Representations of Lagrangian matroids associated to partial duals of maps on surfaces

For a map M cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid correspond to the bases of a Lagrangian subspace L of the standard orthogonal space QE⊕QE∗, where E and E∗ are the edge-sets of M and its dual map. The Lagrangian subspace L is said to be the representation of M; in this paper we study the representations L and Lj of M and its partial duals ∂jM, respectively. We show that Lj can be obtained from L by acting on L with a transposition (j j∗) in the Coxeter group BCn, and that L and Lj form a Lagrangian pair of subspaces, for all j∈E.