Characteristic-Compatible Lifting and Natural Direct-Sum Presentations for q-Matroids
Vaclav Rozhon·Adrian Zámečník
Abstract
A q-matroid is a rank function on the subspaces of a vector space over Fq. A coordinate q-matroid on Fq^n is determined by an ordinary matroid on [n] through their weighted cyclic flats. For q = p^a, we prove that such a q-matroid is representable over a finite extension of Fq if and only if the corresponding ordinary matroid is representable over a field of characteristic p. The sufficiency direction uses a generic diagonal rescaling of an ordinary representation, matroid intersection, and simultaneous specialization over a finite field. We also establish closure of transversal q-matroids under the Ceria–Jurrius direct sum. More precisely, for arbitrary presentations of two transversal q-matroids, the list obtained by inflating every presenting subspace by the other ambient summand presents their direct sum. Its presentation-rank minimum agrees, on every subspace, with the direct-sum rank minimum. Iteration gives a natural presentation and a flattened rank formula for every finite direct sum.Discussion
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