Projective equivalence of \(\Delta\)-matroids with coefficients and symplectic geometries (Q1381327)

``\(\Delta\)-matroids with coefficients in a fuzzy ring'' were introduced by the author [``Pfaffian forms and \(\Delta\)-matroids with coefficients,'' Discrete Math. 148, No. 1-3, 227-252 (1996; Zbl 0843.05012)] as a generalization that unifies the theories of \(\Delta\)-matroids representable by skew-symmetric matrices, and of valuated \(\Delta\)-matroids. This covers, in particular, the cases of matroids representable over fields, of oriented matroids, and of valuated matroids. The current paper introduces a concept of ``projective equivalence'' for \(\Delta\)-matroids with coefficients, again generalizing previously studied special cases, and characterizes it in terms of the inner Tutte group of the underlying matroid, as defined by the author in [``Maurer's homotopy theory and geometric algebra for even \(\Delta\)-matroids,'' Adv. Appl. Math. 17, No. 1, 27-62 (1996; Zbl 0851.05035)]. This is applied to the matroid given by the symplectic projective space of even rank \(\geq 4\) over an arbitrary field \(K\): here the inner Tutte group is \(K^*\), and hence there is only one projective equivalence class of valuations with values in a fixed linearly ordered abelian group.