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Projective Equivalence of $\Delta$-Matroids with Coefficients and Symplectic Geometries
The projective equivalence of matroid representations over fields and of oriented matroids is well-studied. This paper is devoted to the study of projective equivalence of $\Delta $-matroids with coefficients, which covers the concept of projective equivalence of matroids with coefficients and thus in particular the projective equivalence of represented and oriented matroids. A necessary and sufficient condition for the projective equivalence of $\Delta $-matroids with coefficients is established in terms of the inner Tutte group $\mathbf{T}_M^{\left( 0\right) }$ of the underlying combinatorial geometry $M$. The structure of $\mathbf{T}_M^{\left( 0\right)} $ of symplectic projective spaces $M$ of odd dimensions $d\geq 3$ over fields is computed. As a consequence it turns out that in case of finite fields there exists only one projective equivalence class of valuations with values in any linearly ordered abelian group.
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