# Algebraic matroids and Frobenius flocks

## Samenvatting

We show that each algebraic representation of a matroid $M$ in positive characteristic determines a matroid valuation of $M$, which we have named the {\em Lindstr\"om valuation}. If this valuation is trivial, then a linear representation of $M$ in characteristic $p$ can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic $p$ if and only if they are linear in characteristic $p$. To construct the Lindstr\"om valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.
Originele taal-2 Engels 1701.06384v2 1-21 21 arXiv 1701.06384v2 Gepubliceerd - 23 jan 2017

### Bibliografische nota

21 pages, 1 figure

## Trefwoorden

• math.CO
• math.AG
• 05E40 Combinatorial aspects of commutative algebra, 05B35 Matroids, geometric lattices