upper bound on the number of representable orientations of the free spikes.