The directed switching game on Lawrence oriented matroids

The main content of the note is a proof of the conjecture of Hamidoune-Las Vergnas on the directed switching game in the case of Lawrence oriented matroids. C.E. Shannon has introduced the switching game for graphs circa 1960. It has been generalized and solved for matroids by A. Lehman [4]. A switching game on graphs and oriented matroids was introduced by Y. O. Hamidoune and M. Las Vergnas [3]. They have solved it for graphic and cographic oriented matroids. They have stated as a conjecture that the classification of the oriented game is identical to the classification of its non oriented version. This conjecture holds for the graphic and cographic cases, but remains open for more general classes of oriented matroids. In this note, we show that it holds for the class of Lawrence oriented matroids. Definition 1 (Directed switching game on an oriented matroid). Let M be an oriented matroid and e one of its elements. In the directed switching game on M, Maker and Breaker alternately play on M and choose an unplayed element different from e, Maker signs it and Breaker deletes it. Maker wins the game if the final orientation of M contains a positive circuit containing e. By the results of [3], in order to prove that the classification of the directed switching game is identical to the classification of the undirected game, it suffices to prove Conjecture 1. [3] If M is the union of two disjoint bases, then the directed switching game on oM is winning for Maker playing first. Definition 2. The Lawrence oriented matroid defined by an n × r matrix A = (aij) with coefficients in {−1, 1} is the uniform oriented matroid of rank r on n elements such that the sign of an ordered basis (i1, . . . , ir)< is given by: χ(i1, . . . , ir) = r ∏